samples/texts/1513018/page_21.md

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

$$\frac{\mathrm{\partial }{M}_{\alpha }}{\mathrm{\partial }t}+\left(\begin{array}{c}\xi \\ \gamma \end{array}\right)\cdot {\mathrm{\nabla }}_{x,y}{M}_{\alpha }-N_{\alpha +1\mathrm{/}2}+N_{\alpha -1\mathrm{/}2}=Q_{\alpha },\alpha \in \{1,\dots ,N\},(85)\backslash frac\{\backslash partial\; M\_\backslash alpha\}\{\backslash partial\; t\}\; +\; \backslash begin\{pmatrix\}\; \backslash xi\; \backslash \backslash \; \backslash gamma\; \backslash end\{pmatrix\}\; \backslash cdot\; \backslash nabla\_\{x,y\}\; M\_\backslash alpha\; -\; N\_\{\backslash alpha+1/2\}\; +\; N\_\{\backslash alpha-1/2\}\; =\; Q\_\backslash alpha,\; \backslash quad\; \backslash alpha\; \backslash in\; \backslash \{1,\; \backslash dots,\; N\backslash \},\; \backslash qquad\; (85)$$

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

$${\int }_{C_i}(\frac{\mathrm{\partial }{M}_{\alpha }}{\mathrm{\partial }t}+\left(\begin{array}{c}\xi \\ \gamma \end{array}\right)\cdot {\mathrm{\nabla }}_{x,y}{M}_{\alpha })dxdy\approx \mathrm{\mid }{C}_i\mathrm{\mid }\frac{\mathrm{\partial }{M}_{\alpha ,i}}{\mathrm{\partial }t}+\sum _{j\in K_i}{\int }_{{\mathrm{\Gamma }}_{i,j}}{M}_{\alpha ,i,j}dl,(86)\backslash int\_\{C\_i\}\; \backslash left(\; \backslash frac\{\backslash partial\; M\_\backslash alpha\}\{\backslash partial\; t\}\; +\; \backslash begin\{pmatrix\}\; \backslash xi\; \backslash \backslash \; \backslash gamma\; \backslash end\{pmatrix\}\; \backslash cdot\; \backslash nabla\_\{x,y\}\; M\_\backslash alpha\; \backslash right)\; dxdy\; \backslash approx\; |C\_i|\; \backslash frac\{\backslash partial\; M\_\{\backslash alpha,i\}\}\{\backslash partial\; t\}\; +\; \backslash sum\_\{j\; \backslash in\; K\_i\}\; \backslash int\_\{\backslash Gamma\_\{i,j\}\}\; M\_\{\backslash alpha,i,j\}\; dl,\; \backslash qquad\; (86)$$

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

$$M_{\alpha ,i,j}=M_{\alpha ,i}{\zeta }_{i,j}{\mathbf{1}}_{{\zeta }_{i,j}\ge 0}+M_{\alpha ,j}{\zeta }_{i,j}{\mathbf{1}}_{{\zeta }_{i,j}\le 0},M\_\{\backslash alpha,i,j\}\; =\; M\_\{\backslash alpha,i\}\backslash zeta\_\{i,j\}\backslash mathbf\{1\}\_\{\backslash zeta\_\{i,j\}\; \backslash ge\; 0\}\; +\; M\_\{\backslash alpha,j\}\backslash zeta\_\{i,j\}\backslash mathbf\{1\}\_\{\backslash zeta\_\{i,j\}\; \backslash le\; 0\},$$

with $\zeta_{i,j} = (\xi \ \gamma)^T \cdot \mathbf{n}_{i,j}$.

Therefore, the kinetic scheme applied for Eq. (85) is given by

$$f_{\alpha ,i}^{n+1\mathrm{/}{2}^{-}}=(1-\frac{\mathrm{\Delta }{t}^n}{\mathrm{\mid }{C}_i\mathrm{\mid }}\sum _{j\in K_i}{L}_{i,j}{\zeta }_{i,j}{\mathbf{1}}_{{\zeta }_{i,j}\ge 0})M_{\alpha ,i}-\frac{\mathrm{\Delta }{t}^n}{\mathrm{\mid }{C}_i\mathrm{\mid }}\sum _{j\in K_i}{L}_{i,j}{M}_{\alpha ,j}{\zeta }_{i,j}{\mathbf{1}}_{{\zeta }_{i,j}\le 0},(87)f\_\{\backslash alpha,i\}^\{n+1/2^\{-\}\}\; =\; \backslash left(1\; -\; \backslash frac\{\backslash Delta\; t^n\}\{|C\_i|\}\; \backslash sum\_\{j\; \backslash in\; K\_i\}\; L\_\{i,j\}\; \backslash zeta\_\{i,j\}\; \backslash mathbf\{1\}\_\{\backslash zeta\_\{i,j\}\; \backslash ge\; 0\}\backslash right)\; M\_\{\backslash alpha,i\}\; -\; \backslash frac\{\backslash Delta\; t^n\}\{|C\_i|\}\; \backslash sum\_\{j\; \backslash in\; K\_i\}\; L\_\{i,j\}\; M\_\{\backslash alpha,j\}\; \backslash zeta\_\{i,j\}\; \backslash mathbf\{1\}\_\{\backslash zeta\_\{i,j\}\; \backslash le\; 0\},\; \backslash qquad\; (87)$$

$$f_{\alpha ,i}^{n+1^{-}}=f_{\alpha ,i}^{n+1\mathrm{/}{2}^{-}}+\mathrm{\Delta }{t}^n(N_{\alpha +1\mathrm{/}2,i}^{n+1^{-}}-N_{\alpha -1\mathrm{/}2,i}^{n+1^{-}}),(88)f\_\{\backslash alpha,i\}^\{n+1^\{-\}\}\; =\; f\_\{\backslash alpha,i\}^\{n+1/2^\{-\}\}\; +\; \backslash Delta\; t^n\; (N\_\{\backslash alpha+1/2,i\}^\{n+1^\{-\}\}\; -\; N\_\{\backslash alpha-1/2,i\}^\{n+1^\{-\}\}),\; \backslash qquad\; (88)$$

with the exchange terms ${N_{\alpha+1/2,i}^{n+1^{-}}}_{\alpha=0}^N$ defined by

$$N_{\alpha +1\mathrm{/}2,i}^{n+1^{-}}(\xi ,\gamma )=\frac{G_{\alpha +1\mathrm{/}2,i}}{h_i}{f}_{\alpha +1\mathrm{/}2,i}^{n+1^{-}}\mathrm{.}(89)N\_\{\backslash alpha+1/2,i\}^\{n+1^\{-\}\}(\backslash xi,\; \backslash gamma)\; =\; \backslash frac\{G\_\{\backslash alpha+1/2,i\}\}\{h\_i\}\; f\_\{\backslash alpha+1/2,i\}^\{n+1^\{-\}\}.\; \backslash qquad\; (89)$$