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+# Galilean Differential Geometry of Moving Images
+
+Daniel Fagerström
+
+Computational Vision and Active Perception Laboratory (CVAP)
+Department of Numerical Analysis and Computing Science
+KTH (Royal Institute of Technology) SE-100 44 Stockholm, Sweden
+danielf@nada.kth.se
+
+**Abstract.** In this paper we develop a systematic theory about local structure of moving images in terms of Galilean differential invariants. We argue that Galilean invariants are useful for studying moving images as they disregard constant motion that typically depends on the motion of the observer or the observed object, and only describe relative motion that might capture surface shape and motion boundaries. The set of Galilean invariants for moving images also contains the Euclidean invariants for (still) images.
+
+Complete sets of Galilean invariants are derived for two main cases: when the spatio-temporal gradient cuts the image plane and when it is tangent to the image plane. The former case correspond to isophote curve motion and the later to creation and disappearance of image structure, a case that is not well captured by the theory of optical flow.
+
+The derived invariants are shown to be describable in terms of acceleration, divergence, rotation and deformation of image structure.
+
+The described theory is completely based on bottom up computation from local spatio-temporal image information.
+
+## 1 Introduction
+
+The aim of this paper is to describe the local (differential) structure of moving images. By doing this we want to find a set of local differential descriptors that can describe local spatio-temporal pattern much as e.g. gradient strength, Laplacian zero-crossings, blob and ridge detectors, isophote curvature etc describe the local structure in images.
+
+The dominating approach to computational visual motion processing (reviewed in [2, 15]) is to first compute the *optical flow* field, i.e. the velocity vectors of the particles in the visual observer's field of view, projected on its visual sensor area. From this various properties of the surrounding scene can be computed. Ego-motion can, under certain circumstances, be computed from the global shape of the field, object boundaries from discontinuities in the field, and surface shape and motion for rigid objects, can be computed from the local differential structure of the field [12, 13].

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+acceleration extrema is equivalent to requiring $f_{su} = 0$ i.e. finding a $\beta$ that diagonalizes the Hessian matrix in the $\{\partial_s, \partial_u\}$ plane. We will also see that this choice of gauge makes the direction of the spatial tangent, $\partial_u$, constant along $\partial_s$, i.e. $\rho = 0$. From this requirement Guichard [9], derived the same gauge as we use here.
+
+Another choice of spatio-temporal gauge can be found by studying the divergence as a function of $\beta$. The divergence is a linear function of $\beta$ and the disappearance of the divergence, $\delta(\beta) = 0$, is a natural way to fixate the gauge, giving:
+
+$$ \beta_{\delta} = \frac{f_t f_{vv}}{f_v f_{uv}} - \frac{f_{tv}}{f_{uv}}. \qquad (26) $$
+
+This is defined as long as $f_{uv} \neq 0$, i.e. when the flow line curvature in the spatial plane is non-vanishing. It can be shown that the disappearance of the divergence is equivalent to requiring that $f_{sv} = 0$, i.e. finding a $\beta$ such that the Hessian in the $\{\partial_s, \partial_v\}$ plane is diagonalized.
+
+Using (25) and (24) we find the attitude matrix for the acceleration based $\Gamma_3$ tangent gauge,
+
+$$ \begin{pmatrix} \partial_s \\ \partial_u \\ \partial_v \end{pmatrix} = \begin{pmatrix} 1 & \frac{f_t f_{uv}}{f_v f_{uu}} - \frac{f_{tv}}{f_{uu}} & -\frac{f_t}{f_v} \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} \partial_t \\ \partial_u \\ \partial_v \end{pmatrix} = BAi. \qquad (27) $$
+
+The connection matrix can then be found by a tedious but elementary calculation using (3). Using notation from our general discussion about $\Gamma_3$ invariants the elements in the connection matrix (19) becomes:
+
+$$ c_{01} = a^u ds + \delta^u du + \sigma^u dv, \quad c_{02} = a^v ds + \delta^v dv, \quad c_{12} = \kappa du + \mu dv. \qquad (28) $$
+
+Observe that the skew invariant $\sigma^v$ that describe the skew in the gradient direction while moving in the tangent direction, disappear. The spatio-temporal rotation of the frame in the spatial plane $\rho$ disappears as well. We use the conventional notation $\kappa = \kappa^u, \mu = \kappa^v$, for isophote and flow line curvature. We list the resulting scalar invariants in the following theorem.
+
+**Theorem 4.** A complete set of scalar invariants for scalar functions on $\Gamma_3$ at points where the gradient and isophote curvature are non-vanishing are acceleration in the tangent and gradient direction,
+
+$$ a^u = \frac{f_{ss} f_{uv}}{f_v f_{uu}} - \frac{f_{ssu}}{f_{uu}}, \qquad a^v = -\frac{f_{ss}}{f_v}, \qquad (29) $$
+
+divergence in the tangent and gradient direction and skew in the gradient direction while moving in the tangent direction,
+
+$$ \delta^u = -\frac{f_{suu}}{f_{uu}}, \quad \delta^v = -\frac{f_{sv}}{f_v}, \quad \sigma^u = \frac{f_{sv} f_{uv}}{f_v f_{uu}} - \frac{f_{suv}}{f_{uu}}, \qquad (30) $$
+
+as well as isophote and flow line curvature, (see Theorem 1).

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+The invariant $a^v$, is also found in [9] and is denoted *accel*. The reasoning leading to Theorem 4 can be repeated for the divergence based tangent gauge (26).
+
+**Theorem 5.** A complete set of scalar invariants for scalar functions on $\Gamma_3$ at points where the gradient and flow line curvature are non-vanishing are acceleration in the tangent and gradient direction,
+
+$$ a^u = \frac{f_{ss}f_{vv}}{f_v f_{uv}} - \frac{f_{ssv}}{f_{uv}}, \quad a^v = -\frac{f_{ss}}{f_v}, \qquad (31) $$
+
+divergence in the tangent and gradient direction, skew in the gradient direction while moving in the tangent direction,
+
+$$ \delta^u = \frac{f_{su}f_{vv}}{f_v f_{uv}} - \frac{f_{suv}}{f_{uv}}, \quad \delta^v = -\frac{f_{su}}{f_v}, \quad \sigma^u = -\frac{f_{svv}}{f_{uv}}, \qquad (32) $$
+
+and isophote and flow line curvature, (see Theorem 1).
+
+## 6.4 Hessian Gauge
+
+On points where the isophote surface is tangent to the spatial surface, the tangent gauge is not defined. As long as the Hessian is non-degenerate, which generically is the case, we can define an adapted $\Gamma_3$-frame, $\{\partial_r, \partial_p, \partial_q\}$ that diagonalize the Hessian, i.e. $f_{pq} = f_{rp} = f_{rq} = 0$. Using the fact that the spatio-temporal vector in the adapted frame must be on the form,
+
+$$ \partial_r = \partial_t + \beta\partial_x + \gamma\partial_y. \qquad (33) $$
+
+Starting by diagonalizing the Hessian in the spatio-temporal direction we get the constraints $f_{rx} = f_{ry} = 0$, and by using (33) and solving for $\beta$ and $\gamma$, we get
+
+$$ \beta = \frac{f_{ty}f_{xy} - f_{tx}f_{yy}}{f_{xx}f_{yy} - f_{xy}^2}, \quad \gamma = \frac{f_{tx}f_{xy} - f_{ty}f_{xx}}{f_{xx}f_{yy} - f_{xy}^2}. \qquad (34) $$
+
+This gives the first part of the attitude transformation, a spatio-temporal shear $A$. If we project $\partial_r$ on the spatial plane we get the same vector field as when the optical flow constraint equation is used on the gradient of the image [17]. As the next step the frame must be rotated in the spatial plane s.t. the spatial Hessian is diagonalized. Here we can use the results for the Hessian gauge for $E_2$ reviewed in Section 4.2. Combining these steps we get,
+
+$$ \begin{pmatrix} \partial_r \\ \partial_p \\ \partial_q \end{pmatrix} = \begin{pmatrix} 1 & 0 & 0 \\ 0 \cos\phi - \sin\phi \\ 0 \sin\phi \cos\phi \end{pmatrix} \begin{pmatrix} 1 & \beta & \gamma \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} \partial_t \\ \partial_x \\ \partial_y \end{pmatrix} = BAi, \qquad (35) $$
+
+where $\tan 2\phi = f_{xy}/(f_{yy}-f_{xx})$. We proceed using (3) and the same reasoning as for the tangent based frames in the preceding section and arrives to the following theorem.

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+**Theorem 6.** A complete set of scalar invariants for scalar functions on $\Gamma_3$ at points where the Hessian is non-degenerate are as follow:
+
+$$ 
+\begin{align} 
+a^p &= -f_{rrp}/f_{pp} & a^q &= -f_{rrq}/f_{qq} \\ 
+\delta^p &= -f_{rpp}/f_{pp} & \delta^q &= -f_{rqq}/f_{qq} \\ 
+\sigma^p &= -f_{rpq}/f_{pp} & \sigma^q &= -f_{rpq}/f_{qq} \tag{36} \\ 
+\rho &= f_{rpq}/(2f_{pp} - 2f_{qq}) \\ 
+\kappa^p &= f_{ppq}/(2f_{pp} - 2f_{qq}) & \kappa^q &= f_{pqq}/(2f_{pp} - 2f_{qq}). 
+\end{align} 
+$$
+
+Observe that in contrast to the tangent based gauge systems the Hessian
+gauge has all the scalar invariants listed in (19).
+
+7 Conclusion and Discussion
+
+In this paper we have developed a systematic theory about local structure of
+moving images in terms of Galilean differential invariants. We have argued that
+Galilean invariants are useful for studying moving images as it disregard constant
+motion that typically depends on the motion of the observer or the observed ob-
+ject, and only describe relative motion that might capture surface shape and
+motion boundaries. The set of Galilean invariants for moving images also con-
+tains the Euclidean invariants for (still) images.
+
+Comparing to using optic flow as the basic element for describing image mo-
+tion, the above suggested theory is completely bottom up and local, while optic
+flow is based on trying to directly interpreting the image motion in terms of (the
+projection of) motion of object surface points. The estimation of optic flow is
+non-local as it typically is based on gathering statistics about low level features in
+a small spatio-temporal surrounding. There are also Galilean differential invari-
+ants that can capture creation and disappearance of image structure, situations
+that are not covered by the concept of optic flow.
+
+Experimental work is of course needed for evaluating how useful the suggested
+theory is for finding structure in real image sequences. Spatio-temporal images
+derivatives cannot be measured in a point, an integration over a non-vanishing
+spatio-temporal volume is needed [7], i.e. we need filters for measuring deriva-
+tives. As there are no localized filters that are invariant w.r.t. Galilean shear [5], a
+family of velocity adapted filters is needed. For computing a Galilean differential
+invariant, the velocity adapted filter used for measuring it should have the same
+spatio-temporal direction as the spatio-temporally directed gauge coordinate for
+the invariant. This could either be implemented by searching over a precom-
+puted set of spatio-temporally directed derivative filters or by iteratively adapt
+the spatio-temporal direction of the filter. It should be noted that in general,
+gauge adapted derivative filters can be found for several spatio-temporal direc-
+tions at a point, i.e. for real image sequences the invariants can be multi-valued.
+This can be the case for e.g. transparent motion.

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+References
+
+1. L. Alvarez, F. Guichard, P. Lions, and J. Morel. Axioms and fundamental equations of image processing: Multiscale analysis and p.d.e. *Archive for Rational Mechanics and Analysis*, 123(3):199-257, 1993.
+
+2. J. Barron, D. Fleet, and S. Beauchemin. Performance of optical flow techniques. *International Journal of Computer Vision*, 12(1):43-77, 1994.
+
+3. H. Flanders. *Differential forms with applications to the physical sciences*. Dover Publications, Inc., 1989.
+
+4. L. Florack, B. ter Haar Romeny, J. Koenderink, and M. Viergever. Families of tuned scale-space kernels. In *Proc. 2nd European Conference on Computer Vision*, pages 19–23, 1992.
+
+5. L. Florack, B. ter Haar Romeny, J. Koenderink, and M. Viergever. Scale and the differential structure of images. *Image and Vision Computing*, 10:376-388, 1992.
+
+6. L. Florack, B. ter Haar Romeny, J. Koenderink, and M. Viergever. General intensity transformations and differential invariants. *J. of Mathematical Imaging and Vision*, 4:171-187, 1994.
+
+7. L. M. J. Florack. *Image Structure*. Series in Mathematical Imaging and Vision. Kluwer Academic Publishers, Dordrecht, Netherlands, 1997.
+
+8. M. Friedman. *Foundation of Space-Time Theories: Relativistic Physics and Philosophy of Science*. Princeton University Press, 1983.
+
+9. F. Guichard. A morphological, affine, and galilean invariant scale-space for movies. *IEEE Transactions on Image Processing*, 7(3):444-456, 1998.
+
+10. W. C. Hoffman. The Lie algebra of visual perception. *J. Mathematical Psychology* 3 (1966), 65-98; errata, ibid., 4:348-349, 1966.
+
+11. B. Jahne. *Spatio-Temporal Image Processing-Theory and Scientific Applications*. Number 751 in Lecture Notes in Computer Science. Springer, 1993.
+
+12. J. Koenderink and A. van Doorn. Invariant properties of the motion parallax field due to the movement of rigid bodies relative to an observer. *Optica Acta*, 22(9):773-791, 1975.
+
+13. J. Koenderink and A. van Doorn. Local structure of movement parallax of the plane. *J. of the Optical Society of America*, 66(7):717-723, July 1976.
+
+14. J. Koenderink and A. van Doorn. Image processing done right. In A. H. et al., editor, *ECCV 2002*, number 2350 in LNCS, pages 158-172. Springer Verlag, 2002.
+
+15. A. Mitiche and P. Bouthemy. Computation and analysis of image motion: A synthesis of current problems and methods. *International Journal of Computer Vision*, 19(1):29-55, July 1996.
+
+16. M. Spivak. *Differential Geometry*, volume 1-5. Publish or Perish, Inc., Berkeley, California, USA, 1975.
+
+17. S. Uras, F. Girosi, A. Verri, and V. Torre. A computational approach to motion perception. *bc*, 60:79-87, 1988.
+
+18. M. Yamamoto. The image sequence analysis of three-dimensional dynamic scenes. Technical Report 893, Electrotechnical Laboratory, Agency of Industrial Science and Technology, May 1988.
+
+19. C. Zetzsche and E. Barth. Direct detection of flow discontinuities by 3d curvature operators. *Pattern Recognition Letters*, 12:771-779, 1991.

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+Unfortunately the computation of the optical flow field leads to a number of
+well known difficulties. The input is the projected (gray-level) image of the sur-
+roundings as a function of time, i.e. a three-dimensional structure. It is in general
+not possible to uniquely identify what path through the spatio-temporal image
+is a projection of a certain object point. Thus, further assumptions are needed,
+the most common one is the *brightness constancy assumption*, that the projec-
+tion of each object point has a constant gray level. The brightness constancy
+assumption breaks down if the light changes, if the object have non-Lambertian
+reflection, or, if it has specular reflections. However, the problem is still under-
+determined, generically. Except at local extrema in the gray-level image, points
+with a certain gray-level lie along curves, and these curves sweep out surfaces
+in the spatio-temporal image. A point along such a curve can therefore corre-
+spond to any point on the surface at later instants of time. This is refered to as
+the *aperture problem* and is usually treated by invoking additional constraints
+e.g. regularization assumptions, such as smoothly varying brightness patterns,
+or parameterized surface models and trajectory models, leading to least-square
+methods applied in small image regions. Beside the questionable validity of these
+assumptions they lead to inferior results near motion boundaries, i.e. the regions
+that carry most information about object boundaries. The behavior when new
+image structure appears or old structure disappears is also undefined.
+
+An alternative approach for visual motion analysis is to directly analyze the
+geometrical structure of the spatio-temporal input image, thereby avoiding the
+detour through the optic flow estimation step [18, 19, 11]. By using the differential
+geometry of the spatio-temporal image, we get a low level syntactical description
+of the moving image whithout having to rely on the more high level semantic
+concept of object particle motion.
+
+A systematic study of the local image structure, in the context of scale-space
+theory, has been pursued by Florack [6]. The basic idea is to find all descriptors
+of differential image structure that are invariant to rotation and translation (the
+Euclidean group). The choice of Euclidean invariance reflects that the image
+structures should be possible to recognize in spite of (small) camera translations
+and rotations around the optical axis. This theory embeds many of the operators
+previously used in computer vision, such as Canny's edge detector, Laplacian
+zero-crossings, blobs, isophote curvature and as well enabling the discovery of
+new ones.
+
+## 2 Spatio-Temporal Image Geometry
+
+Extending from a theory about spatial images to one about spatio-temporal im-
+ages it is natural to use the concept of absolute time (see e.g. [8] for a more
+elaborate discussion). Each point in space-time can be designated numeric label
+describing what time it occurred. The sets of space-time points that occurred at
+the same time are called planes of simultaneity and the temporal distance be-
+tween two planes of simultaneity can be measured (in the small spatio-temporal
+regions that seeing creatures, operates in, we see no need for handling relativistic

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+effects, there are however other opinions, see [10]). The space-time can be stratified in a sequence of planes of simultaneity, and be given coordinate systems that separates time and space, $(t, x) \in \mathbb{R} \times \mathbb{R}^2$. From the consequences of absolute time, we conclude that we only want to allow for space-time transformations that never mixes the planes of simultaneity.
+
+As a spatio-temporal image restricted to a plane of simultaneity can be considered as a still image the reasons for using Euclidean invariance in the image plane applies to moving images as well. Image properties should not be dependent on when we choose to measure them (invariance under time translations). The local average velocity contains only information about the ego motion and no information about the three dimensional structure of the environment, and is therefore natural to disregard. We thus search for properties that are invariant to the 2+1 dimensional Galilean group. The use of Galilean image geometry has been proposed in e.g. [4, 1, 9]. Using parallel projection as image formation model, the Galilean invariants are those properties of the surrounding that cannot be explained in terms of a relative constant translational motion. A Galilean model of the moving image is also implicitly assumed when divergence, curl and deformation are described as flow field invariants [12].
+
+**Definition 1 (Galilean group).** *The group of Galilean motions $\Gamma_{n+1}$:*
+
+$$ \begin{pmatrix} t' \\ x' \end{pmatrix} = \begin{pmatrix} 1 & v \\ 0 & R \end{pmatrix} \begin{pmatrix} t \\ x \end{pmatrix} + a = \begin{pmatrix} 1 & 0 \\ 0 & R \end{pmatrix} \begin{pmatrix} 1 & v \\ 0 & I \end{pmatrix} \begin{pmatrix} t \\ x \end{pmatrix} + a \quad (1) $$
+
+$x, v \in \mathbb{R}^n, t \in \mathbb{R}, R \in SO(n)$ and $a \in \Gamma_{n+1}$.
+
+Each Galilean motion can be decomposed in a spatial rotation, a spatio-temporal shear (constant velocity) and a space-time translation. It can be shown that planes of simultaneity (constant time) are invariant and has Euclidean geometry, i.e. distances and angles are invariants. The temporal distance between planes of simultaneity is invariant.
+
+## 3 Moving Frames
+
+The Galilean geometry has no metric in traditional sense. That means that metric based differential geometry cannot be used in its normal formulations. We therefore chose to use a Lie group based approach instead (see [14] for a different approach on a geometry with degenerate metric).
+
+According to Klein's famous Erlangen program, given a space $S$ and a group of transformations $G$ over $S$, the geometric structure of $(S, G)$ is all structure that is invariant to transformations in $G$. In the following we will study the differential geometric properties of scalar functions and sub-manifolds (curves and surfaces) in $\mathbb{R}^2$ and $\mathbb{R}^3$ subject to Galilean and in some cases Euclidean transformations.
+
+A convenient way to find geometrical structure is to use Cartan theory about moving frames [3, 16]. A *frame field* is a smooth map from the base space to group

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+elements, $S \to G$. For a Galilean geometry $\Gamma_{n+1}$ the frame field is a mapping $\mathbb{R}^n \to \Gamma_{n+1}$. A frame field can be conceptualized by its action on an arbitrary coordinate system for the tangent space of the base space. For $\Gamma_{n+1}$ we can e.g. attach a Galilean ON-system at each point.
+
+**Definition 2.** A $\Gamma_{n+1}$ coordinate system is an *affine coordinate system* where n vectors lies in the spatial part. A $\Gamma_{n+1}$ ON-system is a $\Gamma_{n+1}$ coordinate system s.t. the spatial part consists of n dimensional ON-coordinate system and the remaining base vector has unit temporal length.
+
+The property of being a $\Gamma_{n+1}$ ON-system is a Galilean invariant. In the sequel we will use the coordinate system view of frame fields as we find it easier to visualise.
+
+The main idea of Cartans theory about moving frames is to put a frame at each point that is connected to the local structure of the sub-manifold or the function in an invariant way. In this way we get a frame field.
+
+For a function $f$ defined on $S$, all expressions over mixed derivatives w.r.t. the Cartan frame at a certain point are by construction geometrical invariants. This class of invariants are called *differential invariants*.
+
+On sub-manifolds, we can find the local geometrical structure from how the frame field varies in the local neighborhood.
+
+Let $i$ be any (global) frame and $e$ a frame connected to the local structure s.t. $e = Ai$, where the attitude transformation $A \in G$ is a function of position. The local variation of $e$ can be described in an invariant way in terms of $e$,
+
+$$de = dAi = dAA^{-1}e = C(A)e, \qquad (2)$$
+
+where the *one-form* (see [3]) $C(A)$ is called the *connection matrix*. In a certain sense, the connection matrix contains all geometric information there is.
+
+Scalar invariants can be generated by contracting the coefficients in the connection matrix on the vectors in the Cartan frame, $c_{ij}e_k$. A useful property of the connection matrix is,
+
+$$C(AB) = C(A) + AC(B)A^{-1}, \qquad (3)$$
+
+which is a direct consequence of the definition.
+
+The level-sets $f^{-1}(c)$ of smooth scalar functions $f$ are sub-manifolds, the geometric structure of those, the *level-set invariants*, are invariant w.r.t. the group of constant monotonic transformations $g \circ f$, $g: \mathbb{R} \to \mathbb{R}$, $g' > 0$.
+
+# 4 Image Geometry
+
+Now we will study Galilean differential geometry of moving images using Cartan frames. Image spaces can be considered being trivial fiber bundle $S \otimes I$, where $S$ is the base space and the fiber $I$ is log intensity [14]. Most of the time we will discuss the image geometry in terms of an arbitrary section of the fiber bundle i.e. functions $f: S \to I$. We will start by reviewing differential geometry for

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+images over $E_2$ to illustrate the method of moving frames and as $E_2$ is a sub geometry of $\Gamma_3$ so that we will need these results later anyway. We continue by studying differential geometry of $\Gamma_2$ and, which is our main goal, differential geometry of images over $\Gamma_3$
+
+For scalar functions over $E_2$ there are two typical situations: the gradient is non-zero almost everywhere and it is zero along curves.
+
+**4.1 Gradient Gauge**
+
+We study the geometry of functions $f$ in $E_2$. For points $p$ where $\nabla f \neq 0$ we attach an ON-frame $\{\partial_u, \partial_v\}$ s.t. $f_u = 0$. $(u, v)$ is a gauge coordinate system.
+
+$$ \left( \begin{array}{c} \partial_u \\ \partial_v \end{array} \right) = \frac{1}{\|\nabla f\|} \left( \begin{array}{cc} f_y - f_x & -f_x \\ f_x & f_y \end{array} \right) \left( \begin{array}{c} \partial_x \\ \partial_y \end{array} \right) = A \left( \begin{array}{c} \partial_x \\ \partial_y \end{array} \right) \quad (4) $$
+
+where $\{\partial_x, \partial_y\}$ is a global ON-frame.
+
+All functions over $\partial_u^i \partial_v^j f$, $i + j \ge 1$ becomes invariants w.r.t. rotations in space and translation in the intensity fibers. From (2) we get the anti-symmetric connection matrix:
+
+$$ C(A) = \begin{pmatrix} 0 & c_{12} \\ -c_{12} & 0 \end{pmatrix}, \qquad (5) $$
+
+where,
+
+$$ c_{12} = \frac{(f_x f_{xy} - f_y f_{xx}) dx + (f_x f_{yy} - f_y f_{xy}) dy}{f_x^2 + f_y^2} = -\frac{f_{uu}}{f_v} du + -\frac{f_{uv}}{f_v} dv. \quad (6) $$
+
+where the expression is simplified by the use of the $\{\partial_u, \partial_v\}$ coordinate system, and the relation $f_u = 0$. By contracting $c_{12}$ on the components in the Cartan frame we arrive at:
+
+**Theorem 1.** A complete set of level-curve invariants for scalar functions on $E_2$ is the level curve curvature, and the flow line curvature,
+
+$$ \kappa = c_{12}\partial_u = -\frac{f_{uu}}{f_v}, \qquad \mu = c_{12}\partial_v = -\frac{f_{uv}}{f_v}. \qquad (7) $$
+
+These are invariants w.r.t. rotation in the plane and monotonic transformations in the intensity fibers.
+
+**4.2 Hessian Gauge**
+
+The ON-frame (4) is not defined on critical points, $\nabla f = 0$, on typical critical points we can instead use an ON-frame $\{\partial_p, \partial_q\}$ that diagonalize the Hessian, i.e. $f_{pq} = 0$ and $|f_{pp}| > |f_{qq}|$.
+
+$$ \left( \begin{array}{c} \partial_p \\ \partial_q \end{array} \right) = \left( \begin{array}{cc} \cos \phi - \sin \phi & -\sin \phi \\ \sin \phi & \cos \phi \end{array} \right) \left( \begin{array}{c} \partial_x \\ \partial_y \end{array} \right) = A \left( \begin{array}{c} \partial_x \\ \partial_y \end{array} \right), \quad (8) $$
+
+where $\tan 2\phi = f_{xy}/(f_{yy} - f_{xx})$. All functions over $\partial_p^i \partial_q^j$, $i+j \ge 2$, becomes invariants w.r.t. the unimodular isotropic group, i.e. rotation in the image plane and addition of a linear light gradient [14]. The Hessian frame $\{\partial_p, \partial_q\}$ is invariant w.r.t. the isotropic group, i.e. all the motion in the isotropic group as well as scaling in the plane and in the intensity fiber [14].

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+5 Functions in $\Gamma_2$
+
+First let us study the general geometrical situation for $\Gamma_2$. The attitude transformation must be of the form:
+
+$$ \begin{pmatrix} \partial_s \\ \partial_u \end{pmatrix} = \begin{pmatrix} 1 & v \\ 0 & 1 \end{pmatrix} \begin{pmatrix} \partial_t \\ \partial_x \end{pmatrix} = A \begin{pmatrix} \partial_t \\ \partial_x \end{pmatrix}, \qquad (9) $$
+
+where $v$, is a function of the spatio-temporal position and $\{\partial_s, \partial_x\}$ is the adapted frame. We immediately see that $\partial_u = \partial_x$. The connection matrix becomes:
+
+$$ C(A) = \begin{pmatrix} 0 & c_{01} \\ 0 & 0 \end{pmatrix} \qquad (10) $$
+
+where $c_{01} = v_t dt + v_x dx$. This could be expressed in the adapted coordinate system instead, giving $c_{01} = v_s ds + v_u du$. If the coefficient in the connection matrix is contracted on the vectors in the adapted frame, we get two scalar invariants, $a = c_{01}\partial_s = v_s$, that describe how the spatio-temporal part of the frame changes in the direction of it self, i.e. it describes the acceleration of the structure that the frame is adapted to. The other scalar invariant, $\delta = c_{01}\partial_u = v_u$, describes how the spatio-temporal part of the adapted frame changes in the spatial direction, i.e. the divergence of the vector field $\partial_s$, restricted to the spatial line.
+
+For scalar functions on $\Gamma_2$, there are three typical situations, the level curves are transverse to the spatial lines almost everywhere, along isolated curves the level curves are tangent to the spatial lines and there are also isolated critical points.
+
+If one uses the constant brightness assumption as binding hypothesis between image patterns and surface motion then the level curves, (or isophotes) corresponds to motion in the traversal case and creation or annihilation of structure in the non-transversal case.
+
+5.1 Spatially Transversal Level Curves
+
+On points where the level curve is transverse to the spatial line, $f_x \neq 0$, we can define a $\Gamma_2$-frame, $\{\partial_s, \partial_x\}$, s.t. $f_s = 0$. Expressed in an arbitrary $\Gamma_2$-frame, $\{\partial_t, \partial_x\}$, $\partial_s$ must be on the form:
+
+$$ \partial_s = \partial_t + \gamma \partial_x, \qquad (11) $$
+
+using $f_s = 0$ and solving for $\gamma$, we get $\gamma = -f_t/f_x$. Hence the attitude matrix becomes,
+
+$$ A = \begin{pmatrix} 1 & -f_t/f_x \\ 0 & 1 \end{pmatrix} \qquad (12) $$
+
+and for the connection matrix (10), we get:
+
+$$ c_{01} = \frac{f_t f_{tx} - f_x f_{tt}}{f_x^2} dt + \frac{f_t f_{xx} - f_x f_{tx}}{f_x^2} dx = -\frac{f_{ss}}{f_x} ds - \frac{f_{sx}}{f_x} dx. \qquad (13) $$

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+Contracting $c_{01}$ on the vectors of the adapted frame we get our scalar invariants, the invariants are summarized in the following theorem.
+
+**Theorem 2.** A complete set of level-curve invariants for spatially transversal level-curves on $\Gamma_2$ is level-curve acceleration the level-curve divergence
+
+$$a = c_{01} \partial_s = -f_{ss}/f_x, \quad \delta = c_{01} \partial_x = -f_{sx}/f_x. \qquad (14)$$
+
+## 5.2 Hessian Invariants
+
+On points where $f_x = 0$, there is no tangent gauge. For points where $f_{xx} \neq 0$, we can define a Hessian gauge, i.e. an adapted Galilean ON-frame $\{\partial_s, \partial_x\}$ s.t. $f_{sx} = 0$. Repeating the steps from the last section, applying (11) on $f_x$, using $f_{sx} = 0$ and solving for $\gamma$, we get the attitude transformation:
+
+$$\begin{pmatrix} \partial_s \\ \partial_x \end{pmatrix} = \begin{pmatrix} 1 & -f_{tx}/f_{xx} \\ 0 & 1 \end{pmatrix} \begin{pmatrix} \partial_t \\ \partial_x \end{pmatrix} = A \begin{pmatrix} \partial_t \\ \partial_x \end{pmatrix}. \qquad (15)$$
+
+and in the connection matrix (10), we get:
+
+$$c_{01} = -\frac{f_{ssx}}{f_{xx}}ds - \frac{f_{sxx}}{f_{xx}} = a ds + \delta dx. \qquad (16)$$
+
+Which we summarize in the following theorem.
+
+**Theorem 3.** A complete set of Hessian invariants for points where $f_{xx} \neq 0$ on $\Gamma_2$ is Hessian acceleration and Hessian divergence
+
+$$a = c_{01} \partial_s = -f_{ssx}/f_{xx}, \quad \delta = c_{01} \partial_x = -f_{sxx}/f_{xx}. \qquad (17)$$
+
+# 6 Functions in $\Gamma_3$
+
+For Galilean 2+1 dimensional geometry, the attitude matrix in general have the form:
+
+$$\begin{pmatrix} \partial_t \\ \partial_u \\ \partial_v \end{pmatrix} = \begin{pmatrix} 1 & v^x & v^y \\ 0 & \cos\theta & -\sin\theta \\ 0 & \sin\theta & \cos\theta \end{pmatrix} \begin{pmatrix} \partial_t \\ \partial_x \\ \partial_y \end{pmatrix} = A_i, \qquad (18)$$
+
+where $v^x, v^y$ and $\theta$ are functions of the spatio-temporal position. It can be shown that the connection matrix expressed in the adapted coordinate system has the form:
+
+$$C(A) = \begin{pmatrix} a^u ds + \delta^u du + \sigma^u dv & a^v ds + \delta^v du + \sigma^v dv \\ 0 & 0 \\ 0 & -(\rho ds + \kappa^u du + \kappa^v dv) \end{pmatrix} = \begin{pmatrix} 0 & c_{01} & c_{02} \\ 0 & 0 & c_{12} \\ 0 & -c_{12} & 0 \end{pmatrix}. \qquad (19)$$
+
+Here $c_{01}$ and $c_{02}$ describes how the spatio-temporal part of the frame moves in different directions, $c_{01}$ describes the motion projected on the $\{\partial_s, \partial_u\}$ plane,

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+and $c_{02}$ the motion projected on the $\{\partial_s, \partial_v\}$ plane. The form $c_{12}$ describes how the spatial frame $\{\partial_u, \partial_v\}$ rotates when moving in different directions.
+
+Contracting the connection forms on the different vectors in the local adapted frame, we get nine different scalar invariants. We continue by giving these invariants an interpretation. If we consider the integral curves from the vector field $\{\partial_s\}$, then $a^u$ describe the acceleration of the integral curve projected on the $\{\partial_s, \partial_u\}$ plane, and $a^v$ the corresponding acceleration on the $\{\partial_s, \partial_v\}$ plane. $\rho$ describes how much the spatial part of the frame rotates in the $\partial_s$ direction. The invariants $\kappa^u$ and $\kappa^v$ describe the curvatures of the integral curves for the vector fields $\{\partial_u\}$ and $\{\partial_v\}$ respectively. The remaining invariants describe how the vector field $\{\partial_s\}$ changes for motions in the spatial plane, $\delta^u$ and $\delta^v$ describe the divergence in the $\partial_u$ and $\partial_v$ directions respectively. $\sigma_u$ describes the skew of the vector field in the $\partial_u$ direction while moving in the $\partial_v$ direction and $\sigma_v$ the skew in the $\partial_v$ direction while moving in the $\partial_u$ direction.
+
+## 6.1 More Descriptive Invariants
+
+Even if the above discussed set of scalar invariants constitute a complete set of scalar invariants for $\Gamma_3$, they are not necessarily the ones that have largest descriptive value. As any invertible transformation of the scalar invariants give rise to a new complete set of scalar invariants, we will develop a set of invariants that are closer to what have been used in other work about moving images.
+
+The acceleration invariants $\{a^u, a^v\}$ could instead be described in a polar coordinate system:
+
+$$a = \sqrt{(a^u)^2 + (a^v)^2}, \quad a_\theta = \arctan(a^v/a^u), \qquad (20)$$
+
+here $a$ is the magnitude of the acceleration, an $a_\theta$ the angle relative to the $\partial_u$ direction. The invariants, $\delta_u, \delta_v, \sigma_u, \sigma_v$ describes how $\partial_s$ changes along motions in the spatial plane. Observe that the vectors in the vector field $\{\partial_s\}$ always have unit length in the temporal direction, therefore the vector field restricted to a certain spatial plane can be projected onto that plane without losing any essential information. The matrix:
+
+$$D = \begin{pmatrix} \delta_u & \sigma_u \\ \sigma_v & \delta_v \end{pmatrix} \qquad (21)$$
+
+is the rate of strain tensor for that projected vector field and it might be more useful to describe the invariants in terms of the Cauchy-Stokes decomposition theorem [12]:
+
+$$D = \frac{\sigma_u - \sigma_v}{2} \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} + \frac{\delta_u + \delta_v}{2} \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} + \frac{1}{2} \begin{pmatrix} \delta_u - \delta_v & \sigma_u + \sigma_v \\ \sigma_u + \sigma_v & \delta_v - \delta_u \end{pmatrix} \quad (22)$$
+
+$$= \frac{\mathrm{curl} D}{2} \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} + \frac{\mathrm{div} D}{2} \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} + \frac{\mathrm{def} D}{2} Q(\phi)^{-1} \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} Q(\phi). \quad (23)$$
+
+First the matrix can be decomposed in an anti symmetric and a symmetric part where the coefficient of the anti symmetric part is called the *curl* that describes

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+the rotational component of the vector field. The symmetric part can in turn be decomposed in a multiple of the identity matrix, the *divergence* part that describe the dilation component of the vector field, and a symmetric matrix with zero trace. The remaining symmetric component of the matrix can be described in terms of the *deformation*, i.e. an area preserving stretching in one direction combined with shrinking in the orthogonal direction, and the direction $\phi$ of the stretching relative to the direction of $\partial_u$.
+
+## 6.2 Choice of Gauge
+
+For Galilean 2 + 1 dimensional geometry isophotes are typically 2 dimensional surfaces. There are two generic cases: points where the isophote surface cuts the spatial surface through the point, and points where the isophote surface is tangent to the spatial surface through the point. The first case can be interpreted as motion of isophote curves in the image, and the second case as creation, annihilation or saddle points.
+
+## 6.3 Tangent Gauge
+
+Our next task is to define an adapted frame for points where the isophote surface cuts the spatial surface. For the spatial plane we can reuse the tangent gauge for $E_2$ in Section 4.1. Starting from an arbitrary frame $i$, we first adapt the spatial sub frame $\{\partial_x, \partial_y\}$, to the gradient and tangent direction in the spatial plane:
+
+$$ \begin{pmatrix} \partial_t \\ \partial_u \\ \partial_v \end{pmatrix} = \frac{1}{\sqrt{f_x^2 + f_y^2}} \begin{pmatrix} 1 & 0 & 0 \\ 0 & f_y & -f_x \\ 0 & f_x & f_y \end{pmatrix} \begin{pmatrix} \partial_t \\ \partial_x \\ \partial_y \end{pmatrix} = Ai. \quad (24) $$
+
+The spatio-temporal vector $\partial_s$ must have unit length in time to be part of a Galilean frame. By requiring $\partial_s$ to lie in the spatio-temporal tangent plane, i.e. $f_s = 0$, it is constrained in one direction. The adapted spatio-temporal direction must have the form:
+
+$$ \partial_s = \partial_t + \beta\partial_u + \gamma\partial_v, $$
+
+in terms of the new frame. Using $0 = f_s = f_t + \gamma f_v$ and solving for $\gamma$ we get that $\gamma = -f_t/f_v$. Still we have one undetermined degree of freedom $\beta \in \mathbb{R}$. For each choice of $\beta$ we have a plane spanned by $\{\partial_s, \partial_v\}$. The image restricted to such a plane is a function on $\Gamma_2$ and can be studied by the methods from Section 5.1. From Theorem 2 there are two scalar invariants: acceleration $a = -f_{ss}/f_v$ and divergence $\delta = -f_{sv}/f_v$. We can see that acceleration becomes a quadratic function of $\beta$ and thus the gauge can be fixed by finding a $\beta$ s.t. $a(\beta)$ is an extremum, i.e. by solving $\partial_\beta a(\beta) = 0$ for $\beta$, which gives:
+
+$$ \beta_a = \frac{f_t f_{uv}}{f_v f_{uu}} - \frac{f_{tu}}{f_{uu}}. \quad (25) $$
+
+Which is defined as long as $f_{uu} \neq 0$, i.e. as long as the isophote curvature in the spatial plane is non-vanishing. It can be shown that requirement of an

samples/texts/1513018/page_1.md

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+Numerical approximation of the 3d hydrostatic
+Navier-Stokes system with free surface
+
+Sebastien Allgeyer, Marie-Odile Bristeau, David Froger, Raouf Hamouda,
+Vincent Jauzein, Anne Mangeney, Jacques Sainte-Marie, Fabien Souillé,
+Martin Vallée
+
+► To cite this version:
+
+Sebastien Allgeyer, Marie-Odile Bristeau, David Froger, Raouf Hamouda, Vincent Jauzein, et al.: Numerical approximation of the 3d hydrostatic Navier-Stokes system with free surface. ESAIM: Mathematical Modelling and Numerical Analysis, EDP Sciences, 2019, 53 (6), 10.1051/m2an/2019044 hal-01393147v5
+
+HAL Id: hal-01393147
+
+https://hal.inria.fr/hal-01393147v5
+
+Submitted on 19 Jul 2019
+
+**HAL** is a multi-disciplinary open access
+archive for the deposit and dissemination of sci-
+entific research documents, whether they are pub-
+lished or not. The documents may come from
+teaching and research institutions in France or
+abroad, or from public or private research centers.
+
+L'archive ouverte pluridisciplinaire **HAL**, est
+destinée au dépôt et à la diffusion de documents
+scientifiques de niveau recherche, publiés ou non,
+émanant des établissements d'enseignement et de
+recherche français ou étrangers, des laboratoires
+publics ou privés.

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+with $\mathbf{U}^N = (u^N, v^N, w^N)^T$. Simple computations give
+
+$$0 = \int_{\mathbb{R}} \mathbf{1}_{z \in L_\alpha(t,x,y)} \nabla \cdot \mathbf{U}^N dz = \frac{\partial h_\alpha}{\partial t} + \frac{\partial}{\partial x} \int_{z_{\alpha-1/2}}^{z_{\alpha+1/2}} u \, dz + \frac{\partial}{\partial y} \int_{z_{\alpha-1/2}}^{z_{\alpha+1/2}} v \, dz - G_{\alpha+1/2} + G_{\alpha-1/2},$$
+
+leading to
+
+$$\frac{\partial h_{\alpha}}{\partial t} + \nabla_{x,y} (h_{\alpha} \mathbf{u}_{\alpha}) = G_{\alpha+1/2} - G_{\alpha-1/2}, \quad (28)$$
+
+with $G_{\alpha \pm 1/2}$ defined by
+
+$$G_{\alpha+1/2} = \frac{\partial z_{\alpha+1/2}}{\partial t} + \mathbf{u}_{\alpha+1/2} \cdot \nabla_{x,y} z_{\alpha+1/2} - w_{\alpha+1/2}.$$
+
+The sum for $\alpha = 1, \dots, N$ of the above relations gives Eq. (18) where the kinematic boundary conditions (4), (5) corresponding to
+
+$$G_{1/2} = G_{N+1/2} = 0, \quad (29)$$
+
+have been used. Similarly, the sum for $j = 1, \dots, \alpha$ of the relations (28) with (29) gives the expression (20) for $G_{\alpha+1/2}$.
+
+Now we consider the Galerkin approximation of Eq. (13) i.e. the quantity
+
+$$\int_{\mathbb{R}} \mathbf{1}_{z \in L_\alpha(t,x,y)} \left( \frac{\partial \mathbf{u}^N}{\partial t} + \nabla_{x,y} (\mathbf{u}^N \otimes \mathbf{u}^N) + \frac{\partial \mathbf{u}^N w^N}{\partial z} + g \nabla_{x,y} \eta \right) dz = 0,$$
+
+leading, after simple computations, to Eq. (19). $\blacksquare$
+
+**Proof of prop. 3.2** In order to obtain (22) we multiply (28) by $g(h+z_b)$ ($|\mathbf{u}_\alpha|^2/2$ and (19) by $\mathbf{u}_\alpha$ then we perform simple manipulations. More precisely, the momentum equation along the x axis multiplied by $u_\alpha$ gives
+
+$$\left(\frac{\partial}{\partial t}(h_\alpha u_\alpha) + \frac{\partial}{\partial x}\left(h_\alpha u_\alpha^2 + \frac{g}{2}hh_\alpha\right) + \frac{\partial}{\partial y}(h_\alpha u_\alpha v_\alpha)\right)u_\alpha = \\ \left(-gh_\alpha \frac{\partial z_b}{\partial x} + u_{\alpha+1/2}G_{\alpha+1/2} - u_{\alpha-1/2}G_{\alpha-1/2}\right)u_\alpha.$$
+
+Considering first the left hand side of the preceding equation excluding the pressure terms, we denote
+
+$$I_{u,\alpha} = \left( \frac{\partial}{\partial t}(h_\alpha u_\alpha) + \frac{\partial}{\partial x}(h_\alpha u_\alpha^2) + \frac{\partial}{\partial y}(h_\alpha u_\alpha v_\alpha) \right) u_\alpha,$$
+
+and using (28) we have
+
+$$I_{u,\alpha} = \frac{\partial}{\partial t} \left( \frac{h_\alpha u_\alpha^2}{2} \right) + \frac{\partial}{\partial x} \left( u_\alpha \frac{h_\alpha u_\alpha^2}{2} \right) + \frac{\partial}{\partial y} \left( v_\alpha \frac{h_\alpha u_\alpha^2}{2} \right) + \frac{u_\alpha^2}{2} \left( \frac{\partial h_\alpha}{\partial t} + \nabla_{x,y} (h_\alpha \mathbf{u}_\alpha) \right).$$
+
+Now we consider the contribution of the pressure terms over the energy balance i.e.
+
+$$I_{p,u,\alpha} = \left( \frac{\partial}{\partial x} \left( \frac{g}{2} hh_{\alpha} \right) + gh_{\alpha} \frac{\partial z_{b}}{\partial x} \right) u_{\alpha},$$
+
+and it comes
+
+$$
+\begin{align*}
+I_{p,u,\alpha} &= gh_{\alpha} \frac{\partial}{\partial x}(h+z_b)u_{\alpha} = g \frac{\partial}{\partial x}(h_{\alpha}(h+z_b)u_{\alpha}) - g(h+z_b) \frac{\partial}{\partial x}(h_{\alpha}u_{\alpha}) \\
+&= \frac{\partial}{\partial x}\left(\left(\frac{g}{2}h_{\alpha}h + \frac{g}{2}h_{\alpha}(h+2z_b)\right)u_{\alpha}\right) - g(h+z_b)\frac{\partial}{\partial x}(h_{\alpha}u_{\alpha}).
+\end{align*}
+$$
+
+Performing the same manipulations over the momentum equation along $y$ and adding the terms $I_{u,\alpha}, I_{v,\alpha}, I_{p,u,\alpha}, I_{p,v,\alpha}$ and (28) multiplied by $g(h+z_b)-|\mathbf{u}_\alpha|^2/2$ gives the result. $\blacksquare$

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+**Proof of prop. 3.3** Using the boundary condition (4), an integration from $z_b$ to $z$ of the divergence free condition (1) easily gives
+
+$$w = -\nabla_{x,y} \int_{z_b}^{z} \mathbf{u} \, dz.$$
+
+Replacing formally in the above equation $\mathbf{u}$ (resp. $w$) by $\mathbf{u}^N$ (resp. $w^N$) defined by (17) and performing an integration over the layer $L_1$ of the obtained relation yields
+
+$$h_1 w_1 = - \int_{z_b}^{z_{3/2}} \nabla_{x,y} \cdot \int_{z_b}^{z} \mathbf{u}_1 \, dz \, dz_1 = h_1 \nabla_{x,y} \cdot (z_b \mathbf{u}_1) - \frac{z_{3/2}^2 - z_b^2}{2} \nabla_{x,y} \cdot \mathbf{u}_1,$$
+
+i.e. $w_1 = \nabla_{x,y} \cdot (z_b \mathbf{u}_1) - z_1 \nabla_{x,y} \cdot \mathbf{u}_1$, corresponding to (27) for $\alpha = 1$. A similar computation for the layers $L_2, \dots, L_N$ proves the result (27) for $\alpha = 2, \dots, N$. A more detailed version of this proof is given in [24]. ■
+
+## 3.2 The layer-averaged Navier-Stokes system
+
+In paragraph 3.1, we have applied the layer-averaging to the Euler system, we now use the same process for the hydrostatic Navier-Stokes system. First, we consider the Navier-Stokes system (1)-(7) for a Newtonian fluid and then with a simplified rheology.
+
+### 3.2.1 Complete model
+
+The layer-averaging process applied to the Navier-Stokes system (1)-(7) leads to the following proposition.
+
+**Proposition 3.4** *The layer-averaged hydrostatic Navier-Stokes system (1)-(7) is given by*
+
+$$ 
+\begin{align}
+& \sum_{\alpha=1}^{N} \frac{\partial h_{\alpha}}{\partial t} + \sum_{\alpha=1}^{N} \nabla_{x,y} \cdot (h_{\alpha} \mathbf{u}_{\alpha}) = 0, \tag{30} \\
+& \frac{\partial h_{\alpha} \mathbf{u}_{\alpha}}{\partial t} + \nabla_{x,y} \cdot (h_{\alpha} \mathbf{u}_{\alpha} \otimes \mathbf{u}_{\alpha}) + \nabla_{x,y} \cdot (\frac{g}{2} h h_{\alpha}) = -gh_{\alpha} \nabla_{x,y} z_{b} \nonumber \\
+& + \mathbf{u}_{\alpha+1/2} G_{\alpha+1/2} - \mathbf{u}_{\alpha-1/2} G_{\alpha-1/2} + \nabla_{x,y} \cdot (h_{\alpha} \boldsymbol{\Sigma}_{\alpha}) \nonumber \\
+& - \boldsymbol{\Sigma}_{\alpha+1/2} \nabla_{x,y} z_{b} z_{\alpha+1/2} + \boldsymbol{\Sigma}_{\alpha-1/2} \nabla_{x,y} z_{b} z_{\alpha-1/2} \nonumber \\
+& + 2\nu_{\alpha+1/2} \frac{\mathbf{u}_{\alpha+1} - \mathbf{u}_{\alpha}}{h_{\alpha+1} + h_{\alpha}} - 2\nu_{\alpha-1/2} \frac{\mathbf{u}_{\alpha} - \mathbf{u}_{\alpha-1}}{h_{\alpha} + h_{\alpha-1}} - \kappa_{\alpha} \mathbf{u}_{\alpha} + W_{\alpha} \mathbf{t}_{s}, \quad \alpha = 1, \dots, N, \tag{31}
+\end{align}
+$$
+
+with
+
+$$ 
+\Sigma_{\alpha+1/2} = 
+\begin{pmatrix}
+\Sigma_{xx,\alpha+1/2} & \Sigma_{xy,\alpha+1/2} \\
+\Sigma_{yx,\alpha+1/2} & \Sigma_{yy,\alpha+1/2}
+\end{pmatrix},
+\qquad (32)
+$$
+
+$$ 
+\Sigma_{xx,\alpha+1/2} = \frac{\nu_{\alpha+1/2}}{h_{\alpha+1} + h_\alpha} \left(h_\alpha \frac{\partial u_\alpha}{\partial x} + h_{\alpha+1} \frac{\partial u_{\alpha+1}}{\partial x}\right) - 2\nu_{\alpha+1/2} \frac{\partial z_{\alpha+1/2}}{\partial x} \frac{u_{\alpha+1}-u_\alpha}{h_{\alpha+1}+h_\alpha}, 
+\qquad (33)
+$$
+
+$$ 
+\Sigma_{xy,\alpha+1/2} = \frac{\nu_{\alpha+1/2}}{h_{\alpha+1} + h_\alpha} \left( h_\alpha \frac{\partial u_\alpha}{\partial y} + h_{\alpha+1} \frac{\partial u_{\alpha+1}}{\partial y} \right) - 2\nu_{\alpha+1/2} \frac{\partial z_{\alpha+1/2}}{\partial y} \frac{u_{\alpha+1}-u_\alpha}{h_{\alpha+1}+h_\alpha}, 
+\qquad (34)
+$$
+
+and similar definitions for $\Sigma_{yx,\alpha+1/2}$, $\Sigma_{yy,\alpha+1/2}$. We also denote
+
+$$ 
+\Sigma_\alpha = 
+\begin{pmatrix}
+\Sigma_{xx,\alpha} & \Sigma_{xy,\alpha} \\
+\Sigma_{yx,\alpha} & \Sigma_{yy,\alpha}
+\end{pmatrix}
+=
+\frac{\Sigma_{\alpha+1/2} + \Sigma_{\alpha-1/2}}{2},
+\qquad (35)
+$$

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+and
+
+$$
+\kappa_{\alpha} = \begin{cases} \kappa & \text{if } \alpha = 1 \\ 0 & \text{if } \alpha \neq 1 \end{cases}, \quad \nu_{\alpha+1/2} = \begin{cases} 0 & \text{if } \alpha = 0, N \\ \nu & \text{if } \alpha = 1, \dots, N-1 \end{cases}, \quad W_{\alpha} = \begin{cases} W & \text{if } \alpha = N \\ 0 & \text{if } \alpha \neq N \end{cases} \tag{36}
+$$
+
+The vertical velocities {$w_\alpha$}$_{\alpha=1}^N$ are defined by (27).
+For smooth solutions, the system (30)-(31) admits the energy balance
+
+$$
+\frac{\partial}{\partial t} \sum_{\alpha=1}^{N} E_{\alpha} + \nabla_{x,y} \cdot \sum_{\alpha=1}^{N} \mathbf{u}_{\alpha} \left( E_{\alpha} + \frac{g}{2} h_{\alpha} h - h_{\alpha} \boldsymbol{\Sigma}_{\alpha} \right) = - \sum_{\alpha=1}^{N-1} \frac{|\mathbf{u}_{\alpha+1} - \mathbf{u}_{\alpha}|^2}{2} |G_{\alpha+1/2}| \\
+- \sum_{\alpha=1}^{N-1} \frac{h_{\alpha+1} + h_{\alpha}}{2\nu} \boldsymbol{\Sigma}_{\alpha+1/2}^2 - \sum_{\alpha=1}^{N-1} 2\nu \frac{|\mathbf{u}_{\alpha+1} - \mathbf{u}_{\alpha}|^2}{h_{\alpha+1} + h_{\alpha}} - \kappa |\mathbf{u}_1|^2 + W \mathbf{u}_N \cdot \mathbf{t}_s, \quad (37)
+$$
+
+with $E_\alpha$ defined by (23) and $\Sigma_{\alpha+1/2}^2 = \sum_{i,j} \Sigma_{i,j,\alpha+1/2}^2$. Relation (37) is consistent with a layer-averaged discretization of the equation (10).
+
+Notice that in (37), we use the notation
+
+$$
+\mathbf{u}_{\alpha}\boldsymbol{\Sigma}_{\alpha} = \begin{pmatrix} u_{\alpha}\boldsymbol{\Sigma}_{xx,\alpha} + v_{\alpha}\boldsymbol{\Sigma}_{yx,\alpha} \\ u_{\alpha}\boldsymbol{\Sigma}_{xy,\alpha} + v_{\alpha}\boldsymbol{\Sigma}_{yy,\alpha} \end{pmatrix}.
+$$
+
+**Proof of prop. 3.4** The proof is given in appendix A. ■
+
+**Remark 3.2** Notice that in the definition (35), since we consider viscous terms we use a centered approximation.
+
+3.2.2 Simplified rheology
+
+The viscous terms in the layer-averaged Navier-Stokes system are difficult to discretize especially when a discrete version of the energy balance has to be preserved. Hence, we propose a modified version of the model given in prop. 3.4.
+
+**Proposition 3.5** The layer-averaged Navier-Stokes can be written under the form
+
+$$
+\sum_{\alpha=1}^{N} \frac{\partial h_{\alpha}}{\partial t} + \sum_{\alpha=1}^{N} \nabla_{x,y}.(h_{\alpha} \mathbf{u}_{\alpha}) = 0, \tag{38}
+$$
+
+$$
+\begin{equation}
+\begin{split}
+& \frac{\partial h_\alpha}{\partial t} + \nabla_{x,y} (h_\alpha \mathbf{u}_\alpha \otimes \mathbf{u}_\alpha) + \nabla_{x,y} (\frac{g}{2} h_\alpha) \\
+& \qquad = -gh_\alpha \nabla_{x,y} z_b \\
+& \qquad + \mathbf{u}_{\alpha+1/2} G_{\alpha+1/2} - \mathbf{u}_{\alpha-1/2} G_{\alpha-1/2} + \nabla_{x,y} (h_\alpha \boldsymbol{\Sigma}_\alpha^0) - \mathbf{T}_\alpha \\
+& \qquad + \Lambda_{\alpha+1/2} (\mathbf{u}_{\alpha+1} - \mathbf{u}_\alpha) - \Lambda_{\alpha-1/2} (\mathbf{u}_\alpha - \mathbf{u}_{\alpha-1}) - \kappa_\alpha \mathbf{u}_\alpha + W_\alpha \mathbf{t}_s,
+\end{split}
+\tag{39}
+\end{equation}
+$$
+
+and
+
+$$
+\Sigma_{xx,\alpha+1/2}^0 = \frac{\nu_{\alpha+1/2}}{h_{\alpha+1} + h_{\alpha}} \left(h_{\alpha} \frac{\partial u_{\alpha}}{\partial x} + h_{\alpha+1} \frac{\partial u_{\alpha+1}}{\partial x}\right), \quad (40)
+$$
+
+$$
+\Sigma^0_{xy,\alpha+1/2} = \frac{\nu_{\alpha+1/2}}{h_{\alpha+1} + h_{\alpha}} \left( h_{\alpha} \frac{\partial u_{\alpha}}{\partial y} + h_{\alpha+1} \frac{\partial u_{\alpha+1}}{\partial y} \right), \quad (41)
+$$
+
+and similar definitions for $\Sigma^0_{yx,\alpha+1/2}$, $\Sigma^0_{yy,\alpha+1/2}$. We also denote
+
+$$
+T_{x,\alpha+1/2} = \frac{2\nu_{\alpha+1/2}}{h_{\alpha+1} + h_{\alpha}} \nabla_{x,y} z_{\alpha+1/2} (h_{\alpha} \nabla_{x,y} u_{\alpha} + h_{\alpha+1} \nabla_{x,y} u_{\alpha+1}), \quad (42)
+$$

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+$$T_{y,\alpha+1/2} = \frac{2\nu_{\alpha+1/2}}{h_{\alpha+1} + h_{\alpha}} \nabla_{x,y} z_{\alpha+1/2} \cdot (h_{\alpha} \nabla_{x,y} v_{\alpha} + h_{\alpha+1} \nabla_{x,y} v_{\alpha+1}), \quad (43)$$
+
+$$\Lambda_{\alpha+1/2} = \nu_{\alpha+1/2} \frac{2 + |\nabla_{x,y} z_{\alpha+1/2}|^2}{h_{\alpha+1} + h_{\alpha}}, \quad (44)$$
+
+$$T_{\alpha+1/2} = \begin{pmatrix} T_{x,\alpha+1/2} \\ T_{y,\alpha+1/2} \end{pmatrix}, \quad T_{\alpha} = \begin{pmatrix} T_{x,\alpha} \\ T_{y,\alpha} \end{pmatrix} = \frac{T_{\alpha+1/2} + T_{\alpha-1/2}}{2}, \quad (45)$$
+
+$$\Sigma_{\alpha+1/2}^{0} = \begin{pmatrix} \Sigma_{xx,\alpha+1/2}^{0} & \Sigma_{xy,\alpha+1/2}^{0} \\ \Sigma_{yx,\alpha+1/2}^{0} & \Sigma_{yy,\alpha+1/2}^{0} \end{pmatrix}, \quad (46)$$
+
+$$\Sigma_\alpha^0 = \begin{pmatrix} \Sigma_{xx,\alpha}^0 & \Sigma_{xy,\alpha}^0 \\ \Sigma_{yx,\alpha}^0 & \Sigma_{yy,\alpha}^0 \end{pmatrix} = \frac{\Sigma_{\alpha+1/2}^0 + \Sigma_{\alpha-1/2}^0}{2}. \quad (47)$$
+
+For smooth solutions, the system (38)-(39) admits the energy balance
+
+$$
+\begin{align}
+& \frac{\partial}{\partial t} \sum_{\alpha=1}^{N} E_{\alpha} + \nabla_{x,y} \cdot \left( \sum_{\alpha=1}^{N} \mathbf{u}_{\alpha} \left( E_{\alpha} + \frac{g}{2} h_{\alpha} h - h_{\alpha} \boldsymbol{\Sigma}_{\alpha}^{0} \right) \right) \nonumber \\
+& = - \sum_{\alpha=1}^{N-1} \frac{|\mathbf{u}_{\alpha+1} - \mathbf{u}_{\alpha}|^2}{2} |G_{\alpha+1/2}| - \sum_{\alpha=1}^{N-1} \frac{2\nu}{h_{\alpha+1} + h_{\alpha}} |\mathbf{u}_{\alpha+1} - \mathbf{u}_{\alpha}|^2 - \kappa |\mathbf{u}_1|^2 + W \mathbf{u}_N \cdot \mathbf{t}_s \nonumber \\
+& \quad - \sum_{\alpha=1}^{N-1} \frac{h_{\alpha+1} + h_{\alpha}}{2\nu} \left( (\Sigma_{xx,\alpha+1/2}^0 + \frac{\partial z_{\alpha+1/2}}{\partial x}(u_{\alpha+1}-u_{\alpha}))^2 + (\Sigma_{xy,\alpha+1/2}^0 + \frac{\partial z_{\alpha+1/2}}{\partial y}(u_{\alpha+1}-u_{\alpha}))^2 \right. \nonumber \\
+& \qquad \left. + (\Sigma_{yx,\alpha+1/2}^0 + \frac{\partial z_{\alpha+1/2}}{\partial x}(v_{\alpha+1}-v_{\alpha}))^2 + (\Sigma_{yy,\alpha+1/2}^0 + \frac{\partial z_{\alpha+1/2}}{\partial y}(v_{\alpha+1}-v_{\alpha}))^2 \right). \tag{48}
+\end{align}
+$$
+
+**Proof of prop. 3.5** Using simple manipulations, the model given in prop. 3.4 corresponds to Eqs. (38)-(47) except Eq. (44) having the form
+
+$$
+\Gamma_{\alpha+1/2} = \nu_{\alpha+1/2} \frac{2 + |\nabla_{x,y} z_{\alpha+1/2}|^2}{h_{\alpha+1} + h_\alpha} - \nu_{\alpha+1/2} \nabla_{x,y} \cdot \left( \frac{h_\alpha \nabla_{x,y} z_{\alpha+1/2}}{h_{\alpha+1} + h_\alpha} \right).
+$$
+
+Notice that
+
+$$
+2 \frac{u_{\alpha+1} - u_{\alpha}}{h_{\alpha+1} + h_{\alpha}} \approx \frac{\partial u}{\partial z},
+$$
+
+for $N$ large and hence can be considered as bounded. Therefore, the second term of $\Gamma_{\alpha+1/2}(u_{\alpha+1} - u_{\alpha})$ vanishes when $h\nu_\alpha \to 0$. Then the quantity $\Gamma_{\alpha+1/2}(u_{\alpha+1} - u_\alpha)$ reduces to the expression of $\Lambda_{\alpha+1/2}(u_{\alpha+1} - u_\alpha)$ with $\Lambda_{\alpha+1/2}$ given by Eq. (44).
+
+The proof of the energy balance (48) is similar to the one given in prop. 3.4. ■
+
+**Remark 3.3** The layer-averaged Navier-Stokes system defined by (38)-(39) has the form
+
+$$
+\frac{\partial U}{\partial t} + \nabla_{x,y} F(U) = S_b(U) + S_e(U, \partial_t U, \partial_x U) + S_{v,f}(U), \quad (49)
+$$
+
+where $U = (h, q_{x,1}, \dots, q_{x,N}, q_{y,1}, \dots, q_{y,N})^T$, and
+
+$$S_b(U) = \left( 0, gh_1 \frac{\partial z_b}{\partial x}, \dots, gh_N \frac{\partial z_b}{\partial x}, gh_1 \frac{\partial z_b}{\partial y}, \dots, gh_N \frac{\partial z_b}{\partial y} \right)^T,$$
+
+with $q_{x,\alpha} = h_\alpha u_\alpha$, $q_{y,\alpha} = h_\alpha v_\alpha$. We denote with $F(U)$ the fluxes of the conservative part, and with $S_e(U, \partial_t U, \partial_x U)$ and $S_{v,f}(U)$ the source terms, representing respectively the momentum exchanges and the viscous, wind and friction effects.
+
+The numerical scheme for the system (49) will be given in Section 5.

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+# 4 Kinetic description for the Euler system
+
+In this section we give a kinetic interpretation of the system (18)-(22). The numerical scheme for the system (18)-(19),(27) will be deduced from the kinetic description.
+
+## 4.1 Preliminaries
+
+We begin this section by recalling the classical kinetic approach – used in [46] for example – for the 1d Saint-Venant system
+
+$$ 
+\begin{aligned}
+\partial_t h + \partial_x(hu) &= 0, \\
+\partial_t(hu) + \partial_x(hu^2 + g \frac{h^2}{2}) + gh\partial_x z_b &= 0,
+\end{aligned} 
+\quad (50) $$
+
+with the water depth $h(t,x) \ge 0$, the water velocity $u(t,x) \in \mathbb{R}$ and a slowly varying topography $z_b(x)$.
+
+The kinetic Maxwellian is given by
+
+$$ M(U, \xi) = \frac{1}{g\pi} \left(2gh - (\xi - u)^2\right)_+^{1/2}, \quad (51) $$
+
+where $U = (h, hu)^T$, $\xi \in \mathbb{R}$ and $x_+ = \max(0, x)$ for any $x \in \mathbb{R}$. It satisfies the following moment relations,
+
+$$ \int_{\mathbb{R}} \left(\frac{1}{\xi}\right) M(U, \xi) d\xi = U, \qquad \int_{\mathbb{R}} \xi^2 M(U, \xi) d\xi = hu^2 + g \frac{h^2}{2}. \quad (52) $$
+
+These definitions allow us to obtain a *kinetic representation* of the Saint-Venant system.
+
+**Lemma 4.1** If the topography $z_b(x)$ is Lipschitz continuous, the pair of functions $(h, hu)$ is a weak solution to the Saint-Venant system (50) if and only if $M(U, \xi)$ satisfies the kinetic equation
+
+$$ \partial_t M + \xi \partial_x M - g(\partial_x z_b) \partial_\xi M = Q, \quad (53) $$
+
+for some “collision term” $Q(t, x, \xi)$ that satisfies, for a.e. $(t, x)$,
+
+$$ \int_{\mathbb{R}} Q d\xi = \int_{\mathbb{R}} \xi Q d\xi = 0. \quad (54) $$
+
+**Proof** If (53) and (54) are satisfied, we can multiply (53) by $(1, \xi)^T$, and integrate with respect to $\xi$. Using (52) and (54) and integrating by parts the term in $\partial_\xi M$, we obtain (50). Conversely, if $(h, hu)$ is a weak solution to (50), just define $Q$ by (53); it will satisfy (54) according to the same computations. $\blacksquare$
+
+The standard way to use Lemma 4.1 is to write a kinetic relaxation equation [4, 14, 15], like
+
+$$ \partial_t f + \xi \partial_x f - g(\partial_x z_b) \partial_\xi f = \frac{M-f}{\epsilon}, \quad (55) $$
+
+where $f(t,x,\xi) \ge 0$, $M = M(U,\xi)$ with $U(t,x) = \int (1,\xi)^T f(t,x,\xi) d\xi$, and $\epsilon > 0$ is a relaxation time. In the limit $\epsilon \to 0$ we recover formally the formulation (53), (54). We refer to [14] for general considerations on such kinetic relaxation models without topography, the case with topography being introduced in [46]. Note that the notion of *kinetic representation* as (53), (54) differs from the so called *kinetic formulations* where a large set of entropies is involved, see [45]. For systems of conservation laws, these kinetic formulations include non-adveptive terms that prevent from writing down simple approximations. In general, kinetic relaxation approximations can be compatible with just a single entropy. Nevertheless this is enough for proving the convergence as $\epsilon \to 0$, see [13].

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+## 4.2 Kinetic interpretation
+
+In this paragraph, we give a kinetic interpretation of the model (18)-(19), (25). To build the Gibbs equilibria, we choose the function
+
+$$ \chi_0(z_1, z_2) = \frac{1}{4\pi} \mathbf{1}_{z_1^2+z_2^2 \le 4}. \quad (56) $$
+
+This choice corresponds to the 2d version of the kinetic maxwellian used in 1d (see remark 4.2) and we have
+
+$$ M_{\alpha} = M(U_{\alpha}, \xi, \gamma) = \frac{h_{\alpha}}{c^2} \chi_0 \left( \frac{\xi - u_{\alpha}}{c}, \frac{\gamma - v_{\alpha}}{c} \right), \quad (57) $$
+
+with $c = \sqrt{\frac{g}{2}h}$
+
+$$ U_{\alpha} = (h_{\alpha}, h_{\alpha}u_{\alpha}, h_{\alpha}v_{\alpha})^{T}, \quad (58) $$
+
+and where $(\xi, \gamma) \in \mathbb{R}^2$. In other words, we have $M_\alpha = \frac{l_\alpha}{2g\pi} \mathbf{1}_{(\xi-u_\alpha)^2 + (\gamma-v_\alpha)^2 \le 2gh}$.
+
+**Remark 4.2** Starting from the 2d maxwellian in the single layer case i.e.
+
+$$ M_{sv} = \frac{1}{2g\pi} \mathbf{1}_{(\xi-u)^2 + (\gamma-v)^2 \le 2gh}, \quad (59) $$
+
+and computing its integral w.r.t. the variable $\gamma$ yields
+
+$$ \int_{\mathbb{R}} M_{sv} d\gamma = \int_{v-\sqrt{(2gh-(\xi-u)^2)+}}^{v+\sqrt{(2gh-(\xi-u)^2)+}} \frac{1}{2g\pi} d\gamma = \frac{1}{g\pi} \sqrt{(2gh - (\xi - u)^2)}, $$
+
+that is exactly the expression (51).
+
+The quantity $M_\alpha$ satisfies the following moment relations
+
+$$ \int_{\mathbb{R}^2} \begin{pmatrix} 1 \\ \xi \\ \gamma \end{pmatrix} M(U_\alpha, \xi, \gamma) d\xi d\gamma = \begin{pmatrix} h_\alpha \\ h_\alpha u_\alpha \\ h_\alpha v_\alpha \end{pmatrix}, \quad \int_{\mathbb{R}^2} \begin{pmatrix} \xi^2 \\ \xi\gamma \\ \gamma^2 \end{pmatrix} M(U_\alpha, \xi, \gamma) d\xi d\gamma = \begin{pmatrix} h_\alpha u_\alpha^2 + g^{\frac{h_\alpha h}{2}} \\ h_\alpha u_\alpha v_\alpha \\ h_\alpha v_\alpha^2 + g^{\frac{h_\alpha h}{2}} \end{pmatrix}. \quad (60) $$
+
+The interest of the function $\chi_0$ and hence the particular form (57) lies in its link with a kinetic entropy. Consider the kinetic entropy
+
+$$ H(f, \xi, \gamma, z_b) = \frac{\xi^2 + \gamma^2}{2} f + g z_b f, \quad (61) $$
+
+where $f \ge 0$, $(\xi, \gamma) \in \mathbb{R}^2$, $z_b \in \mathbb{R}$. Then one can check the relations
+
+$$ \int_{\mathbb{R}^2} \begin{pmatrix} 1 \\ \xi \\ \gamma \end{pmatrix} H(M_\alpha, \xi, \gamma) d\xi d\gamma = \begin{pmatrix} E_\alpha = \frac{h_\alpha}{2}(u_\alpha^2 + v_\alpha^2) + g\frac{h_\alpha}{2}h_\alpha(h+2z_b) \\ u_\alpha(E_\alpha + g\frac{h_\alpha}{2}h_\alpha h) \\ v_\alpha(E_\alpha + g\frac{h_\alpha}{2}h_\alpha h) \end{pmatrix}. \quad (62) $$
+
+Let us introduce the Gibbs equilibria $N_{\alpha+1/2}$ defined by for $\alpha = 0, \dots, N$ by
+
+$$ N_{\alpha+1/2} = N(U_{\alpha+1/2}, \xi) = \frac{G_{\alpha+1/2}}{c^2} \chi_0 \left( \frac{\xi - u_{\alpha+1/2}}{c}, \frac{\gamma - v_{\alpha+1/2}}{c} \right) \\ = \frac{G_{\alpha+1/2}}{g\pi h} \mathbf{1}_{(\xi-u_{\alpha+1/2})^2 + (\gamma-v_{\alpha+1/2})^2 \le 2gh} = \frac{G_{\alpha+1/2}}{h} M_{\alpha+1/2}, \quad (63) $$

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+where $G_{\alpha+1/2}$ is defined by (20) and $u_{\alpha+1/2}, v_{\alpha+1/2}$ are given by (21). The quantity $N_{\alpha+1/2}$ satisfies
+the following moment relations
+
+$$
+\int_{\mathbb{R}^2} \begin{pmatrix} 1 \\ \xi \\ \gamma \end{pmatrix} N_{\alpha+1/2} d\xi d\gamma = \begin{pmatrix} G_{\alpha+1/2} \\ u_{\alpha+1/2} G_{\alpha+1/2} \\ v_{\alpha+1/2} G_{\alpha+1/2} \end{pmatrix}, \quad \int_{\mathbb{R}^2} \begin{pmatrix} \xi^2 \\ \frac{\gamma}{2} \\ \frac{\gamma}{2} \end{pmatrix} N_{\alpha+1/2} d\xi d\gamma = \left( \begin{pmatrix} \frac{u_{\alpha+1/2}^2}{2} + \frac{g}{4}h \\ \frac{v_{\alpha+1/2}^2}{2} + \frac{g}{4}h \\ \frac{v_{\alpha+1/2}^2}{2} + \frac{g}{4}h \end{pmatrix} \right) G_{\alpha+1/2}. \tag{64}
+$$
+
+Notice that from (20), we can give a kinetic interpretation on the exchange terms under the form
+
+$$
+G_{\alpha+1/2} = - \sum_{j=1}^{N} \left( \sum_{p=1}^{\alpha} l_p - \mathbf{1}_{j \le \alpha} \right) \int_{\mathbb{R}^2} \begin{pmatrix} \xi \\ \gamma \end{pmatrix} \cdot \nabla_{x,y} M_j d\xi d\gamma, \quad (65)
+$$
+
+for $\alpha = 1, \dots, N$.
+
+Then we have the two following results.
+
+**Proposition 4.1** The functions **u**<sup>N</sup> defined by (17) and h are strong solutions of the system (18)-(19) if and only if the sets of equilibria {M<sub>α</sub>}<sup>N</sup><sub>α=1</sub>, {N<sub>α+1/2</sub>}<sup>N</sup><sub>α=0</sub> are solutions of the kinetic equations defined by
+
+$$
+(\mathcal{B}_\alpha) \quad \frac{\partial M_\alpha}{\partial t} + \begin{pmatrix} \xi \\ \gamma \end{pmatrix} \cdot \nabla_{x,y} M_\alpha - g \nabla_{x,y} z_b \cdot \nabla_{\xi,\gamma} M_\alpha - N_{\alpha+1/2} + N_{\alpha-1/2} = Q_\alpha, \quad (66)
+$$
+
+for $\alpha = 1, \dots, N$. The quantities $Q_\alpha = Q_\alpha(t, x, y, \xi, \gamma)$ are "collision terms" equal to zero at the macroscopic level, i.e. they satisfy a.e. for values of $(t, x, y)$
+
+$$
+\int_{\mathbb{R}^2} Q_\alpha d\xi d\gamma = \int_{\mathbb{R}^2} \xi Q_\alpha d\xi d\gamma = \int_{\mathbb{R}^2} \gamma Q_\alpha d\xi d\gamma = 0. \qquad (67)
+$$
+
+**Proposition 4.2** The solutions of (66) are entropy solutions if
+
+$$
+\frac{\partial H(M_\alpha)}{\partial t} + \begin{pmatrix} \xi \\ \gamma \end{pmatrix} \cdot \nabla_{x,y} H(M_\alpha) - g \nabla_{x,y} z_b \cdot \nabla_{\xi,\gamma} H(M_\alpha) \leq (H(N_{\alpha+1/2}) - H(N_{\alpha-1/2})), \quad (68)
+$$
+
+with the notation $H(M) = H(M, \xi, \gamma, z_b)$ and $H$ defined by (61). The integration in $\xi, \gamma$ of relation (68) gives
+
+$$
+\frac{\partial E_{\alpha}}{\partial t} + \nabla_{x,y} . u_{\alpha} (E_{\alpha} + \frac{g}{2} h_{\alpha} h) \leq l_{\alpha} \left( \frac{|u_{\alpha+1/2}|^2}{2} + gz_b \right) G_{\alpha+1/2} - l_{\alpha} \left( \frac{|u_{\alpha-1/2}|^2}{2} + gz_b \right) G_{\alpha-1/2}.
+$$
+
+**Proof of proposition 4.1** The proof relies on averages w.r.t the variables $\xi, \gamma$ of Eq. (66) against the vector $(1, \xi, \gamma)^T$. Using relations (60), (63), (64) and the properties of the collision terms (67), the quantities
+
+$$
+\int_{\mathbb{R}^2} (\mathcal{B}_\alpha) d\xi d\gamma, \quad \int_{\mathbb{R}^2} \xi (\mathcal{B}_\alpha) d\xi d\gamma, \quad \text{and} \quad \int_{\mathbb{R}^2} \gamma (\mathcal{B}_\alpha) d\xi d\gamma,
+$$
+
+respectively give Eqs. (28) and (19). The sum for $\alpha = 1$ to $N$ of Eqs. (28) with (20) gives (18) that
+completes the proof. $\blacksquare$

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+**Proof of prop. 4.2** The proof is obtained multiplying (66) by $H'_{\alpha}(\overline{M}_{\alpha}, \xi, \gamma, z_b)$. Indeed, it is easy to see that
+
+$$H'_{\alpha}(M_{\alpha}, \xi, \gamma, z_b) \frac{\partial M_{\alpha}}{\partial v} = \frac{\partial}{\partial v} H_{\alpha}(M_{\alpha}, \xi, \gamma, z_b),$$
+
+for $v = t, x, y, \xi, \gamma$. Likewise for the quantity $H'_{\alpha}(M_{\alpha}, \xi, \gamma, z_b)N_{\alpha+1/2}$, we have
+
+$$H'_{\alpha}(M_{\alpha}, \xi, \gamma, z_b) N_{\alpha+1/2} = H(N_{\alpha+1/2}, \xi, \gamma, z_b).$$
+
+So finally, Eq. (66) multiplied by $H'_{\alpha}(M_{\alpha}, \xi, \gamma, z_b)$ gives
+
+$$\frac{\partial H_{\alpha}}{\partial t} + \begin{pmatrix} \xi \\ \gamma \end{pmatrix} \cdot \nabla_{x,y} H_{\alpha} - g \nabla_{x,y} z_b \cdot \nabla_{\xi,\gamma} H_{\alpha} \leq \left( \frac{\xi^2 + \gamma^2}{2} + gz_b \right) (N_{\alpha+1/2} - N_{\alpha-1/2}).$$
+
+It remains to calculate the sum of the preceding relations from $\alpha = 1, \dots, N$ and to integrate the obtained relation in $\xi, \gamma$ over $\mathbb{R}^2$ that completes the proof. ■
+
+**Remark 4.3** If we introduce a $(2N+1) \times N$ matrix $K(\xi, \gamma)$ defined by
+
+$$K_{1,j} = 1, \quad K_{i+1,j} = \xi \delta_{i,j}, \quad K_{i+N+1,j+N} = \gamma \delta_{i,j},$$
+
+for $i, j = 1, \dots, N$ with $\delta_{i,j}$ the Kronecker symbol. Then, using Prop. 4.1, we can write
+
+$$U = \int_{\mathbb{R}^2} K(\xi, \gamma) M(\xi, \gamma) d\xi d\gamma, \quad F(U) = \int_{\mathbb{R}^2} \begin{pmatrix} \xi \\ \gamma \end{pmatrix} K(\xi, \gamma) M(\xi, \gamma) d\xi d\gamma, \qquad (69)$$
+
+$$S_e(U) = \int_{\mathbb{R}^2} K(\xi, \gamma) N(\xi, \gamma) d\xi d\gamma, \qquad (70)$$
+
+with $M(\xi, \gamma) = (M(U_1, \xi, \gamma), \dots, M(U_N, \xi, \gamma))^T$ and
+
+$$N(\xi, \gamma) = \begin{pmatrix} N_{3/2}(\xi, \gamma) - N_{1/2}(\xi, \gamma) \\ \vdots \\ N_{N+1/2}(\xi, \gamma) - N_{N-1/2}(\xi, \gamma) \end{pmatrix}.$$
+
+Hence, using the above notations, the layer-averaged Euler system (18)-(19) can be written under the form
+
+$$\int_{\mathbb{R}^2} K(\xi, \gamma) \left( \frac{\partial M(\xi, \gamma)}{\partial t} + \begin{pmatrix} \xi \\ \gamma \end{pmatrix} \cdot \nabla_{x,y} M(\xi, \gamma) - g \nabla_{x,y} z_b \cdot \nabla_{\xi,\gamma} M - N(\xi, \gamma) \right) d\xi d\gamma = 0.$$
+
+# 5 Numerical scheme
+
+The numerical scheme for the model (49) proposed in this section extends the results presented by some of the authors in [5, 9, 8, 4]. Compared to these previous results, it has the following advantages
+
+* it gives a 3d approximation of the Navier-Stokes system whereas 2d situations $(x, y)$ and $(x, z)$ where considered in [5, 9, 8],
+
+* the implicit treatment of the vertical exchanges terms gives a bounded CFL condition even when the water depth vanishes,

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+* the kinetic interpretation, on which is based the numerical scheme, is also valid for the vertical exchange terms – that was not the case in [9, 8] – and allows to derive a robust and accurate numerical scheme,
+
+* the numerical approximation of the system given in (49) is endowed with strong stability properties (well-balancing, positivity of the water depth, ...),
+
+* convergence curves towards a 3d non-stationary analytical solution with wet-dry interfaces have been obtained (see paragraph 6.2.1).
+
+First, we focus on the Euler part of the system (49) then in paragraph 5.6, a numerical scheme for the viscous terms is proposed.
+
+Notice that, as a consequence of the layer-averaged discretization, the system (49) and the Boltzmann type equation (66) are only 2d $(x, y)$ partial differential equations with source terms. Hence, the spacial approximation of the considered PDEs is performed on a 2d planar mesh.
+
+## 5.1 Semi-discrete (in time) scheme
+
+We consider discrete times $t^n$ with $t^{n+1} = t^n + \Delta t^n$. For the time discretisation of the layer-averaged Navier-Stokes system (49) we adopt the following scheme
+
+$$U^{n+1} = U - \Delta t^n (\nabla_{x,y} F(U) - S_b(U)) + \Delta t^n S_e^{n+1} + \Delta t^n S_{v,f}^{n+p}, \quad (71)$$
+
+where the superscript $n$ has been omitted and the integer $p = 0, 1/2, 1$ will be precised below.
+
+Using the expressions (38)-(39) for the layer averaged model, the semi-discrete in time scheme (71) writes
+
+$$h^{n+1} = h^{n+1/2} = h - \Delta t^n \sum_{\alpha=1}^{N} \nabla_{x,y} (h_\alpha \mathbf{u}_\alpha), \quad (72)$$
+
+$$(h_\alpha \mathbf{u}_\alpha)^{n+1/2} = h_\alpha \mathbf{u}_\alpha - \Delta t^n \left( \nabla_{x,y} (h_\alpha \mathbf{u}_\alpha \otimes \mathbf{u}_\alpha) + \nabla_{x,y} \left(\frac{g}{2} h h_\alpha\right) + g h_\alpha \nabla_{x,y} z_b \right), \quad (73)$$
+
+$$(h_\alpha \mathbf{u}_\alpha)^{n+1} = (h_\alpha \mathbf{u}_\alpha)^{n+1/2} - \Delta t^n \left( \mathbf{u}_{\alpha+1/2}^{n+1} G_{\alpha+1/2} - \mathbf{u}_{\alpha-1/2}^{n+1} G_{\alpha-1/2} + \nabla_{x,y} (h_\alpha^{n+p} \boldsymbol{\Sigma}_\alpha^{0,n+p}) \right. \\ \left. - \mathbf{T}_\alpha^{n+p} + \Lambda_{\alpha+1/2} \frac{\mathbf{u}_{\alpha+1}^{n+p} - \mathbf{u}_\alpha^{n+p}}{h_{\alpha+1}^{n+p} + h_\alpha^{n+p}} - \Lambda_{\alpha-1/2} \frac{\mathbf{u}_\alpha^{n+p} - \mathbf{u}_{\alpha-1}^{n+p}}{h_\alpha^{n+p} + h_{\alpha-1}^{n+p}} - \kappa_\alpha \mathbf{u}_\alpha^{n+p} + W_\alpha^{n+p} \mathbf{t}_s \right), (74)$$
+
+$$G_{\alpha+1/2} = - \sum_{j=1}^{N} \left( \sum_{p=1}^{\alpha} l_p - \mathbf{1}_{j \le \alpha} \right) \nabla_{x,y} (h_j \mathbf{u}_j), \quad (75)$$
+
+for $\alpha = 1, \dots, N$. The vertical velocities $\{w_\alpha\}_{\alpha=1}^N$ are defined by (27). The first two equations (72)-(73) consist in an explicit time scheme where the horizontal fluxes and the topography source term are taken into account whereas in Eq. (74) an implicit treatment of the exchange terms between layers is proposed. The implicit part of the scheme requires to solve a linear problem (see lemma. 5.1) but, on the contrary of previous work of some of the authors [9], it implies that the CFL condition (101) no more depends on the exchange terms.
+
+When $\nu = \kappa = 0$ in Eq. (74), Eqs. (72)-(74) correspond to the layer-averaged of the Euler system. The choice $p=1$ (resp. $p=1/2$) in Eq. (74) corresponds to an implicit (resp. semi-implicit) treatment of the viscous and friction terms whereas the choice $p=0$ implies an explicit treatment and requires a CFL condition. Notice that
+
+* the implicit or semi-implicit treatment of the viscous terms requires to solve a system that is linear since $h^{n+1}$ is known from the mass conservation equation,

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+* the explicit discretisation is simpler to implement but implies a restrictive CFL when the spatial discretisation (vertical and horizontal) becomes fine. In the context of geophysical flows, this limitation is rarely severe. In particular, the CFL condition induced by the explicit treatment of the viscous terms is less restrictive than the one coming from the advection part (see Eq. (101)) in all the numerical tests of Section 6.
+
+## 5.2 Space discretization
+
+Let $\Omega$ denote the computational domain with boundary $\Gamma$, which we assume is polygonal. Let $T_h$ be a triangulation of $\Omega$ for which the vertices are denoted by $P_i$ with $S_i$ the set of interior nodes and $G_i$ the set of boundary nodes.
+
+For the space discretization of the system(72)-(74), we use a finite volume technique for the Euler part – that is described below – and a finite element approach –$\mathbb{P}_1$ on $T_h$ – for the viscous part that is described in paragraph 5.6.
+
+## 5.3 Finite volume formalism for the Euler part
+
+In this paragraph and in paragraph (5.4), we propose a space discretization for the model (72)-(74) without the viscous and friction terms i.e. the system
+
+$$h^{n+1} = h^{n+1/2} = h - \Delta t^n \sum_{\alpha=1}^{N} \nabla_{x,y} (h_\alpha \mathbf{u}_\alpha), \quad (76)$$
+
+$$(h_\alpha \mathbf{u}_\alpha)^{n+1/2} = h_\alpha \mathbf{u}_\alpha - \Delta t^n \left( \nabla_{x,y} (h_\alpha \mathbf{u}_\alpha \otimes \mathbf{u}_\alpha) + \nabla_{x,y} \left(\frac{g}{2} h h_\alpha\right) + g h_\alpha \nabla_{x,y} z_b \right), \quad (77)$$
+
+$$(h_\alpha \mathbf{u}_\alpha)^{n+1} = (h_\alpha \mathbf{u}_\alpha)^{n+1/2} - \Delta t^n \left( \mathbf{u}_{\alpha+1/2}^{n+1} G_{\alpha+1/2} - \mathbf{u}_{\alpha-1/2}^{n+1} G_{\alpha-1/2} \right), \quad (78)$$
+
+completed with (75).
+
+We recall now the general formalism of finite volumes on unstructured meshes.
+
+The dual cells $C_i$ are obtained by joining the centers of mass of the triangles surrounding each vertex $P_i$. We use the following notations (see Fig. 2):
+
+• $K_i$, set of subscripts of nodes $P_j$ surrounding $P_i$,
+
+• $|C_i|$, area of $C_i$,
+
+• $\Gamma_{ij}$, boundary edge between the cells $C_i$ and $C_j$,
+
+• $L_{ij}$, length of $\Gamma_{ij}$,
+
+• $\mathbf{n}_{ij}$, unit normal to $\Gamma_{ij}$, outward to $C_i$ ($\mathbf{n}_{ji} = -\mathbf{n}_{ij}$).
+
+If $P_i$ is a node belonging to the boundary $\Gamma$, we join the centers of mass of the triangles adjacent to the boundary to the middle of the edge belonging to $\Gamma$ (see Fig. 2) and we denote
+
+• $\Gamma_i$, the two edges of $C_i$ belonging to $\Gamma$,
+
+• $L_i$, length of $\Gamma_i$ (for sake of simplicity we assume in the following that $L_i = 0$ if $P_i$ does not belong to $\Gamma$),
+
+• $\mathbf{n}_i$, the unit outward normal defined by averaging the two adjacent normals.

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+Numerical approximation of the 3d hydrostatic
+Navier-Stokes system with free surface.
+
+S. Allgeyer*, M.-O. Bristeau†,‡, D. Froger†,‡, R. Hamouda†,‡, V. Jauzein§ A. Mangeney¶†,‡,
+J. Sainte-Marie†,‡,∥ F. Souillé†,‡ and M. Vallée¶
+
+July 19, 2019
+
+# Contents
+
+<table><tr><td>1</td><td>Introduction</td><td>3</td></tr><tr><td>2</td><td>The hydrostatic Navier-Stokes system</td><td>4</td></tr><tr><td>2.1</td><td>Boundary conditions</td><td>5</td></tr><tr><td>2.1.1</td><td>Bottom and free surface</td><td>5</td></tr><tr><td>2.1.2</td><td>Fluid boundaries and solid walls.</td><td>5</td></tr><tr><td>2.2</td><td>Energy balance</td><td>6</td></tr><tr><td>2.3</td><td>The hydrostatic Euler system</td><td>6</td></tr><tr><td>3</td><td>The layer-averaged model</td><td>6</td></tr><tr><td>3.1</td><td>The layer-averaged Euler system.</td><td>7</td></tr><tr><td>3.2</td><td>The layer-averaged Navier-Stokes system</td><td>10</td></tr><tr><td>3.2.1</td><td>Complete model.</td><td>10</td></tr><tr><td>3.2.2</td><td>Simplified rheology</td><td>11</td></tr><tr><td>4</td><td>Kinetic description for the Euler system</td><td>13</td></tr><tr><td>4.1</td><td>Preliminaries</td><td>13</td></tr><tr><td>4.2</td><td>Kinetic interpretation.</td><td>14</td></tr><tr><td>5</td><td>Numerical scheme</td><td>16</td></tr><tr><td>5.1</td><td>Semi-discrete (in time) scheme.</td><td>17</td></tr><tr><td>5.2</td><td>Space discretization.</td><td>18</td></tr><tr><td>5.3</td><td>Finite volume formalism for the Euler part.</td><td>18</td></tr><tr><td>5.4</td><td>Discrete kinetic equation.</td><td>20</td></tr><tr><td>5.4.1</td><td>Without topography.</td><td>20</td></tr><tr><td>5.4.2</td><td>With topography.</td><td>22</td></tr></table>
+
+*Research School of Earth Sciences, Australian National University, Canberra, ACT, Australia
+
+†Inria Paris, 2 rue Simone Iff, CS 42112, 75589 Paris Cedex 12, France
+
+‡Sorbonne Université, Lab. Jacques-Louis Lions, 4 Place Jussieu, F-75252 Paris cedex 05
+
+§SAUR, Research & development department, Pôle technologique, 2 rue de la Bresle, 78310 Maurepas, France
+
+¶Univ. Paris Diderot, Sorbonne Paris Cité, Institut de Physique du Globe de Paris, Seismology Group, 1 rue Jussieu, Paris F-75005, France
+
+∥Corresponding author: Jacques.Sainte-Marie@inria.fr

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+Figure 2: (a) Dual cell $C_i$ and (b) Boundary cell $C_i$.
+
+We define the piecewise constant functions $U^n(x, y)$ on cells $C_i$ corresponding to time $t^n$ and
+$z_b(x, y)$ as
+
+$$
+U^n(x, y) = U_i^n, \quad z_b(x, y) = z_{b,i}, \quad \text{for } (x, y) \in C_i, \tag{79}
+$$
+
+with $U_i^n = (h_i^n, q_{x,1,i}^n, \dots, q_{x,N,i}^n, q_{y,1,i}^n, \dots, q_{y,N,i}^n)^T$ i.e.
+
+$$
+U_i^n \approx \frac{1}{|C_i|} \int_{C_i} U(t^n, x, y) dxdy, \quad z_{b,i} \approx \frac{1}{|C_i|} \int_{C_i} z_b(x, y) dxdy.
+$$
+
+We will also use the notation
+
+$$
+U_{\alpha,i}^{n} \approx \frac{1}{|C_i|} \int_{C_i} U_{\alpha}(t^n, x, y) dx dy,
+$$
+
+with $U_\alpha$ defined by (58). A finite volume scheme for solving the system (76)-(77) is a formula of the form
+
+$$
+U_i^{n+1/2} = U_i - \sum_{j \in K_i} \sigma_{i,j} \mathcal{F}_{i,j} - \sigma_i \mathcal{F}_{e,i}, \quad (80)
+$$
+
+where using the notations of (71)
+
+$$
+\sum_{j \in K_i} L_{i,j} \mathcal{F}_{i,j} \approx \int_{C_i} \nabla_{x,y} F(U) dxdy, \qquad (81)
+$$
+
+with
+
+$$
+\sigma_{i,j} = \frac{\Delta t^n L_{i,j}}{|C_i|}, \quad \sigma_i = \frac{\Delta t^n L_i}{|C_i|}.
+$$
+
+Here we consider first-order explicit schemes where
+
+$$
+\mathcal{F}_{i,j} = F(U_i, U_j, z_{b,i} - z_{b,j}, \mathbf{n}_{i,j}). \tag{82}
+$$
+
+and
+
+$$
+\mathcal{F}_{i,j} = F(U_i, U_j, z_{b,i} - z_{b,j}, \mathbf{n}_{i,j}) = \begin{pmatrix} F(U_{1,i}, U_{1,j}, z_{b,i} - z_{b,j}, \mathbf{n}_{i,j}) \\ \vdots \\ F(U_{N,i}, U_{N,j}, z_{b,i} - z_{b,j}, \mathbf{n}_{i,j}) \end{pmatrix} \tag{83}
+$$
+
+and for the boundary nodes
+
+$$
+\mathcal{F}_{e,i} = F(U_i, U_{e,i}, \mathbf{n}_i) = \begin{pmatrix} F(U_{1,i}, U_{1,e,i}, \mathbf{n}_i) \\ \vdots \\ F(U_{N,i}, U_{N,e,i}, \mathbf{n}_i) \end{pmatrix}. \quad (84)
+$$

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+Relation (80) tells how to compute the values $U_i^{n+1/2}$ knowing $U_i$ and discretized values $z_{b,i}$ of the topography. Following (81), the term $\mathcal{F}_{i,j}$ in (80) denotes an interpolation of the normal component of the flux $F(U).\mathbf{n}_{i,j}$ along the edge $C_{i,j}$. The functions $F(U_i, U_j, z_{b,i} - z_{b,j}, \mathbf{n}_{i,j}) \in \mathbb{R}^{2N+1}$ are the numerical fluxes, see [17].
+
+In the next paragraph we define $\mathcal{F}(U_i, U_j, z_{b,i} - z_{b,j}, \mathbf{n}_{i,j})$ using the kinetic interpretation of the system. The computation of the value $U_{i,e}$, which denotes a value outside $C_i$ (see Fig. 2-(b)), defined such that the boundary conditions are satisfied, and the definition of the boundary flux $F(U_i, U_{e,i}, \mathbf{n}_i)$ are described paragraph 5.7. Notice that we assume a flat topography on the boundaries i.e. $z_{b,i} = z_{b,i,e}$.
+
+## 5.4 Discrete kinetic equation
+
+The choice of a kinetic scheme is motivated by several arguments. First, the kinetic interpretation is a suitable starting point for building a stable numerical scheme. We will prove in paragraph 5.4 that the proposed kinetic scheme preserves positivity of the water depth and ensures a discrete local maximum principle for a tracer concentration (temperature, salinity...). Second, the construction of the kinetic scheme does not need the computation of the system eigenvalues. This point is very important here since these eigenvalues are not available in explicit analytical form, and they are hardly accessible even numerically. Furthermore, as previously mentioned, hyperbolicity of the multilayer model may not hold, and the kinetic scheme allows overcoming this difficulty.
+
+### 5.4.1 Without topography
+
+In a first step we consider a situation with flat bottom. Following prop. 4.1, the model (18)-(19) reduces, for each layer, to a classical Saint-Venant system with exchange terms and its kinetic interpretation (see Eq. (66)) is given by
+
+$$ \frac{\partial M_\alpha}{\partial t} + \begin{pmatrix} \xi \\ \gamma \end{pmatrix} \cdot \nabla_{x,y} M_\alpha - N_{\alpha+1/2} + N_{\alpha-1/2} = Q_\alpha, \quad \alpha \in \{1, \dots, N\}, \qquad (85) $$
+
+with the notations defined in paragraph 4.2.
+
+Let $C_i$ be a cell, see Fig. 2. The integral over $C_i$ of the convective part of the kinetic equation (85) gives
+
+$$ \int_{C_i} \left( \frac{\partial M_\alpha}{\partial t} + \begin{pmatrix} \xi \\ \gamma \end{pmatrix} \cdot \nabla_{x,y} M_\alpha \right) dxdy \approx |C_i| \frac{\partial M_{\alpha,i}}{\partial t} + \sum_{j \in K_i} \int_{\Gamma_{i,j}} M_{\alpha,i,j} dl, \qquad (86) $$
+
+with $M_{\alpha,i} = M(U_{\alpha,i}, \xi, \gamma)$, $\mathbf{n}_{i,j}$ being the outward normal to the cell $C_i$. The quantity $M_{\alpha,i,j}$ is defined by the classical kinetic upwinding
+
+$$ M_{\alpha,i,j} = M_{\alpha,i}\zeta_{i,j}\mathbf{1}_{\zeta_{i,j} \ge 0} + M_{\alpha,j}\zeta_{i,j}\mathbf{1}_{\zeta_{i,j} \le 0}, $$
+
+with $\zeta_{i,j} = (\xi \ \gamma)^T \cdot \mathbf{n}_{i,j}$.
+
+Therefore, the kinetic scheme applied for Eq. (85) is given by
+
+$$ f_{\alpha,i}^{n+1/2^{-}} = \left(1 - \frac{\Delta t^n}{|C_i|} \sum_{j \in K_i} L_{i,j} \zeta_{i,j} \mathbf{1}_{\zeta_{i,j} \ge 0}\right) M_{\alpha,i} - \frac{\Delta t^n}{|C_i|} \sum_{j \in K_i} L_{i,j} M_{\alpha,j} \zeta_{i,j} \mathbf{1}_{\zeta_{i,j} \le 0}, \qquad (87) $$
+
+$$ f_{\alpha,i}^{n+1^{-}} = f_{\alpha,i}^{n+1/2^{-}} + \Delta t^n (N_{\alpha+1/2,i}^{n+1^{-}} - N_{\alpha-1/2,i}^{n+1^{-}}), \qquad (88) $$
+
+with the exchange terms $\{N_{\alpha+1/2,i}^{n+1^{-}}\}_{\alpha=0}^N$ defined by
+
+$$ N_{\alpha+1/2,i}^{n+1^{-}}(\xi, \gamma) = \frac{G_{\alpha+1/2,i}}{h_i} f_{\alpha+1/2,i}^{n+1^{-}}. \qquad (89) $$

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+Following (21) we can write
+
+$$f_{\alpha+1/2,i}^{n+1-} = \begin{cases} f_{\alpha,i}^{n+1-} & \text{if } G_{\alpha+1/2} \le 0 \\ f_{\alpha+1,i}^{n+1-} & \text{if } G_{\alpha+1/2} > 0 \end{cases}$$
+
+leading to
+
+$$N_{\alpha+1/2,i}^{n+1-}(\xi, \gamma) = \frac{|G_{\alpha+1/2,i}|_+}{h_i} f_{\alpha+1,i}^{n+1-} - \frac{|G_{\alpha+1/2,i}|_-}{h_i} f_{\alpha,i}^{n+1-}.$$
+
+Notice that the previous definition is consistent with (63). From (65), we get
+
+$$G_{\alpha+1/2,i} = -\frac{1}{|C_i|} \sum_{k=1}^{N} \left( \sum_{p=1}^{\alpha} l_p - \mathbf{1}_{k \le \alpha} \right) \sum_{j \in K_i} L_{i,j} \int_{\mathbb{R}^2} \left( M_{k,i} \zeta_{i,j} \mathbf{1}_{\zeta_{i,j} \ge 0} + M_{k,j} \zeta_{i,j} \mathbf{1}_{\zeta_{i,j} \le 0} \right) d\xi d\gamma.$$
+
+By analogy with the computations in (60), we can recover the macroscopic quantities $U_{\alpha,i}^{n+1}$ at time $t^{n+1}$ by integration of the relation (88)
+
+$$U_{\alpha,i}^{n+1} = \int_{\mathbb{R}^2} \begin{pmatrix} 1 \\ \xi \\ \gamma \end{pmatrix} f_{\alpha,i}^{n+1-} d\xi d\gamma. \quad (90)$$
+
+The scheme (87) and the definition (90) allow to complete the definition of the macroscopic scheme (80),(83), (84) with the numerical flux given by the flux vector splitting formula [15]
+
+$$ 
+\begin{aligned}
+\mathcal{F}_{i,j} &= F^+(U_i^n, \mathbf{n}_{i,j}) + F^-(U_j^n, \mathbf{n}_{i,j}) \\
+&= \int_{\mathbb{R}^2} \mathcal{K}(\xi, \gamma) M_i \zeta_{i,j} \mathbf{1}_{\zeta_{i,j} \ge 0} d\xi d\gamma + \int_{\mathbb{R}^2} \mathcal{K}(\xi, \gamma) M_j \zeta_{i,j} \mathbf{1}_{\zeta_{i,j} \le 0} d\xi d\gamma,
+\end{aligned}
+\quad (91) $$
+
+where $\mathcal{K}(\xi, \gamma)$ is defined in Remark 4.3 and $M_i = (M_{1,i}, \dots, M_{N,i})^T$.
+
+Using (89), we rewrite the step (88) under the form
+
+$$ (\mathbf{I}_N + \Delta t \mathbf{G}_{N,i}) f^{n+1-} = f^{n+1/2-}, $$
+
+where $I_N$ is the identity matrix of size $N$ and $\mathbf{G}_{N,i}$ is defined by
+
+$$ G_{N,i} = \begin{pmatrix}
+-\frac{|G_{3/2,i}|_-}{h_{1,i}^{n+1}} & -\frac{|G_{3/2,i}|_+}{h_{1,i}^{n+1}} & 0 & 0 & \cdots & 0 \\
+\frac{|G_{3/2,i}|_-}{h_{2,i}^{n+1}} & \ddots & \ddots & 0 & \cdots & 0 \\
+0 & \ddots & \ddots & \ddots & 0 & 0 \\
+\vdots & 0 & \frac{|G_{\alpha-1/2,i}|_-}{h_{\alpha,i}^{n+1}} & -\frac{|G_{\alpha+1/2,i}|_- - |G_{\alpha-1/2,i}|_+}{h_{\alpha,i}^{n+1}} & -\frac{|G_{\alpha+1/2,i}|_+}{h_{\alpha,i}^{n+1}} & 0 \\
+\vdots & \ddots & 0 & \ddots & \ddots & -\frac{|G_{N-1/2,i}|_+}{h_{N-1,i}^{n+1}} \\
+0 & \cdots & 0 & 0 & \frac{|G_{N-1/2,i}|_-}{h_{N,i}^{n+1}} & \frac{|G_{N-1/2,i}|_+}{h_{N,i}^{n+1}}
+\end{pmatrix}. $$
+
+Hence, the resolution of the discrete kinetic equation (88) requires to inverse the matrix
+
+$$ \begin{pmatrix} I_N + \Delta t G_{N,i} & 0 \\ 0 & I_N + \Delta t G_{N,i} \end{pmatrix} $$
+
+and we have the following lemma.
+
+**Lemma 5.1** The matrix $I_N + \Delta t G_{N,i}$

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+(i) is invertible for any $h_i^{n+1} > 0$,
+
+(ii) $(\mathbf{I}_N + \Delta t \mathbf{G}_{N,i})^{-1}$ has only positive coefficients,
+
+(iii) for any vector $T$ with non negative entries i.e. $T_\alpha \ge 0$, for $1 \le \alpha \le N$, one has
+
+$$
+\|(I_N + \Delta t G_{N,i})^{-t} T\|_{\infty} \le \|T\|_{\infty}.
+$$
+
+**Remark 5.2** Compared to an explicit treatment of the vertical exchanges terms as presented in [8, 9], the implicit scheme (88) requires to invert for each cell a small matrix whose size corresponds to the number of layers. Depending on the type of simulation carried out, it increases the computational costs e.g. for the tsunami simulation the difference is around 20%. But it is worth noticing that the explicit treatment of the vertical exchanges terms can lead to severe constraints on the CFL condition since the quantity
+
+$$
+\frac{|G_{\alpha+1/2,i}|}{h_i},
+$$
+
+is not bounded, see [9, Prop. 5.2].
+
+**Proof of lemma 5.1** (i) For any $h_i^{n+1} > 0$, the matrix $\mathbf{I}_N + \Delta t \mathbf{G}_{N,i}$ is a strictly dominant diagonal matrix and hence it is invertible.
+
+(ii) Denoting $\mathbf{G}_{N,i}^d$ (resp. $\mathbf{G}_{N,i}^{nd}$) the diagonal (resp. non diagonal) part of $\mathbf{G}_{N,i}$ we can write
+
+$$
+\mathbf{I}_N + \Delta t \mathbf{G}_{N,i} = (\mathbf{I}_N + \Delta t \mathbf{G}_{N,i}^d) (\mathbf{I}_N - (\mathbf{I}_N + \Delta t \mathbf{G}_{N,i}^d)^{-1} (-\Delta t \mathbf{G}_{N,i}^{nd})),
+$$
+
+where all the entries of the matrix $\mathbf{J}_{N,i} = (\mathbf{I}_N + \Delta t \mathbf{G}_{N,i}^d)^{-1}(-\Delta t \mathbf{G}_{N,i}^{nd})$, are non negative and less than 1. And hence, we can write
+
+$$
+(\mathbf{I}_N + \Delta t \mathbf{G}_{N,i})^{-1} = \sum_{k=0}^{\infty} J_{N,i}^k,
+$$
+
+proving all the entries of $(\mathbf{I}_N + \Delta t \mathbf{G}_{N,i})^{-1}$ are non negative.
+
+(ii) Let us consider the vector **1** whose entries are all equal to 1. Since we have
+
+$$
+(\mathbf{I}_N + \Delta t \mathbf{G}_{N,i})^t \mathbf{1} = \mathbf{1},
+$$
+
+we also have $\mathbf{1} = (\mathbf{I}_N + \Delta t \mathbf{G}_{N,i})^{-t} \mathbf{1}$. Now let $T$ be a vector whose entries $\{T_\alpha\}_{1\le\alpha\le N}$ are non-negative, then
+
+$$
+(\mathbf{I}_N + \Delta t \mathbf{G}_{N,i})^{-t} \mathbf{T} \le (\mathbf{I}_N + \Delta t \mathbf{G}_{N,i})^{-t} \mathbf{1} \| \mathbf{T} \|_{\infty} = \mathbf{1} \| \mathbf{T} \|_{\infty},
+$$
+
+that completes the proof. ■
+
+5.4.2 With topography
+
+The hydrostatic reconstruction scheme (HR scheme for short) for the Saint-Venant system has been introduced in [3] in the 1d case and described in 2d for unstructured meshes in [5]. The HR in the context of the kinetic description for the Saint-Venant system has been studied in [4].
+
+In order to take into account the topography source and to preserve relevant equilibria, the HR
+leads to a modified version of (80) under the form
+
+$$
+U_i^{n+1/2} = U_i^n - \sum_{j \in K_i} \sigma_{i,j} \mathcal{F}_{i,j}^* - \sigma_i \mathcal{F}_{i,e} + \sum_{j \in K_i} \sigma_{i,j} S_{i,j}^*, \quad (92)
+$$

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+where
+
+$$
+\mathcal{F}_{i,j}^* = F(U_{i,j}^*, U_{j,i}^*, \mathbf{n}_{i,j}), \quad \mathcal{S}_{i,j}^* = S(U_i, U_{i,j}^*, \mathbf{n}_{i,j}) =
+\begin{pmatrix}
+0 \\
+\frac{g}{2} l_1 (h_{i,j}^{*2} - h_i^2) \mathbf{n}_{i,j} \\
+: \\
+\frac{g}{2} l_N (h_{i,j}^{*2} - h_i^2) \mathbf{n}_{i,j}
+\end{pmatrix},
+\quad (93)
+$$
+
+with
+
+$$
+\begin{align}
+z_{b,i,j}^* &= \max(z_{b,i}, z_{b,j}), & h_{i,j}^* &= \max(h_i + z_{b,i} - z_{b,i,j}^*, 0), \nonumber \\
+U_{i,j}^* &= (h_{i,j}^*, l_1 h_{i,j}^* u_{1,i}, \dots, l_N h_{i,j}^* u_{N,i}), & l_1 h_{i,j}^* v_{1,i}, \dots, l_N h_{i,j}^* v_{N,i})^T. \tag{94}
+\end{align}
+$$
+
+We would like here to propose a kinetic interpretation of the HR scheme, which means to interpret the above numerical fluxes as averages with respect to the kinetic variables of a scheme written on a kinetic function $f$. More precisely, we would like to approximate the solution to (66) by a kinetic scheme such that the associated macroscopic scheme is exactly (92)-(93) with homogeneous numerical flux $\mathcal{F}$ given by (91). We denote $M_{\alpha,i,j}^* = M(U_{\alpha,i,j}^*, \xi, \gamma)$ for any $\alpha = 1, \dots, N$ and we consider the scheme
+
+$$
+\begin{equation}
+\begin{split}
+f_{\alpha,i}^{n+1/2-} &= M_{\alpha,i} - \frac{\Delta t^n}{|C_i|} \sum_{j \in K_i} L_{i,j} \zeta_{i,j} \mathbf{1}_{\zeta_{i,j} \ge 0} M_{\alpha,i,j}^* - \frac{\Delta t^n}{|C_i|} \sum_{j \in K_i} L_{i,j} M_{\alpha,j,i}^* \zeta_{i,j} \mathbf{1}_{\zeta_{i,j} \le 0}, \\
+&\quad - \frac{\Delta t^n}{|C_i|} \sum_{j \in K_i} L_{i,j} (M_{\alpha,i} - M_{\alpha,i,j}^*) \theta_{\alpha,i,j},
+\end{split}
+\tag{95}
+\end{equation}
+$$
+
+$$
+f_{\alpha,i}^{n+1-} = f_{\alpha,i}^{n+1/2-} + \Delta t^n (N_{\alpha+1/2,i}^{*,n+1-} - N_{\alpha-1/2,i}^{*,n+1-}), \quad (96)
+$$
+
+where
+
+$$
+\theta_{\alpha,i,j} = \begin{pmatrix} \xi - u_{\alpha,i} \\ \gamma - v_{\alpha,i} \end{pmatrix} . n_{i,j}.
+$$
+
+For the exchange terms, by analogy with (89) we define
+
+$$
+N_{\alpha+1/2, i}^{*,n+1-}(\xi, \gamma) = \frac{G_{\alpha+1/2, i}^{*}}{h_i} f_{\alpha+1/2, i}^{n+1-}, \quad (97)
+$$
+
+and using (65) we get
+
+$$
+G_{\alpha+1/2, i}^* = - \frac{1}{|C_i|} \sum_{k=1}^{N} \left( \sum_{p=1}^{\alpha} l_p - 1_{k \le \alpha} \right) \sum_{j \in K_i} L_{i,j} \int_{\mathbb{R}^2} \left( M_{k,i,j}^* \zeta_{i,j} 1_{\zeta_{i,j} \ge 0} + M_{k,i,j}^* \zeta_{i,j} 1_{\zeta_{i,j} \le 0} \right) d\xi d\gamma.
+$$
+
+It is easy to see that in the previous formula, we have the moment relations
+
+$$
+\int_{\mathbb{R}^2} (M_{\alpha,i} - M_{\alpha,i,j}^*) \theta_{\alpha,i,j} d\xi d\gamma = 0, \quad (98)
+$$
+
+$$
+\int_{\mathbb{R}^2} \binom{\xi}{\gamma} (M_{\alpha,i}-M_{\alpha,i,j}^*) \theta_{\alpha,i,j} d\xi d\gamma = \frac{g}{2} l_\alpha (h_{i,j}^{*2} - h_i^2) n_{i,j}, \quad (99)
+$$
+
+Using again (90), the integration of the set of equations (95)-(96), for $\alpha = 1, \dots, N$, multiplied by $K(\xi, \gamma)$ with respect to $\xi, \gamma$ then gives the HR scheme (92)-(93) with (91),(94). Thus as announced, (95)-(96) is a kinetic interpretation of the HR scheme in 3d for an unstructured mesh.
+
+There exists a velocity $v_m \ge 0$ such that for all $\alpha, i$,
+
+$$
+|\xi| \geq v_m \text{ or } |\gamma| \geq v_m \Rightarrow M(U_{\alpha,i}, \xi, \gamma) = 0. \quad (100)
+$$

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+This means equivalently that $|u_{\alpha,i}| + |v_{\alpha,i}| + \sqrt{2gh_i} \le v_m$. We consider a CFL condition strictly less than one,
+
+$$ \sigma_i v_m \le \beta < \frac{1}{2} \quad \text{for all } i, \qquad (101) $$
+
+where $\sigma_i = \Delta t^n \sum_{j \in K_i} L_{i,j} / |C_i|$, and $\beta$ is a given constant.
+Then the following proposition holds.
+
+**Proposition 5.1** *Under the CFL condition (101), the scheme (95)-(96) verifies the following properties.*
+
+(i) The macroscopic scheme derived from (95)-(96) using (90) is a consistent discretization of the layer-averaged Euler system (18)-(19).
+
+(ii) The kinetic function remains nonnegative i.e.
+
+$$ f_{\alpha,i}^{n+1-} \ge 0, \quad \forall (\xi, \gamma) \in \mathbb{R}^2, \forall i, \forall \alpha. $$
+
+(iii) The scheme (95)-(96) is kinetic well balanced i.e. at rest
+
+$$ f_{\alpha,i}^{n+1-} = M_{\alpha,i}, \quad \forall (\xi, \gamma) \in \mathbb{R}^2, \forall i, \forall \alpha = 1, \dots, N. \qquad (102) $$
+
+**Proof of prop. 5.1** (i) Since the Boltzmann type equations (66) are almost linear transport equations with source terms, the discrete kinetic scheme (95)-(96) is clearly a consistent discretization of (66). And therefore using the kinetic interpretation given in prop. 4.1, the macroscopic scheme obtained from (95)-(96) using (90) is a consistent discretization of the layer-averaged Euler system (18)-(19).
+
+(ii) In (95)-(96) we have
+
+$$ \frac{\Delta t^n}{|C_i|} \sum_{j \in K_i} L_{i,j} M_{\alpha,j,i}^* \zeta_{i,j} 1_{\zeta_{i,j} \le 0} \le 0, $$
+
+and the HR (94) ensures $M_{\alpha,i}^* \le M_{\alpha,i}$, $\forall (\xi, \gamma) \in \mathbb{R}^2$, $\forall \alpha$ leading to
+
+$$ f_{\alpha,i}^{n+1/2-} \ge \left( 1 - \frac{\Delta t^n}{|C_i|} \sum_{j \in K_i} L_{i,j} (\zeta_{i,j} 1_{\zeta_{i,j} \ge 0} + \theta_{\alpha,i,j} 1_{\theta_{\alpha,i,j} \ge 0}) \right) M_{\alpha,i}, $$
+
+But $\zeta_{i,j} 1_{\zeta_{i,j} \ge 0} \le \max\{|\xi|, |\gamma|\}$, $\theta_{\alpha,i,j} 1_{\theta_{\alpha,i,j} \ge 0} \le \max\{|\xi - u_{\alpha,i}|, |\gamma - v_{\alpha,i}|\}$ and therefore
+
+$$ \frac{\Delta t^n}{|C_i|} \sum_{j \in K_i} L_{i,j} (\zeta_{i,j} 1_{\zeta_{i,j} \ge 0} + \theta_{\alpha,i,j} 1_{\theta_{\alpha,i,j} \ge 0}) \le \sigma_i (\max\{|\xi|, |\gamma|\} + \max\{|\xi - u_{\alpha,i}|, |\gamma - v_{\alpha,i}|\}) \le 1, $$
+
+where (100),(101) have been used, proving $f_{\alpha,i}^{n+1/2-} \ge 0$ for any $\xi \in \mathbb{R}$ and any $\alpha \in \{1, \dots, N\}$. Now using the results of lemma 5.1, it ensures that $f_{\alpha,i}^{n+1-}$ defined by (96) satisfies $f_{\alpha,i}^{n+1-} \ge 0$ for any $(\xi, \gamma) \in \mathbb{R}^2$ and any $\alpha \in \{1, \dots, N\}$, proving (ii).
+
+(iii) Considering the situation at rest i.e. $u_{\alpha,i} = v_{\alpha,i} = 0$, $\forall \alpha, i$ and $h_i + z_{b,i} = h_j + z_{b,j}$, $\forall i, j$ we have
+
+$$ M_{\alpha,i} = M_{\alpha,i,j}^*, \quad \forall \alpha, i, j. $$
+
+From (95)-(96), this gives (102). ■

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+## 5.5 Macroscopic scheme
+
+The numerical scheme for the system (76)-(78) is given by (90),(95), (96) and requires to calculate fluxes having the form
+
+$$F_{sv}(U) = \begin{pmatrix} F_h \\ F_{hu} \\ F_{hv} \end{pmatrix} = \int_{n_x\xi+n_y\gamma \ge 0} \begin{pmatrix} 1 \\ \xi \\ \gamma \end{pmatrix} (n_x\xi + n_y\gamma) M(U, \xi, \gamma) d\xi d\gamma.$$
+
+with $M$ given by (59), $n_x$ and $n_y$ being the components of a normal unit vector **n**. Defining the change of variables
+
+$$\xi = u + cz_1, \quad \gamma = v + cz_2,$$
+
+we can write
+
+$$F_{sv}(U) = h \int_{n_x(cz_1+u)+n_y(cz_2+v) \ge 0} (n_x(cz_1+u) + n_y(cz_2+v)) \begin{pmatrix} 1 \\ u+cz_1 \\ v+cz_2 \end{pmatrix} \chi_0(z_1, z_2) dz_1 dz_2,$$
+
+where $\chi_0$ is defined by (56). A second change of variables $y_1 = n_x z_1 + n_y z_2$, $y_2 = n_x z_2 - n_y z_1$, $\tilde{u} = n_x u + n_y v$ gives
+
+$$F_{sv}(U) = h \int_{\{y_1 \ge -\frac{\tilde{u}}{c}\} \times \mathbb{R}} (\tilde{u} + cy_1) \begin{pmatrix} 1 \\ u + cn_x y_1 \\ v + cn_y y_1 \end{pmatrix} \chi_0(y_1, y_2) dy_1 dy_2, \quad (103)$$
+
+since $\chi_0$ is odd. The details of the computations of formula (103) is given in appendix B.
+
+Using the properties obtained at the kinetic level for the resolution of the system (18)-(19), the following proposition holds.
+
+**Proposition 5.2** *Under the CFL condition* (101), the scheme (90),(95), (96) satisfies the following properties.
+
+(i) The macroscopic scheme derived from (95)-(96) using (90) is a consistent discretization of the layer-averaged Euler system (18)-(19).
+
+(ii) The water depth remains nonnegative i.e.
+
+$$h_i^{n+1} \ge 0, \quad \forall i, \quad \text{when } h_i^n \ge 0 \quad \forall i.$$
+
+(iii) The scheme (90),(95), (96) is well-balanced i.e. it preserves the so-called “lake at rest” solution.
+
+**Proof of prop. 5.2** The proof is similar to the one given in prop. 5.1. ■
+
+## 5.6 The discrete layer-averaged Navier-Stokes system
+
+In this paragraph, we detail the space discretization of the viscous terms. Several expressions have been obtained for the viscous terms, see paragraph 3.2. In this paragraph, we give a numerical scheme for the model (74), rewriting it under the form
+
+$$(h_\alpha \mathbf{u}_\alpha)^{n+1} = \widetilde{h_\alpha \mathbf{u}_\alpha} + \Delta t^n S_{v,f}(U), \quad (104)$$
+
+with $S_{v,f}(U) = (S_{v,f,1}, \dots, S_{v,f,N})^T$ and
+
+$$\widetilde{h_\alpha \mathbf{u}_\alpha} = h_\alpha \mathbf{u}_\alpha - \Delta t^n \left( \nabla_{x,y} (h_\alpha \mathbf{u}_\alpha \otimes \mathbf{u}_\alpha) - \nabla_{x,y} \left(\frac{g}{2} h h_\alpha\right) - g h_\alpha \nabla_{x,y} z_b \right)$$

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+$$
++ \mathbf{u}_{\alpha+1/2}^{n+1} G_{\alpha+1/2} - \mathbf{u}_{\alpha-1/2}^{n+1} G_{\alpha-1/2} ),
+$$
+
+$$
+S_{v,f,\alpha} = \nabla_{x,y} (h_{\alpha} \Sigma_{\alpha}^{0}) + \Gamma_{\alpha+1/2} (\mathbf{u}_{\alpha+1} - \mathbf{u}_{\alpha}) - \Gamma_{\alpha-1/2} (\mathbf{u}_{\alpha} - \mathbf{u}_{\alpha-1}) - \kappa_{\alpha} \mathbf{u}_{\alpha} + W_{\alpha} \mathbf{t}_{s},
+$$
+
+with the definitions (40),(41),(46), (47),(44) for $\mathbf{T}_\alpha^0$, $\Gamma_{\alpha\pm 1/2}$. It remains to give a fully discrete scheme for the viscous and friction terms $\{S_{v,f,\alpha}\}$.
+
+The discretization of (104) is done using a finite element / finite difference approximation obtained as follows. We depart from the triangulation defined in paragraph (5.2) and we use the cells values of the variables – inherited from the finite volume framework – to define a $\mathbb{P}_1$ approximation of the variables.
+
+Notice that, compared to the advection and pressure terms, the discretization of the viscous
+terms raises less difficulties and we propose a stable scheme that will be extended to more general
+rheology terms [24] and more completely analyzed in a forthcoming paper.
+
+Using a classical $\mathbb{P}_1$ finite element type approximation with mass lumping of Eq. (104), we get
+
+$$
+\begin{equation}
+\begin{split}
+\mathbf{U}_{\alpha}^{n+1} = {}& \tilde{\mathbf{U}}_{\alpha} - \Delta t^n (\mathcal{K}_{\alpha+1} \mathbf{U}_{\alpha+1} + \mathcal{K}_{\alpha} \mathbf{U}_{\alpha} + \mathcal{K}_{\alpha-1} \mathbf{U}_{\alpha-1}) \\
+& + \Delta t^n \mathcal{G}_{\alpha+1/2} (\mathbf{U}_{\alpha+1} - \mathbf{U}_{\alpha}) - \Delta t^n \mathcal{G}_{\alpha-1/2} (\mathbf{U}_{\alpha} - \mathbf{U}_{\alpha-1}) - \Delta t^n \kappa_{\alpha} \mathbf{U}_{\alpha} + \Delta t^n W_{\alpha} \mathbf{t}_s,
+\end{split}
+\tag{105}
+\end{equation}
+$$
+
+with the matrices
+
+$$
+\begin{align*}
+K_{\alpha,ji} &= \frac{\nu}{2} \int_{\Omega} \left( \frac{h_{\alpha}}{h_{\alpha+1} + h_{\alpha}} + \frac{h_{\alpha}}{h_{\alpha} + h_{\alpha-1}} \right) \nabla_{x,y} \varphi_i \cdot \nabla_{x,y} \varphi_j \, dxdy, \\
+K_{\alpha\pm 1,ji} &= \frac{\nu_{\alpha\pm 1/2}}{2} \int_{\Omega} \frac{h_{\alpha\pm 1}}{h_{\alpha+1} + h_{\alpha}} \nabla_{x,y}\varphi_i \cdot \nabla_{x,y}\varphi_j \, dxdy, \\
+G_{\alpha+1/2,ji} &= \nu_{\alpha+1/2} \int_{\Omega} \frac{1 + |\nabla_{x,y} z_{\alpha+1/2}|^2}{h_{\alpha+1} + h_{\alpha}} \varphi_i \cdot \varphi_j \, dxdy,
+\end{align*}
+$$
+
+where $\varphi_i$, $\varphi_j$ are the basis functions. We have presented an explicit in time version of (105) that is stable under a classical CFL condition. An implicit or semi-implicit version of (105) can also be used.
+
+The main purpose of this paper is to propose a stable and robust numerical approximation of the incompressible Euler system with free surface. Voluntarily, we give few details concerning the numerical approximation of the dissipative terms:
+
+• the viscous and friction terms are dissipative and hence a reasonable approximation leads to a stable numerical scheme.
+
+* In this paper, we consider a simplified Newtonian rheology for the fluid, the numerical approximation of the general (layer-averaged) rheology [24] will be studied in a forthcoming paper.
+
+**5.7 Boundary conditions**
+
+The contents of this paragraph slightly differ from previous works of one of the authors [27] and valid for the classical Saint-Venant system. First, we focus on the boundary conditions for the layer-averaged Euler system i.e. the system (72)-(73) for $v = 0$, $k = 0$ and then for the viscous part.
+
+5.7.1 Layer-averaged Euler system
+
+In this paragraph we detail the computation of the boundary flux $\mathcal{F}(\mathbf{U}_i, \mathbf{U}_{e,i}, \mathbf{n}_i)$ appearing in (80), (83), (84).
+The variable $\mathbf{U}_{i,e}^n$ can be interpreted as an approximation of the solution in a ghost cell adjacent to
+the boundary. As before we introduce the vector
+
+$$
+U_{i,e} = (h_{i,e}^n, (hu)_{1,i,e}, \dots, (hu)_{N,i,e}, (hv)_{1,i,e}, \dots, (hv)_{N,i,e})^T,
+$$

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+and we will use the flux vector splitting form associated to the kinetic formulation (91)
+
+$$ \mathcal{F}(U_i, U_{i,e}, \mathbf{n}_i) = F^+(U_i, \mathbf{n}_i) + F^-(U_{i,e}, \mathbf{n}_i) \quad (106) $$
+
+with $U_{i,e}^n$ defined according to the boundary type.
+
+**Solid wall** If we consider a node $i_0$ belonging to a solid wall, we prescribe a slip condition written
+
+$$ \mathbf{u}_{\alpha} \cdot \mathbf{n}_{i_0} = 0, \quad (107) $$
+
+for $\alpha = 1, \dots, N$. We assume the continuity of the water depth $h_{i_0,e} = h_{i_0}$ and of the tangential component of velocity.
+
+From (103) with (107) we obtain
+
+$$ F_h^+(U_{\alpha,i_0}) + F_h^-(U_{\alpha,i_0,e}) = 0, $$
+
+and
+
+$$ \left( \begin{array}{c} F_{hu}^{+}(U_{\alpha,i_0}) + F_{hu}^{-}(U_{\alpha,i_0,e}) \\ F_{hv}^{+}(U_{\alpha,i_0}) + F_{hv}^{-}(U_{\alpha,i_0,e}) \end{array} \right) \cdot \mathbf{n}_{i_0} = \frac{gh_{\alpha,i_0}h_{i_0}}{2}, \quad \left( \begin{array}{c} F_{hu}^{+}(U_{\alpha,i_0}) + F_{hu}^{-}(U_{\alpha,i_0,e}) \\ F_{hv}^{+}(U_{\alpha,i_0}) + F_{hv}^{-}(U_{e,\alpha,i_0}) \end{array} \right) \cdot \mathbf{t}_{i_0} = 0, $$
+
+for $\alpha = 1, \dots, N$ for a vector $\mathbf{t}_{i_0}$ orthogonal to $\mathbf{n}_{i_0}$. The condition (107) is therefore prescribed weakly but a posteriori, in order to be sure that (107) is satisfied, we can apply
+
+$$ \mathbf{u}_{\alpha,i_0,e} = \mathbf{u}_{\alpha,i_0} - (\mathbf{u}_{\alpha,i_0} \cdot \mathbf{n}_{i_0})\mathbf{n}_{i_0}. $$
+
+**Fluid boundary** Even if the considered model is more complex than the Shallow water system, we can consider that the type of the flow depends, for each layer, on the value of the Froud number $Fr_\alpha = |\mathbf{u}_\alpha|/\sqrt{gh}$, a flow is said torrential, for $|\mathbf{u}_\alpha| > \sqrt{gh}$ and fluvial, for $|\mathbf{u}_\alpha| < \sqrt{gh}$.
+
+Generally, for the fluid boundaries, the conditions prescribed by the user depend on the type of the flow defined by this criterion.
+
+We have also to notice that with **n** the outward unit normal to the boundary edge, an inflow boundary corresponds to $\mathbf{u}_\alpha \cdot \mathbf{n} < 0$ and an outflow one to $\mathbf{u}_\alpha \cdot \mathbf{n} > 0$.
+
+We will treat the following cases: for a fluvial flow boundary, we distinguish the cases where the flux or the water depth are given, while for a torrential flow we distinguish the inflow or outflow boundaries.
+
+**Fluvial boundary. Flux given** We consider first a fluvial boundary, so we assume that
+
+$$ |\mathbf{u}_\alpha| < \sqrt{gh}. \quad (108) $$
+
+If for each layer, the flux $\mathbf{q}_{g,\alpha}$ is given, then we wish to impose
+
+$$ (F_h(\mathbf{U}_{\alpha,i}) + F_h(\mathbf{U}_{e,\alpha,i})) \cdot \mathbf{n}_i = \mathbf{q}_{g,\alpha} \cdot \mathbf{n}_i, \quad F_h(\mathbf{U}_{e,\alpha,i}) \cdot \mathbf{t}_i = q_{g,\alpha} \cdot \mathbf{t}_i, \quad (109) $$
+
+with $\mathbf{n}_i \cdot \mathbf{t}_i = 0$. The value $\mathbf{q}_{g,\alpha}$ depends on the value of the prescribed flux along the vertical axis.
+
+If one directly imposes (109), it leads to instabilities (especially because the numerical values are not necessarily in the regime of validity of this condition). We propose to discretize it in a weak form. We denote
+
+$$ a_1 = \mathbf{q}_{g,\alpha} \cdot \mathbf{n}_i - F_h(\mathbf{U}_{\alpha,i}) \cdot \mathbf{n}_i. \quad (110) $$
+
+If $a_1 \ge 0$, we prescribe
+
+$$ F_h(\mathbf{U}_{e,\alpha,i}) = 0, \quad F_{hu}(\mathbf{U}_{e,\alpha,i}) = 0, \quad \text{and} \quad F_{hv}(\mathbf{U}_{e,\alpha,i}) = 0. $$

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+If $a_1 < 0$, we have to write a third equation to be able to compute the three components of $\mathbf{U}_{e,\alpha}$ and by analogy with what is done for the Saint-Venant system – where the Riemann invariant related to the outgoing characteristic is preserved – we assume the quantity $\mathbf{u}_\alpha \cdot \mathbf{n}$ is constant though the interface, i.e.
+
+$$ \mathbf{u}_{e,\alpha,i} \cdot \mathbf{n}_i - 2\sqrt{\overline{gh_{e,i}}} = \mathbf{u}_{\alpha,i} \cdot \mathbf{n}_i - 2\sqrt{\overline{gh_i}}. \quad (111) $$
+
+As (108) is satisfied, the eigenvalue $\overline{u_{e,\alpha,i} \cdot n_i} - 2\sqrt{\overline{gh_{e,i}}}$ is positive.
+
+We use the equations (109) and (111) to compute $h_{e,i}$ and $\mathbf{u}_{e,\alpha,i} \cdot \mathbf{n}_i$. We denote $a_2 = \mathbf{u}_{\alpha,i} \cdot \mathbf{n}_i - 2\sqrt{\overline{gh_i}}$ and
+
+$$ m = \frac{\mathbf{u}_{e,\alpha,i} \cdot \mathbf{n}_i}{\sqrt{\overline{gh_{e,i}}}}. \quad (112) $$
+
+Then the equation (111) gives
+
+$$ \sqrt{\overline{gh_{e,i}}} (m-2) = a_2 \quad (113) $$
+
+and using the definition of $F_h$ (see (133)) with (109), (110) we have
+
+$$ \frac{h_{e,i}}{\pi} \int_{z \le \frac{-\mathbf{u}_{e,\alpha,i} \cdot \mathbf{n}_i}{\sqrt{\frac{\overline{gh_{e,i}}}{2}}}} \left( \mathbf{u}_{e,\alpha,i} \cdot \mathbf{n}_i + \sqrt{\frac{\overline{gh_{e,i}}}{2}} z \right) \sqrt{1 - \frac{z^2}{4}} dz = a_1, \quad (114) $$
+
+or using (111)
+
+$$ \psi(h_{e,i}) = a_1, \quad (115) $$
+
+with
+
+$$ \psi(h_{e,i}) = \frac{h_{e,i}}{\pi} \int_{z \le \frac{-(2\sqrt{\overline{gh_{e,i}}}+a_2)}{\sqrt{\frac{\overline{gh_{e,i}}}{2}}}} \left( 2\sqrt{\overline{gh_{e,i}}} + a_2 + \sqrt{\frac{\overline{gh_{e,i}}}{2}} z \right) \sqrt{1 - \frac{z^2}{4}} dz. $$
+
+It is easy to see that $h \mapsto \psi(h)$ is a growing function of $h$ with $\psi(0) = 0$ and $\psi(+\infty) = +\infty$ and therefore, Eq. (114) admits a unique solution for any $a_1 > 0$. Using (113), Eq. (115) is equivalent to solve for $m$
+
+$$ \Psi(m) = a_2, \quad (116) $$
+
+with
+
+$$ \Psi(m) = K \frac{m-2}{\phi(m)^{1/3}}, $$
+
+and $K = (\sqrt{2}ga_1)^{1/3}$
+
+$$ \phi(m) = \frac{1}{\pi} \int_{z \le -\sqrt{2m}} (\sqrt{2}m + z) \sqrt{1 - \frac{z^2}{4}} dz. $$
+
+In practice, we use a Newton-Raphson algorithm to solve an equivalent form of Eq. (116), namely
+
+$$ m - 2 - \frac{a_2}{K} \phi(m)^{1/3} = 0. $$
+
+Once the above equation has been solved, from (112), (113) we deduce
+
+$$ h_{e,i} = \frac{1}{g} \left( \frac{a_2}{m-2} \right)^2, \quad \mathbf{u}_{e,\alpha,i} \cdot \mathbf{n}_i = \frac{a_2 m}{m-2} = m \sqrt{\overline{gh_{e,i}}}. $$
+
+**Remark 5.3** Notice that in the procedure proposed to calculate $\mathbf{u}_{e,\alpha,i}$, $h_{e,i}$, even if $h_{e,i}$ represents a total water depth, a different value of $h_{e,i}$ is calculated for each layer $\alpha$. $h_{e,i}$ is only used to ensure (109).

samples/texts/1513018/page_3.md

@@ -0,0 +1,109 @@
+<table>
+  <tbody>
+    <tr>
+      <td>5.5 Macroscopic scheme</td>
+      <td>25</td>
+    </tr>
+    <tr>
+      <td>5.6 The discrete layer-averaged Navier-Stokes system</td>
+      <td>25</td>
+    </tr>
+    <tr>
+      <td>5.7 Boundary conditions</td>
+      <td>26</td>
+    </tr>
+    <tr>
+      <td>&nbsp;&nbsp;&nbsp;&nbsp;5.7.1 Layer-averaged Euler system</td>
+      <td>26</td>
+    </tr>
+    <tr>
+      <td>&nbsp;&nbsp;&nbsp;&nbsp;5.7.2 Layer-averaged Navier-Stokes system</td>
+      <td>30</td>
+    </tr>
+    <tr>
+      <td>5.8 Toward second order schemes</td>
+      <td>30</td>
+    </tr>
+    <tr>
+      <td>&nbsp;&nbsp;&nbsp;&nbsp;5.8.1 Second order reconstruction for the layer-averaged Euler system</td>
+      <td>30</td>
+    </tr>
+    <tr>
+      <td>&nbsp;&nbsp;&nbsp;&nbsp;5.8.2 Modified Heun scheme</td>
+      <td>30</td>
+    </tr>
+    <tr>
+      <td><b>6 Numerical applications</b></td>
+      <td><b>31</b></td>
+    </tr>
+    <tr>
+      <td>&nbsp;&nbsp;&nbsp;&nbsp;6.1 Stationary analytical solution</td>
+      <td>31</td>
+    </tr>
+    <tr>
+      <td>&nbsp;&nbsp;&nbsp;&nbsp;6.2 Non-stationary analytical solutions</td>
+      <td>33</td>
+    </tr>
+    <tr>
+      <td>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;6.2.1 Radially-symmetrical parabolic bowl</td>
+      <td>33</td>
+    </tr>
+    <tr>
+      <td>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;6.2.2 Draining of a tank</td>
+      <td>35</td>
+    </tr>
+    <tr>
+      <td>&nbsp;&nbsp;&nbsp;&nbsp;6.3 Simulation of a tsunami</td>
+      <td>36</td>
+    </tr>
+    <tr>
+      <td>&nbsp;&nbsp;&nbsp;&nbsp;6.4 Monai valley benchmark</td>
+      <td>37</td>
+    </tr>
+    <tr>
+      <td>&nbsp;&nbsp;&nbsp;&nbsp;6.5 Hydrodynamics in a raceway</td>
+      <td>39</td>
+    </tr>
+    <tr>
+      <td>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;6.5.1 Experimental measurements</td>
+      <td>39</td>
+    </tr>
+    <tr>
+      <td>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;6.5.2 Simulation results</td>
+      <td>40</td>
+    </tr>
+    <tr>
+      <td><b>7 Conclusion</b></td>
+      <td><b>42</b></td>
+    </tr>
+  </tbody>
+</table>
+
+Abstract
+
+In this paper we propose a stable and robust strategy to approximate the 3d incompressible
+hydrostatic Euler and Navier-Stokes systems with free surface.
+
+Compared to shallow water approximation of the Navier-Stokes system, the idea is to use
+a Galerkin type approximation of the velocity field with piecewise constant basis functions in
+order to obtain an accurate description of the vertical profile of the horizontal velocity. Such a
+strategy has several advantages. It allows
+
+* to rewrite the Navier-Stokes equations under the form of a system of conservation laws with source terms,
+
+* the easy handling of the free surface, which does not require moving meshes,
+
+* the possibility to take advantage of robust and accurate numerical techniques developed in extensive amount for Shallow Water type systems.
+
+Compared to previous works of some of the authors, the three dimensional case is studied
+in this paper. We show that the model admits a kinetic interpretation including the vertical
+exchanges terms, and we use this result to formulate a robust finite volume scheme for its nu-
+merical approximation. All the aspects of the discrete scheme (fluxes, boundary conditions,...)
+are completely described and the stability properties of the proposed numerical scheme (well-
+balancing, positivity of the water depth,...) are discussed. We validate the model and the
+discrete scheme with some numerical academic examples (3d non stationary analytical solu-
+tions) and illustrate the capability of the discrete model to reproduce realistic tsunami waves
+propagation, tsunami runup and complex 3d hydrodynamics in a raceway.
+
+**Keywords** Free surface flows, Navier-Stokes equations, Euler system, Free surface, 3d model,
+Hydrostatic assumption, Kinetic description, Finite volumes.

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+**Fluvial boundary. Water depth given** We verify that the flow is actually fluvial, i.e.
+
+$$
+(\mathbf{u}_{\alpha,i} \cdot \mathbf{n}_i - \sqrt{gh_i})(\mathbf{u}_{\alpha,i} \cdot \mathbf{n}_i + \sqrt{gh_i}) \le 0. \quad (117)
+$$
+
+Since the water depth is given, we write
+
+$$
+h_{e,i} = h_{g,i}. \tag{118}
+$$
+
+We assume the continuity of the tangential component
+
+$$
+(h\mathbf{u})_{e,\alpha,i} \cdot \mathbf{t}_i = (h\mathbf{u})_{\alpha,i} \cdot \mathbf{t}_i, \quad (119)
+$$
+
+with $\mathbf{t}_i \cdot \mathbf{n}_i = 0$. To define completely $\mathbf{u}_{e,\alpha,i}$, we assume, as in the previous case, that the Riemann invariant is constant along the outgoing characteristic (111), so we obtain
+
+$$
+\mathbf{u}_{e,\alpha,i} \cdot \mathbf{n}_i = \mathbf{u}_{\alpha,i} \cdot \mathbf{n}_i + 2\sqrt{g}(\sqrt{h_i} - \sqrt{h_{g,i}}). \quad (120)
+$$
+
+Sometimes it appears that the numerical values do not satisfy the condition (117), then the flow is
+in fact torrential and
+
+• if **u**<sub>α,i</sub>·**n**<sub>i</sub} > 0 , the condition (118) cannot be satisfied (see Sec. 6.2.4),
+
+• if $\mathbf{u}_{\alpha,i} \cdot \mathbf{n}_i < 0$, one condition is missing and we prescribe $\mathbf{u}_{e,\alpha,i} \cdot \mathbf{n}_i = \mathbf{u}_{\alpha,i} \cdot \mathbf{n}_i$.
+
+**Torrential inflow boundary** For a torrential inflow boundary we assume that the water depth and the flux are given, then we prescribe
+
+$$
+h_{e,i} = h_{g,i}, \quad (h\mathbf{u})_{e,\alpha,i} \cdot \mathbf{t}_i = (h\mathbf{u})_{g,\alpha,i} \cdot \mathbf{t}_i,
+$$
+
+and
+
+$$
+(F_h(\mathbf{U}_{\alpha,i}) + F_h(\mathbf{U}_{e,\alpha,i})) \cdot \mathbf{n}_i = \mathbf{q}_{g,\alpha} \cdot \mathbf{n}_i = (h\mathbf{u})_{g,\alpha,i} \cdot \mathbf{n}_i.
+$$
+
+In this case we have to compute $(h\mathbf{u})_{e,\alpha,i} \cdot \mathbf{n}_i$ or $\mathbf{u}_{e,\alpha,i} \cdot \mathbf{n}_i$. We consider an inflow boundary, so $(h\mathbf{u})_{g,\alpha,i} \cdot \mathbf{n}_i < 0$ therefore using the notation (110) we have $a_1 < 0$. By analogy with the previous section we denote
+
+$$
+m = \frac{\mathbf{u}_{e,\alpha,i} \cdot \mathbf{n}_i}{\sqrt{gh_{g,i}}},
+$$
+
+then the equation for *m* is (see (112)-(114))
+
+$$
+\phi(m) = \sqrt{\frac{2}{g} \frac{a_1}{h_{g,i}^{3/2}}}.
+$$
+
+As in the paragraph entitled *Flux given*, the above equation has a unique solution $m < 2$ for $a_1 < 0$.
+
+**Torrential outflow boundary** In the case of a torrential outflow boundary, we do not prescribe any condition. We assume that the two Riemann invariants are constant along the outgoing characteristics leading to
+
+$$
+\begin{align*}
+\mathbf{u}_{e,\alpha,i} \cdot \mathbf{n}_i - 2\sqrt{gh_{e,i}} &= \mathbf{u}_{\alpha,i} \cdot \mathbf{n}_i - 2\sqrt{gh_i}, \\
+\mathbf{u}_{e,\alpha,i} \cdot \mathbf{n}_i + 2\sqrt{gh_{e,i}} &= \mathbf{u}_{\alpha,i} \cdot \mathbf{n}_i + 2\sqrt{gh_i},
+\end{align*}
+$$
+
+and we deduce $h_{e,i} = h_i$, $\mathbf{u}_{e,\alpha,i} \cdot \mathbf{n}_i = \mathbf{u}_{\alpha,i} \cdot \mathbf{n}_i$. We assume that we also have $(h\mathbf{u})_{e,\alpha,i} \cdot \mathbf{t}_i = (h\mathbf{u})_{\alpha,i} \cdot \mathbf{t}_i$.

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+### 5.7.2 Layer-averaged Navier-Stokes system
+
+Because of the fractional step we use, the boundary conditions for the layer-averaged Euler system are, to some extent, independent from the one used for the rheology terms.
+
+For the resolution of Eq. (104), boundary conditions associated with the operator
+
+$$ \nabla_{x,y} \cdot (h_{\alpha} \mathbf{T}_{\alpha}^{0}), $$
+
+have to be specified and usually we prescribe homogeneous Neumann boundary conditions (corresponding to an imposed stress). Of course, in particular cases, Dirichlet or Robin type boundary conditions can also be considered.
+
+## 5.8 Toward second order schemes
+
+In order to improve the accuracy of the results the first-order scheme defined in paragraphs 5.3-5.5 can be extended to a formally second-order one using a MUSCL like extension (see [52]).
+
+### 5.8.1 Second order reconstruction for the layer-averaged Euler system
+
+In the definition of the flux (93), we replace the piecewise constant values $U_{i,j}$, $U_{j,i}$ by more accurate reconstructions deduced from piecewise linear approximations, namely the values $\tilde{U}_{i,j}$, $\tilde{U}_{j,i}$ reconstructed on both sides of the interface. The reconstruction procedure is similar to the one used and described in [5, paragraph. 5.1].
+
+The second order reconstruction is only applied for the horizontal fluxes. For the exchange terms along the vertical axis involving the quantities $G_{\alpha \pm 1/2}$, we keep the first order approximation. Despite this, we recover over the simulations (see paragraphs 6.1, 6.2) a second order type convergence curve. For this reason, we call this reconstruction "second order".
+
+### 5.8.2 Modified Heun scheme
+
+The explicit time scheme (72)-(73) used in the previous paragraphs corresponds to a first order explicit Euler scheme. The second-order accuracy in time is usually recovered by the Heun method [16] that is a slight modification of the second order Runge-Kutta method. More precisely, for a dynamical system written under the form
+
+$$ \frac{\partial y}{\partial t} = f(y), \qquad (121) $$
+
+the Heun scheme consists in defining $y^{n+1}$ by
+
+$$ y^{n+1} = y(t^n + \Delta t^n) = \frac{y^n + \tilde{y}^{n+2}}{2}, \qquad (122) $$
+
+with
+
+$$ \tilde{y}^{n+1} = y^n + \Delta t^n f(y^n, t^n), \quad \tilde{y}^{n+2} = \tilde{y}^{n+1} + \Delta t^n f(\tilde{y}^{n+1}, t^{n+1}). \qquad (123) $$
+
+But the scheme defined by (122) does not preserve the invariant domains. Indeed, the time step being given by a CFL condition, $\Delta t^n$ in the relation (123) should be replaced by $\tilde{\Delta} t^{n+1}$ i.e. the time step satisfying the CFL condition and calculated using $\tilde{y}^{n+1}$. Thus in situations where the time step strongly varies from one iteration to another, the Heun scheme does not preserve the positivity of the scheme.
+
+To overcome this difficulty, we propose an improvement of the Heun scheme

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+**Proposition 5.3** The scheme defined by $y^{n+1} = (1-\gamma)y^n + \gamma\tilde{y}^{n+2}$ with
+
+$$ \tilde{y}^{n+1} = y^n + \Delta t_1^n f(y^n), \quad \tilde{y}^{n+2} = \tilde{y}^{n+1} + \Delta t_2^n f(\tilde{y}^{n+1}), $$
+
+and
+
+$$ \Delta t^n = \frac{2\Delta t_1^n \Delta t_2^n}{\Delta t_1^n + \Delta t_2^n}, \quad \gamma = \frac{(\Delta t^n)^2}{2\Delta t_1^n \Delta t_2^n}, $$
+
+is second order and compatible with a CFL constraint. Since $\gamma \ge 0$, $y^{n+1}$ is a convex combination of $y^n$ and $\tilde{y}^{n+2}$ so the scheme preserves the positivity. For the previous relations $\Delta t_1^n$ and $\Delta t_2^n$ respectively satisfy the CFL conditions associated with $y^n$ and $\tilde{y}^{n+1}$.
+
+When $\Delta t_1^n = \Delta t_2^n = \Delta t^n$, the scheme reduces to the classical Heun scheme with $\alpha = \gamma = 1/2$.
+
+**Proof of proposition 5.3** Using (121), a Taylor expansion of $y(t^n + \Delta t^n)$ gives
+
+$$ y(t^n + \Delta t^n) = y^n + \Delta t^n f(y^n) + \frac{(\Delta t^n)^2}{2} f(y^n) f'(y^n) + O((\Delta t^n)^3). $$
+
+Using the definitions given in the proposition, we have
+
+$$ \tilde{y}^{n+2} = y^n + (\Delta t_1^n + \Delta t_2^n) f(y^n) + \Delta t_1^n \Delta t_2^n f(y^n) f'(y^n) + O(\Delta t_2^n (\Delta t_1^n)^2), $$
+
+and a simple calculus gives $\alpha y^n + \beta \tilde{y}^{n+1} + \gamma \tilde{y}^{n+2} - y(t^n + \Delta t^n) = O((\Delta t^n)^3)$, that completes the proof. $\blacksquare$
+
+# 6 Numerical applications
+
+In this section, we use the numerical scheme to simulate several test cases: analytical solutions or in situ measurements, stationary or non-stationary solutions, for the Euler and Navier-Stokes systems. The obtained results emphasize the accuracy of the numerical procedure in a wide range of typical applications and its applicability to a real tsunami case. We also propose simulations of the hydrodynamic regime in a raceway agitated by a paddlewheel and confront the results with experimental measurements.
+
+The numerical simulations presented in this section have been obtained with the code Freshkiss3d [49] where the numerical scheme presented in this paper is implemented.
+
+## 6.1 Stationary analytical solution
+
+First, we compare our numerical model with stationary analytical solutions for the free surface Euler system proposed by some of the authors in [21].
+
+We consider as geometrical domain a channel $(x, y) \in [0, x_{max}] \times [0, 2]$. The analytical solution given in [21, Prop. 3.1] and defined by
+
+$$ z_b = \bar{z}_b - h_0 - \frac{\alpha^2 \beta^2}{2g \sin^2(\beta h_0)}, \qquad (124) $$
+
+$$ u_{\alpha,\beta} = \frac{\alpha\beta}{\sin(\beta h_0)} \cos(\beta(z-z_b)), \qquad (125) $$
+
+$$ v_{\alpha,\beta} = 0, $$
+
+$$ w_{\alpha,\beta} = \alpha\beta \left( \frac{\partial z_b}{\partial x} \frac{\cos(\beta(z-z_b))}{\sin(\beta h_0)} + \frac{\partial h_0}{\partial x} \frac{\sin(\beta(z-z_b)) \cos(\beta h_0)}{\sin^2(\beta h)} \right), $$

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+with $\alpha = 1 \text{ m}^2 \cdot \text{s}^{-1}$, $\beta = 1 \text{ m}^{-1}$, $\bar{z}_b = \text{cst}$, $x_{\text{max}} = 20 \text{ m}$ and
+
+$$h_0(x, y) = \frac{1}{2} + \frac{3}{2} \frac{1}{1 + (x - \frac{1}{2}x_{\text{max}})^2} - \frac{1}{2} \frac{1}{2 + (x - \frac{2}{3}x_{\text{max}})^2}, \quad (126)$$
+
+is a stationary regular analytical solution of the incompressible and hydrostatic Euler system with free surface (12)-(13),(4),(5) with $p^a = 0$.
+
+In order to obtain the simulated solution, we consider the topography defined by (124), (126) and we impose the following boundary conditions:
+
+* solid wall for the two boundaries $y=0 \text{ m}$ and $y=2 \text{ m}$,
+
+* given water depth $h_0(x_{\text{max}}, y)$ at $x = x_{\text{max}} = 20 \text{ m}$,
+
+* given flux defined by (125) at $x=0 \text{ m}$.
+
+We have performed the simulations for several unstructured meshes having 290 nodes and 2 layers, 597 nodes and 4 layers, 1010 nodes and 8 layers, 2112 nodes and 17 layers, see Remark 6.1.
+
+**Remark 6.1** In each case where a convergence curve towards an analytical solution is presented, we have proceeded as follows. First, we choose a sequence of unstructured meshes for the considered horizontal geometrical domain. Then the number of layers is adapted so that each 3d element of the mesh can be approximatively considered as a regular polyhedron.
+
+On Fig. 3(a), we have depicted the features of the analytical solution we use for the convergence test, it clearly appears on Fig. 3-(a) that the velocity profile of the chosen analytical solution varies along the z axis. The $L_2$ errors for the convergence test and the corresponding convergence rate are given in Tab. 1. Figure 3-(b) gives the convergence curve towards the analytical solution i.e. the $\log(L^2 - \text{error})$ of the water depth – at time $T = 300$ seconds when the stationary regime is reached – versus $\log(h_{a0}/h_a)$ for the first and second-order schemes and they are compared to the theoretical order (we denote by $h_a$ the average edge length and $h_{a0}$ the average edge length of the coarser mesh).
+
+Figure 3: (a) Surface level of the analytical solution (124)-(126) and horizontal velocity $u_{\alpha,\beta}$, (b) error between the analytical solution and the simulated one with the six meshes, first order (space and time) and second order extension (space and time) schemes. The first and second order theoretical curves correspond to the dashed lines.

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+<table><thead><tr><th rowspan="2">N<sub>dof</sub></th><th colspan="4">First order scheme</th><th colspan="4">Second order scheme</th></tr><tr><th>L<sub>2</sub> error h</th><th>rate</th><th>L<sub>2</sub> error u</th><th>rate</th><th>L<sub>2</sub> error h</th><th>rate</th><th>L<sub>2</sub> error u</th><th>rate</th></tr></thead><tbody><tr><td>580</td><td>0.0405</td><td>-</td><td>0.0699</td><td>-</td><td>0.006</td><td>-</td><td>0.0092</td><td>-</td></tr><tr><td>2360</td><td>0.0265</td><td>1.02</td><td>0.0457</td><td>0.68</td><td>2.716 10<sup>-3</sup></td><td>1.92</td><td>3.214 10<sup>-3</sup></td><td>1.69</td></tr><tr><td>8080</td><td>0.0210</td><td>0.96</td><td>0.0332</td><td>0.88</td><td>1.721 10<sup>-3</sup></td><td>1.89</td><td>1.521 10<sup>-3</sup></td><td>2.07</td></tr><tr><td>34204</td><td>0.0179</td><td>0.91</td><td>0.0256</td><td>0.97</td><td>1.24 10<sup>-3</sup></td><td>1.84</td><td>1.24 10<sup>-3</sup></td><td>1.80</td></tr></tbody></table>
+
+Table 1: Stationary solution: $L_2$ error table and convergence rate for the velocity and the water depth for each mesh.
+
+**Remark 6.2** Following the results given in [21, paragraph 3.4], it is possible to obtain stationary analytical solutions with discontinuities for the Euler system. In this case, the unknowns are not given by algebraic expressions but are obtained through the resolution of an ODE involving only the water depth $h$.
+
+The numerical scheme has been used in the context of such a discontinuous analytical solution. As planned, for the first and second order schemes, we recover a first order convergence of the simulated solution towards the analytical one because of the discontinuity of the reference solution.
+
+## 6.2 Non-stationary analytical solutions
+
+In a recent paper [25], some of the authors have proposed time-dependent 3d analytical solutions for the Euler and Navier-Stokes equations, some of them concern hydrostatic models. We confront our numerical scheme to these situations where analytical solutions are available.
+
+### 6.2.1 Radially-symmetrical parabolic bowl
+
+The Thacker' analytical solution [50], corresponds to a periodic oscillation in a parabolic bowl. In [25] an extension of the Thacker' radially-symmetrical solution to the situation where the velocity field depends on the vertical coordinate is proposed. This means the proposed solution, described hereafter in prop. 6.1, is analytical for the 3d incompressible hydrostatic Euler system but does not correspond to a shallow water regime.
+
+**Proposition 6.1** For some $t_0 \in \mathbb{R}$, $(\alpha, \beta, \gamma) \in \mathbb{R}_{+*}^3$ such that $\gamma < 1$ let us consider the functions $h, u, v, w, p$ defined for $t \ge t_0$ by
+
+$$h(t, x, y) = \max \left\{ 0, \frac{1}{r^2} f \left( \frac{r^2}{\gamma \cos(\omega t) - 1} \right) \right\}, \quad (127)$$
+
+$$u(t, x, y, z) = x \left( \beta \left( z - z_b - \frac{h}{2} \right) + \frac{\omega \gamma \sin(\omega t)}{2(1 - \gamma \cos(\omega t))} \right), \quad (128)$$
+
+$$v(t, x, y, z) = y \left( \beta \left( z - z_b - \frac{h}{2} \right) + \frac{\omega \gamma \sin(\omega t)}{2(1 - \gamma \cos(\omega t))} \right), \quad (129)$$
+
+$$p(t, x, y, z) = g(h + z_b - z), \quad (130)$$
+
+with $\omega = \sqrt{4\alpha g}$, $r = \sqrt{x^2 + y^2}$ and with a bottom topography defined by
+
+$$z_b(x, y) = \alpha \frac{r^2}{2}, \quad (131)$$
+
+and the function $f$ given by
+
+$$f(z) = -\frac{4g}{\beta^2} + \frac{2}{\beta^2} \sqrt{4g^2 + cz + \beta^2 \alpha g (\gamma^2 - 1) z^2},$$

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+$c$ being a negative constant such that $c \le 4g^2/(\gamma - 1)$. From Eq. (1), the vertical velocity $w$ can be expressed under the form
+
+$$w(t, x, y, z) = -\frac{\partial}{\partial x} \int_{z_b}^{z} u dz - \frac{\partial}{\partial y} \int_{z_b}^{z} v dz.$$
+
+Then $h, u, v, w, p$ as defined previously satisfy the 3D hydrostatic Euler system (12)-(13) completed with (4),(5).
+
+The geometrical domain is defined by $(x, y) \in [-L/2, L/2]^2$ and the chosen parameters are $\alpha = 2$, $\beta = 1$, $\gamma = 0.3$, $c = -1$, $L = 1$, the considered analytical solution is depicted on Fig. 4. The initial conditions correspond to (127)-(130) at time $t = t_0 = 0$ s.
+
+Figure 4: 3D axisymmetrical parabolic bowl: (a) free surface at $t = 0$ (red), $t = T/4$ (dark grey), $t = T/2$ (blue), with the period $T$ defined by $T = 2\pi/\omega$, (b) velocity norm and vectors at $t = 0, T/6, 2T/6, T/2$, in $(x, y = 0, z)$ slice plane.
+
+In order to evaluate the convergence rate of the simulated solution $h_{sim}$ towards the analytical one $h_{anal}$, we have performed a convergence test, the errors and convergence rates appear over Tab. 2. The five unstructured meshes we have considered have respectively 1273 nodes and a single layer, 11104 nodes and 6 layers, 30441 nodes and 15 layers, 59473 nodes and 30 layers and 98137 nodes and 50 layers, see Remark 6.1. We have plotted (see Fig. 5) the $\log(L^2 - \text{error})$ over the water depth at time $T = 2\pi/\omega$ seconds versus $\log(h_{a0}/h_a)$ for the first and second-order schemes and they are compared to the theoretical order.
+
+<table><thead><tr><th rowspan="2">N<sub>dof</sub></th><th colspan="4">First order scheme</th><th colspan="4">Second order scheme</th></tr><tr><th>L<sub>2</sub> error h</th><th>rate</th><th>L<sub>2</sub> error u</th><th>rate</th><th>L<sub>2</sub> error h</th><th>rate</th><th>L<sub>2</sub> error u</th><th>rate</th></tr></thead><tbody><tr><td>1273</td><td>0.0215</td><td>-</td><td>0.0175</td><td>-</td><td>3.19 &times; 10<sup>-4</sup></td><td>-</td><td>0.41 &times; 10<sup>-3</sup></td><td>-</td></tr><tr><td>66624</td><td>0.0065</td><td>1.05</td><td>0.0051</td><td>1.05</td><td>0.26 &times; 10<sup>-4</sup></td><td>2.01</td><td>0.35 &times; 10<sup>-4</sup></td><td>2.07</td></tr><tr><td>456615</td><td>0.0052</td><td>0.75</td><td>0.0039</td><td>0.71</td><td>0.95 &times; 10<sup>-5</sup></td><td>2.79</td><td>1.30 &times; 10<sup>-5</sup></td><td>2.89</td></tr><tr><td>1784190</td><td>0.0040</td><td>0.81</td><td>0.0029</td><td>0.80</td><td>0.67 &times; 10<sup>-5</sup></td><td>1.20</td><td>0.90 &times; 10<sup>-5</sup></td><td>1.15</td></tr><tr><td>4906850</td><td>0.0030</td><td>0.87</td><td>0.0021</td><td>0.86</td><td>0.43 &times; 10<sup>-5</sup></td><td>1.37</td><td>0.58 &times; 10<sup>-5</sup></td><td>1.40</td></tr></tbody></table>
+
+Table 2: 3D axisymmetrical parabolic bowl: $L_2$ error table and convergence rate for the velocity and the water depth for each mesh.
+
+The analytical solution of prop. 6.1 is non stationary and hence, the errors due to the time scheme are combined with the one induced by the space discretization. Moreover this test case has

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+Figure 5: 3D axisymmetrical parabolic bowl: error between the analytical water depth (a), the analytical horizontal velocity (b) and the simulated ones with the five unstructured meshes. The curves for the first order scheme (space and time) and its second order extension (space and time) are compared to the first and second order theoretical curves (dashed lines).
+
+a lot of wet/dry interfaces where the second order reconstruction in space cannot be applied. This can explain the differences between the theoretical and observed slopes for the convergence tests of the second order schemes, see Tab. 2 and Fig. 5.
+
+### 6.2.2 Draining of a tank
+
+Considering the Navier-Stokes system (1)-(3) completed with the boundary conditions (5)-(8), the following proposition holds, see [25] for more details about the proposed analytical solution.
+
+**Proposition 6.2** For some $t_0 \in \mathbb{R}$, $t_1 \in \mathbb{R}^*_+$, $(\alpha, \beta) \in \mathbb{R}^2_+$ such that $\alpha\beta > L$, let us consider the functions $h, u, v, w, p, \phi$ defined for $t \ge t_0$ by
+
+$$ 
+\begin{align*}
+h(t, x, y) &= \alpha f(t), \\
+u(t, x, y, z) &= \beta \left( (z - z_b) - \frac{\alpha}{2} f(t) \right) + f(t) \left( x \cos^2(\theta) + y \sin^2(\theta) \right), \\
+v(t, x, y, z) &= \beta \left( (z - z_b) - \frac{\alpha}{2} f(t) \right) + f(t) \left( x \cos^2(\theta) + y \sin^2(\theta) \right), \\
+w(t, x, y, z) &= f(t)(z_b - z), \\
+p(t, x, y, z) &= p^a(t, x, y) - 2\nu f(t) + g(h - (z - z_b)),
+\end{align*}
+$$
+
+where $f(t) = 1/(t - t_0 + t_1)$ and with a flat bottom $z_b(x,y) = z_{b,0} = \text{cst}$ and $p^a(t,x,y) = p^{a,1}(t)$, with $p^{a,1}(t)$ a given function.
+
+Then $h, u, v, w, p$ as defined previously satisfy the 3d hydrostatic Navier-Stokes system (1)-(3)
+completed with the boundary conditions (6), (4), (7), (5) and $\kappa = \frac{2\nu\alpha\beta}{h(t,x,y)[\alpha\beta - 2(x\cos^2(\theta) + y\sin^2(\theta))]}$ in (6),
+$W = \nu\beta/2$ with $t_s = \frac{1}{\sqrt{2}}(1,1,0)^t$ in (7). The appropriate boundary conditions for $x \in \{-L/2, L/2\}$
+or $y \in \{-L/2, L/2\}$ are also determined by the expressions of $h, v, u, w$ given above.
+
+Choosing the viscosity $\nu = 0$, the variables $h, u, v, w, p$ become analytical solutions of the 3d hydro-
+static Euler system (12)-(13) completed with the boundary conditions (5), (4) and $p(t, x, y, \eta(t, x, y)) =$
+0.
+
+**Proof of prop. 6.2** The proof of prop. 6.2 relies on very simple computations and is not detailed here. ■

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+We have performed the simulations for several unstructured meshes of the geometrical domain $(x, y) \in [0, 5] \times [0, 1]$ and an adapted number of layers so that each 3d element of the mesh can be approximatively considered as a regular polyhedron, the considered meshes have 483 nodes and 3 layers, 700 nodes and 6 layers, 1306 nodes and 10 layers, 2781 nodes and 20 layers.
+
+For $L = 2$ m, $\alpha = 1$ m.s, $t_0 = 0$ s, $t_1 = 0.5$ s, $\beta = 2.5$ s⁻¹, $\theta = 0$, $\nu = 0$ m²·s⁻¹, $p^{a,1} = 0$ m²·s⁻² on Fig. 6-(a), we have depicted the features of the analytical solution – at time $T = 0.5$ second – we use for the convergence test. Figure 6-(b) gives the convergence curve towards the analytical solution i.e. the log($L^2$-error) of the water depth – at time $T = 1$ second – versus log($h_{a0}/h_a$) for the first and second-order schemes and they are compared to the theoretical order (we denote by $h_a$ the average edge length and $h_{a0}$ the average edge length of the coarser mesh). Notice that in this test case, the errors due to the space and time discretization are combined, this explains why the theoretical orders of convergence are not exactly obtained. Moreover, the boundary conditions (inflow prescribed) play an important role and since their numerical treatment is only at the first order in space, this also explains the difference between the theoretical and observed orders of convergence. With the mesh having 2781 nodes, we have tested the influence of the number of layers, see Fig. 6-(c). When the numbers of layers increase, we recover the analytical velocity profile.
+
+Figure 6: Analytical solution given in prop. 6.2 : (a) slice of the fluid domain for $y = 0.5$ m and velocity field at time $T = 0.5$ s, (b) convergence curve towards the reference solution, first order (space and time) and second order extension (space and time) schemes. The first and second order theoretical curves correspond to the dashed lines. (c) Analytical horizontal velocity $u$ along $z$ at abscissa $x = L/2$ m and time $T$ and its simulated values for different numbers of layers.
+
+## 6.3 Simulation of a tsunami
+
+In this section, we test our discrete model in the case of a real tsunami propagation for which field measurements are available (free surface variations recorded by buoys). Even if in such cases, involving long wave propagation, 2d shallow water models can be used instead of 3d description, and we test here the capacity of our model and of the numerical procedure to handle this complex situation.
+
+Simulation of tsunami waves generated by earthquakes is very important in Earth science for hazard assessment and for recovering earthquake characteristics. Indeed, tsunami waves can be analyzed to recover the earthquake source that generated the tsunami and are now classically used in joint inversion methods. It has been shown that tsunami waves provide strong constraints on the spatial distribution of the source, especially in the case of shallow slip [36]. In some cases, far-field tsunami gauges may help constrain the earthquake source process even though they are affected by the compressibility of the water column and of the Earth [36, 54].

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+The 2014/04/01 Iquique earthquake struck off the coast of Chile at 20:46 local time (23:46 UTC), with a moment magnitude of 8.1. The epicenter of the earthquake was approximately 95 kilometers (59 mi) northwest of Iquique, as shown in Fig. 7.
+
+We have carried out simulations of the tsunami induced by the earthquake using
+
+* a topography obtained from the National Oceanic and Atmospheric Administration (NOAA, [47]) using the ETOPO1 data (1-arc minute global relief model),
+
+* an unstructured mesh whose dimensions – a square of 2224.2 km² – correspond to the domain covered by Fig. 7,
+
+* a source corresponding to the seafloor displacement induced by the earthquake (Fig. 7). This source is obtained by computing the 3D final displacements of the seafloor generated by the earthquake coseismic slip. This coseismic slip has been itself retrieved by inversion of numerous geodetic and seismic data, according to the model determined by [51]. The source is activated at time $t_0$, just after the earthquake occurrence ($t_0$ is here 2014/04/01,23h47mn25s)
+
+We did not consider here the Coriolis force, the tides and the ocean currents. The results shown in Fig. 8 have been obtained with a mesh containing 545821 nodes and 5 layers (computation time was 35 minutes with a Mac book air 1.7 GHz Intel core i7). We compare the numerical solutions – provided by the first order scheme (space and time) and the second order scheme (space and time) – with the DART measurements (obtained from the NOAA website http://www.ndbc.noaa.gov/dart.shtml). A series of simulations have been performed using several meshes and we present “converged” results in the sense that a finer mesh would give the same results. This is illustrated in Fig. 8-(d), where we plot the simulation results obtained with three meshes having respectively 311687 nodes (coarse mesh), 545821 nodes (fine mesh), and 985327 nodes (very fine mesh): the curves corresponding to the fine (cyan) and very fine (blue) curves are very similar.
+
+Fig. 8-(a),(b),(c) shows that the second order scheme significantly improves the results both for the amplitude and phase of the water waves. The second order scheme is able to very accurately reproduce the shape of the first wave at the closest DART buoy 32401, located at 287 km from the epicenter. The two following peaks in the waveform are quite well reproduced up to about $1.755 \times 10^5$ s. This is also the case at the DART buoy 32402, located 853 km from the epicenter. The arrival time of the first wave is very well reproduced at the three DART buoys, slightly better with the second order scheme. At the most distant buoy 32412 (1650 km from the source), the first order scheme is not able to reproduce the recorded wave. The second order scheme reproduces the first wave quite well but not the rest of the waveform, possibly due to Earth curvature effects that are not taken into account here. Globally, the low frequency content of the signals is better explained by the model than the high frequency fluctuations. These high frequency fluctuations may be related to effects not accounted for here, such as spatio-temporal heterogeneity of the real source, small wavelength fluctuation of the topography, and possibly non-hydrostatic effects [26, 1].
+
+## 6.4 Monai valley benchmark
+
+In 2004, as part of a workshop organized by the US National Science Foundation, an experiment has been set up, reproducing the impact of a tsunami wave on the shore of the Okushiri island, in the Monai village area. The objective of the experiment was to provide a set of well organized data, reproducing the 1993 tsunami event, to validate numerical codes for Tsunami simulation [42, 39].
+
+This test case has been simulated by various numerical tools, it is well suited to test the numerical treatment of wet/dry interfaces. We reproduce hereafter the results obtained with Freshkiss3d [49].
+
+The geometrical domain of 5.448 m × 3.402 m is depicted over Fig. 9 where the bathymetry and the free surface elevation at time $t = 16$ s are presented. The topography, the input wave

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+Figure 7: (Left) Location of the zone of interest, located offshore Northern Chile; (Center) Bathymetric map showing the earthquake epicenter (red cone) and the location of the three DART buoys (black boxes); (Right) Vertical displacement (in centimeter) of the topography due to the earthquake. Horizontal displacements (not shown here) are also taken into account in the simulation.
+
+Figure 8: a), b) c) : Comparison between the sea level variations recorded by the 3 DART buoys and the corresponding simulations (1st and 2nd order schemes). d) Effect of the mesh size on the simulation accuracy for DART buoy 32401. In a), b), c), d), observations have been detided using a low pass Butterworth filter (order 4 and cutoff frequency of 4 hours), and the simulated waveforms have been filtered with the same Butterworth filter.

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+# 1 Introduction
+
+In this paper we present layer-averaged Euler and Navier-Stokes models for the numerical simulation of incompressible free surface flows over variable topographies. We are mainly interested in applications to geophysical water flows such as tsunamis, lakes, rivers, estuarine waters, hazardous flows in the context either of advection dominant flows or of wave propagation.
+
+The simulation of these flows requires stable, accurate, conservative schemes able to sharply resolve stratified flows, to handle efficiently complex topographies and free surface deformations, and to capture robustly wet/dry fronts. In addition, the application to realistic three-dimensional problems demands efficient methods with respect to computational cost. The present work is aimed at building a simulation tool endowed with these properties.
+
+Due to computational issues associated with the free surface Navier-Stokes or Euler equations, the simulations of geophysical flows are often carried out with shallow water type models of reduced complexity. Indeed, for vertically averaged models such as the Saint-Venant system [11], efficient and robust numerical techniques (relaxation schemes [19], kinetic schemes [45, 4], ...) are available and avoid to deal with moving meshes. In order to describe and simulate complex flows where the velocity field cannot be approximated by its vertical mean, multilayer models have been developed [2, 6, 7, 18, 29, 28, 44, 53, 20]. Unfortunately these models are physically relevant for non miscible fluids. In [34, 9, 8, 48, 24, 33], some authors have proposed a simpler and more general formulation for multilayer model with mass exchanges between the layers. The obtained model has the form of a conservation law with source terms and presents remarkable differences with respect to classical models for non miscible fluids. In the multilayer approach with mass exchanges, the layer partition is merely a discretization artefact, and it is not physical. Therefore, the internal layer boundaries do not necessarily correspond to isopycnic surfaces. A critical distinguishing feature of our model is that it allows fluid circulation between layers. This changes dramatically the properties of the model and its ability to describe flow configurations that are crucial for the foreseen applications, such as recirculation zones.
+
+Compared to previous works of some of the authors [9, 8, 24], that handled only the 2d configurations, this paper deals with the 3d case on unstructured meshes reinforcing the need of efficient numerical schemes. The key points of this paper are the following
+
+* A formulation of the 3d Navier-Stokes system under the form of a set of conservation laws with source terms on a fixed 2d domain.
+
+* Compared to previous works of some of the authors [9, 8], we propose a complete kinetic interpretation of the model allowing to derive a robust and accurate numerical scheme. Notice that the kinetic interpretation is valid for the implicit treatment of the vertical exchange terms arising in the multilayer description.
+
+* Choosing a Newtonian rheology for the fluid, we propose energy-consistent – at the continuous and discrete levels – models extending previous results [24] in the 3d context.
+
+* Compared to 2d $(x, z)$ situations, the 3d approximation with unstructured meshes raises new difficulties (boundary conditions, computational costs, upwinding,...). The fact that the proposed strategy is also efficient in 3d proves – to some extent – its applicability. We give a complete description of all the ingredients of the numerical scheme. Even if some parts have been already published in 2d, the objective is to have a self-contained paper for 3d applications.
+
+* The numerical approximation of the 3d Navier-Stokes system is endowed with strong stability properties (consistency, well-balancing, positivity of the water depth, wet/dry interfaces treatment,...).

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+and the time series of surface elevation at different gauges are available here [43]. For the three
+gauges located respectively at (x=4.521, y=1.196), (x=4.521, y=1.696), and (x=4.521, y=2.196),
+we compare the simulation results and the experimental data, see Fig. 10. The unstructured mesh
+used has 8 layers and 35 000 nodes for the horizontal mesh (corresponding to a mean edge length of
+0.028 m), notice that it is significantly coarser than the recommended grid sizes of Δx = Δy = 0.014
+m. Since the test case corresponds to a shallow water flow, the results we obtain are similar to those
+provided with the resolution of the classical Saint-Venant system, see (Clawpack [31], Hysea [40]...).
+
+Figure 9: (a) Bathymetry profile for the experimental setup, the three red cones represent the location of the three gauges and (b) free surface elevation at time t = 16 s.
+
+Figure 10: Comparison between the free surface variations recorded by the 3 gauges and the corresponding simulations (2^nd^ order schemes in space and time).
+
+**6.5 Hydrodynamics in a raceway**
+
+In this section, we test our model in the case of raceways used as High Rate Algal Ponds developed
+to produce microalgae biomass [32] or treat wastewater as enhanced stabilization ponds.
+
+**6.5.1 Experimental measurements**
+
+Raceways are annular shaped ponds where the water is mixed with a paddlewheel, see Fig. 11-(a).
+In this experiment, the raceway has a length of 4.2 m and a width of 1.2 m with perfect circular
+shape at each extremity. The height of water is 0.5 m for a total volume of water circulated of 2.37
+m³, see Fig. 11-(b). The paddlewheel has a diameter of 1.36 m and a width of 0.5 m with 6 blades
+equally distributed. Based on plastic material, blades are reinforced by two circular plates at each
+side with holes in between each blade in order to avoid air trapping. The paddlewheel was placed at

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+the beginning of the straight part of the raceway (rotating axe at 0.66 m after the end of the curve) and a depth of 0.2 m leaving 0.3 m between the downiest part of the paddlewheel and the bottom of the raceway. The rotation frequency of the paddlewheel is maintained in order to obtain a speed at the circumference of 0.6 m·s⁻¹.
+
+In order to measure the water velocities at different localized points of the pond, a correlation wedge flow sensor from NIVUS (model POA-V2XXK) was used. It measures continuously the speed in a window of 5° (inclination of the measure is 40°) of 16 layers from the bottom until a maximum height of 1 m. The accuracy alleged by the supplier is 0.5% for speed between 0.05 to 0.5 m·s⁻¹ and 1% over (max 6 m·s⁻¹). Hence, for the points $A_{in}$, $A_{out}$, $B_{in}$, $B_{out}$, $C_{in}$, $C_{mid}$, $C_{out}$, $P_{in}$, $P_{out}$, $Q_{in}$, $Q_{out}$, $R_{in}$ and $R_{out}$ depicted over Fig. 11-(b) and having, in the plane (O, x, y), the coordinates (given in meter)
+
+* $A_{in}: (0.75, 0.11), A_{out}: (0.5, 0.51)$
+
+* $B_{in}: (1.75, 0.11), B_{out}: (1.5, 0.51)$
+
+* $C_{in}: (2.75, 0.11), C_{mid}: (2.5, 0.31), C_{out}: (2.5, 0.51)$
+
+* $P_{in}: (2.75, -0.11), P_{out}: (2.5, -0.51)$
+
+* $Q_{in}: (1.75, -0.11), Q_{out}: (1.5, -0.51)$
+
+* $R_{in}: (0.75, -0.11), R_{out}: (0.5, -0.51)$
+
+we can measure the horizontal velocities at several elevations from the bottom to the free surface.
+
+Such experimental measurements allow to describe finely the hydrodynamic regime within the raceway. The measured average horizontal velocity is 0.21 m·s⁻¹. A gradient of velocity can be observed from the bottom to the top of water as well as from the inner to the outer border. A dead zone is observed at 2.5 m close to the outer border ($B_{out}$). Profile is much more linear in the second straight part (return) with an acceleration of water close to the outer border ($P_{out}$) and a deceleration in the inner border with a dead zone after the curve close to the inner border ($P_{in}$). Finally, an ascendant flow is observed at the beginning of the second straight part ($P_{out}$) with high speed coming from the bottom to the top of the profile along the outer border ($Q_{out}$).
+
+### 6.5.2 Simulation results
+
+The simulation has been carried out with a mesh having 3330 nodes and 25 layers. Starting from a raceway at rest, after 50 seconds a stationary regime is reached – especially far from the paddlewheel. The viscosity of the fluid is $\nu = 0.005$ m²·s⁻¹ and the bottom friction $\kappa = 0.002$ m²·s⁻¹. The paddlewheel and its modeling in the multilayer framework is described in [12] and not presented here.
+
+A global view of the hydrodynamics in the raceway at time $T = 50$ seconds is given over Fig. 12 and the comparisons between the simulation results and the experimental measurements are given over Fig. 13. Figure 13 enables to formulate three main comments:
+
+* the velocity field in the raceway is really 3d in the sense that two closed points can have very different velocity fields, see e.g. points $A_{out}$ and $A_{in}$ or $P_{out}$ and $P_{in}$.
+
+* The results are in good agreement with the measurements.
+
+* At some points of the raceway ($A_{in}$ near the paddlewheel or $C_{out}$ near the turn), the non-hydrostatic effects can be significant and this can explain the discrepancy between the simulations and the measurements.

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+Figure 11: (a) A typical raceway for cultivating microalgae, notice the paddlewheel which mixes the culture suspension, picture from INRA (ANR Symbiose project) and (b) geometry of the experimental raceway, location of the paddlewheel and position of the sensors.
+
+Figure 12: The raceway with the simulated velocity field $u$ (upper view).