samples/texts/1513018/page_17.md

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

$$H_{\alpha }^{\mathrm{\prime }}(M_{\alpha },\xi ,\gamma ,z_b)\frac{\mathrm{\partial }{M}_{\alpha }}{\mathrm{\partial }v}=\frac{\mathrm{\partial }}{\mathrm{\partial }v}{H}_{\alpha }(M_{\alpha },\xi ,\gamma ,z_b),H\text{'}\_\{\backslash alpha\}(M\_\{\backslash alpha\},\; \backslash xi,\; \backslash gamma,\; z\_b)\; \backslash frac\{\backslash partial\; M\_\{\backslash alpha\}\}\{\backslash partial\; v\}\; =\; \backslash frac\{\backslash partial\}\{\backslash partial\; v\}\; H\_\{\backslash alpha\}(M\_\{\backslash alpha\},\; \backslash xi,\; \backslash gamma,\; z\_b),$$

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

$$H_{\alpha }^{\mathrm{\prime }}(M_{\alpha },\xi ,\gamma ,z_b)N_{\alpha +1\mathrm{/}2}=H(N_{\alpha +1\mathrm{/}2},\xi ,\gamma ,z_b)\mathrm{.}H\text{'}\_\{\backslash alpha\}(M\_\{\backslash alpha\},\; \backslash xi,\; \backslash gamma,\; z\_b)\; N\_\{\backslash alpha+1/2\}\; =\; H(N\_\{\backslash alpha+1/2\},\; \backslash xi,\; \backslash gamma,\; z\_b).$$

So finally, Eq. (66) multiplied by $H'{\alpha}(M{\alpha}, \xi, \gamma, z_b)$ gives

$$\frac{\mathrm{\partial }{H}_{\alpha }}{\mathrm{\partial }t}+\left(\begin{array}{c}\xi \\ \gamma \end{array}\right)\cdot {\mathrm{\nabla }}_{x,y}{H}_{\alpha }-g{\mathrm{\nabla }}_{x,y}{z}_b\cdot {\mathrm{\nabla }}_{\xi ,\gamma }{H}_{\alpha }\le (\frac{{\xi }^2+{\gamma }^2}{2}+g{z}_b)(N_{\alpha +1\mathrm{/}2}-N_{\alpha -1\mathrm{/}2})\mathrm{.}\backslash frac\{\backslash partial\; H\_\{\backslash alpha\}\}\{\backslash partial\; t\}\; +\; \backslash begin\{pmatrix\}\; \backslash xi\; \backslash \backslash \; \backslash gamma\; \backslash end\{pmatrix\}\; \backslash cdot\; \backslash nabla\_\{x,y\}\; H\_\{\backslash alpha\}\; -\; g\; \backslash nabla\_\{x,y\}\; z\_b\; \backslash cdot\; \backslash nabla\_\{\backslash xi,\backslash gamma\}\; H\_\{\backslash alpha\}\; \backslash leq\; \backslash left(\; \backslash frac\{\backslash xi^2\; +\; \backslash gamma^2\}\{2\}\; +\; gz\_b\; \backslash right)\; (N\_\{\backslash alpha+1/2\}\; -\; N\_\{\backslash alpha-1/2\}).$$

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

$$K_{1,j}=1,K_{i+1,j}=\xi {\delta }_{i,j},K_{i+N+1,j+N}=\gamma {\delta }_{i,j},K\_\{1,j\}\; =\; 1,\; \backslash quad\; K\_\{i+1,j\}\; =\; \backslash xi\; \backslash delta\_\{i,j\},\; \backslash quad\; K\_\{i+N+1,j+N\}\; =\; \backslash gamma\; \backslash delta\_\{i,j\},$$

for $i, j = 1, \dots, N$ with $\delta_{i,j}$ the Kronecker symbol. Then, using Prop. 4.1, we can write

$$U={\int }_{{\mathbb{R}}^2}K(\xi ,\gamma )M(\xi ,\gamma )d\xi d\gamma ,F(U)={\int }_{{\mathbb{R}}^2}\left(\begin{array}{c}\xi \\ \gamma \end{array}\right)K(\xi ,\gamma )M(\xi ,\gamma )d\xi d\gamma ,(69)U\; =\; \backslash int\_\{\backslash mathbb\{R\}^2\}\; K(\backslash xi,\; \backslash gamma)\; M(\backslash xi,\; \backslash gamma)\; d\backslash xi\; d\backslash gamma,\; \backslash quad\; F(U)\; =\; \backslash int\_\{\backslash mathbb\{R\}^2\}\; \backslash begin\{pmatrix\}\; \backslash xi\; \backslash \backslash \; \backslash gamma\; \backslash end\{pmatrix\}\; K(\backslash xi,\; \backslash gamma)\; M(\backslash xi,\; \backslash gamma)\; d\backslash xi\; d\backslash gamma,\; \backslash qquad\; (69)$$

$$S_e(U)={\int }_{{\mathbb{R}}^2}K(\xi ,\gamma )N(\xi ,\gamma )d\xi d\gamma ,(70)S\_e(U)\; =\; \backslash int\_\{\backslash mathbb\{R\}^2\}\; K(\backslash xi,\; \backslash gamma)\; N(\backslash xi,\; \backslash gamma)\; d\backslash xi\; d\backslash gamma,\; \backslash qquad\; (70)$$

with $M(\xi, \gamma) = (M(U_1, \xi, \gamma), \dots, M(U_N, \xi, \gamma))^T$ and

$$N(\xi ,\gamma )=\left(\begin{array}{c}{N}_{3\mathrm{/}2}(\xi ,\gamma )-N_{1\mathrm{/}2}(\xi ,\gamma )\\ \mathrm{⋮}\\ N_{N+1\mathrm{/}2}(\xi ,\gamma )-N_{N-1\mathrm{/}2}(\xi ,\gamma )\end{array}\right)\mathrm{.}N(\backslash xi,\; \backslash gamma)\; =\; \backslash begin\{pmatrix\}\; N\_\{3/2\}(\backslash xi,\; \backslash gamma)\; -\; N\_\{1/2\}(\backslash xi,\; \backslash gamma)\; \backslash \backslash \; \backslash vdots\; \backslash \backslash \; N\_\{N+1/2\}(\backslash xi,\; \backslash gamma)\; -\; N\_\{N-1/2\}(\backslash xi,\; \backslash gamma)\; \backslash end\{pmatrix\}.$$

Hence, using the above notations, the layer-averaged Euler system (18)-(19) can be written under the form

$${\int }_{{\mathbb{R}}^2}K(\xi ,\gamma )(\frac{\mathrm{\partial }M(\xi ,\gamma )}{\mathrm{\partial }t}+\left(\begin{array}{c}\xi \\ \gamma \end{array}\right)\cdot {\mathrm{\nabla }}_{x,y}M(\xi ,\gamma )-g{\mathrm{\nabla }}_{x,y}{z}_b\cdot {\mathrm{\nabla }}_{\xi ,\gamma }M-N(\xi ,\gamma ))d\xi d\gamma =0.\backslash int\_\{\backslash mathbb\{R\}^2\}\; K(\backslash xi,\; \backslash gamma)\; \backslash left(\; \backslash frac\{\backslash partial\; M(\backslash xi,\; \backslash gamma)\}\{\backslash partial\; t\}\; +\; \backslash begin\{pmatrix\}\; \backslash xi\; \backslash \backslash \; \backslash gamma\; \backslash end\{pmatrix\}\; \backslash cdot\; \backslash nabla\_\{x,y\}\; M(\backslash xi,\; \backslash gamma)\; -\; g\; \backslash nabla\_\{x,y\}\; z\_b\; \backslash cdot\; \backslash nabla\_\{\backslash xi,\backslash gamma\}\; M\; -\; N(\backslash xi,\; \backslash gamma)\; \backslash right)\; d\backslash xi\; d\backslash gamma\; =\; 0.$$

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,