samples/texts/1513018/page_23.md

(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

$$\mathrm{\parallel }(I_N+\mathrm{\Delta }t{G}_{N,i}{)}^{-t}T{\mathrm{\parallel }}_{\mathrm{\infty }}\le \mathrm{\parallel }T{\mathrm{\parallel }}_{\mathrm{\infty }}\mathrm{.}\backslash |(I\_N\; +\; \backslash Delta\; t\; G\_\{N,i\})^\{-t\}\; T\backslash |\_\{\backslash infty\}\; \backslash le\; \backslash |T\backslash |\_\{\backslash infty\}.$$

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

$$\frac{\mathrm{\mid }{G}_{\alpha +1\mathrm{/}2,i}\mathrm{\mid }}{h_i},\backslash frac\{|G\_\{\backslash alpha+1/2,i\}|\}\{h\_i\},$$

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

$${\mathbf{I}}_N+\mathrm{\Delta }t{\mathbf{G}}_{N,i}=({\mathbf{I}}_N+\mathrm{\Delta }t{\mathbf{G}}_{N,i}^d)({\mathbf{I}}_N-({\mathbf{I}}_N+\mathrm{\Delta }t{\mathbf{G}}_{N,i}^d{)}^{-1}(-\mathrm{\Delta }t{\mathbf{G}}_{N,i}^{nd})),\backslash mathbf\{I\}\_N\; +\; \backslash Delta\; t\; \backslash mathbf\{G\}\_\{N,i\}\; =\; (\backslash mathbf\{I\}\_N\; +\; \backslash Delta\; t\; \backslash mathbf\{G\}\_\{N,i\}^d)\; (\backslash mathbf\{I\}\_N\; -\; (\backslash mathbf\{I\}\_N\; +\; \backslash Delta\; t\; \backslash mathbf\{G\}\_\{N,i\}^d)^\{-1\}\; (-\backslash Delta\; t\; \backslash mathbf\{G\}\_\{N,i\}^\{nd\})),$$

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

$$({\mathbf{I}}_N+\mathrm{\Delta }t{\mathbf{G}}_{N,i}{)}^{-1}=\sum _{k=0}^{\mathrm{\infty }}{J}_{N,i}^k,(\backslash mathbf\{I\}\_N\; +\; \backslash Delta\; t\; \backslash mathbf\{G\}\_\{N,i\})^\{-1\}\; =\; \backslash sum\_\{k=0\}^\{\backslash infty\}\; J\_\{N,i\}^k,$$

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

$$({\mathbf{I}}_N+\mathrm{\Delta }t{\mathbf{G}}_{N,i}{)}^t\mathbf{1}=\mathbf{1},(\backslash mathbf\{I\}\_N\; +\; \backslash Delta\; t\; \backslash mathbf\{G\}\_\{N,i\})^t\; \backslash mathbf\{1\}\; =\; \backslash mathbf\{1\},$$

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

$$({\mathbf{I}}_N+\mathrm{\Delta }t{\mathbf{G}}_{N,i}{)}^{-t}\mathbf{T}\le ({\mathbf{I}}_N+\mathrm{\Delta }t{\mathbf{G}}_{N,i}{)}^{-t}\mathbf{1}\mathrm{\parallel }\mathbf{T}{\mathrm{\parallel }}_{\mathrm{\infty }}=\mathbf{1}\mathrm{\parallel }\mathbf{T}{\mathrm{\parallel }}_{\mathrm{\infty }},(\backslash mathbf\{I\}\_N\; +\; \backslash Delta\; t\; \backslash mathbf\{G\}\_\{N,i\})^\{-t\}\; \backslash mathbf\{T\}\; \backslash le\; (\backslash mathbf\{I\}\_N\; +\; \backslash Delta\; t\; \backslash mathbf\{G\}\_\{N,i\})^\{-t\}\; \backslash mathbf\{1\}\; \backslash |\; \backslash mathbf\{T\}\; \backslash |\_\{\backslash infty\}\; =\; \backslash mathbf\{1\}\; \backslash |\; \backslash mathbf\{T\}\; \backslash |\_\{\backslash infty\},$$

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

$$U_i^{n+1\mathrm{/}2}=U_i^n-\sum _{j\in K_i}{\sigma }_{i,j}{\mathcal{F}}_{i,j}^{\ast }-{\sigma }_i{\mathcal{F}}_{i,e}+\sum _{j\in K_i}{\sigma }_{i,j}{S}_{i,j}^{\ast },(92)U\_i^\{n+1/2\}\; =\; U\_i^n\; -\; \backslash sum\_\{j\; \backslash in\; K\_i\}\; \backslash sigma\_\{i,j\}\; \backslash mathcal\{F\}\_\{i,j\}^*\; -\; \backslash sigma\_i\; \backslash mathcal\{F\}\_\{i,e\}\; +\; \backslash sum\_\{j\; \backslash in\; K\_i\}\; \backslash sigma\_\{i,j\}\; S\_\{i,j\}^*,\; \backslash quad\; (92)$$