samples/texts/1513018/page_25.md

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,

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.

$$f_{\alpha ,i}^{n+1-}\ge 0,\mathrm{\forall }(\xi ,\gamma )\in {\mathbb{R}}^2,\mathrm{\forall }i,\mathrm{\forall }\alpha \mathrm{.}f\_\{\backslash alpha,i\}^\{n+1-\}\; \backslash ge\; 0,\; \backslash quad\; \backslash forall\; (\backslash xi,\; \backslash gamma)\; \backslash in\; \backslash mathbb\{R\}^2,\; \backslash forall\; i,\; \backslash forall\; \backslash alpha.$$

(iii) The scheme (95)-(96) is kinetic well balanced i.e. at rest

$$f_{\alpha ,i}^{n+1-}=M_{\alpha ,i},\mathrm{\forall }(\xi ,\gamma )\in {\mathbb{R}}^2,\mathrm{\forall }i,\mathrm{\forall }\alpha =1,\dots ,N\mathrm{.}(102)f\_\{\backslash alpha,i\}^\{n+1-\}\; =\; M\_\{\backslash alpha,i\},\; \backslash quad\; \backslash forall\; (\backslash xi,\; \backslash gamma)\; \backslash in\; \backslash mathbb\{R\}^2,\; \backslash forall\; i,\; \backslash forall\; \backslash alpha\; =\; 1,\; \backslash dots,\; N.\; \backslash qquad\; (102)$$

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

$$\frac{\mathrm{\Delta }{t}^n}{\mathrm{\mid }{C}_i\mathrm{\mid }}\sum _{j\in K_i}{L}_{i,j}{M}_{\alpha ,j,i}^{\ast }{\zeta }_{i,j}{1}_{{\zeta }_{i,j}\le 0}\le 0,\backslash frac\{\backslash Delta\; t^n\}\{|C\_i|\}\; \backslash sum\_\{j\; \backslash in\; K\_i\}\; L\_\{i,j\}\; M\_\{\backslash alpha,j,i\}^*\; \backslash zeta\_\{i,j\}\; 1\_\{\backslash zeta\_\{i,j\}\; \backslash le\; 0\}\; \backslash le\; 0,$$

and the HR (94) ensures $M_{\alpha,i}^* \le M_{\alpha,i}$, $\forall (\xi, \gamma) \in \mathbb{R}^2$, $\forall \alpha$ leading to

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

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

$$\frac{\mathrm{\Delta }{t}^n}{\mathrm{\mid }{C}_i\mathrm{\mid }}\sum _{j\in K_i}{L}_{i,j}({\zeta }_{i,j}{1}_{{\zeta }_{i,j}\ge 0}+{\theta }_{\alpha ,i,j}{1}_{{\theta }_{\alpha ,i,j}\ge 0})\le {\sigma }_i(\max \{\mathrm{\mid }\xi \mathrm{\mid },\mathrm{\mid }\gamma \mathrm{\mid }\}+\max \{\mathrm{\mid }\xi -u_{\alpha ,i}\mathrm{\mid },\mathrm{\mid }\gamma -v_{\alpha ,i}\mathrm{\mid }\})\le 1,\backslash frac\{\backslash Delta\; t^n\}\{|C\_i|\}\; \backslash sum\_\{j\; \backslash in\; K\_i\}\; L\_\{i,j\}\; (\backslash zeta\_\{i,j\}\; 1\_\{\backslash zeta\_\{i,j\}\; \backslash ge\; 0\}\; +\; \backslash theta\_\{\backslash alpha,i,j\}\; 1\_\{\backslash theta\_\{\backslash alpha,i,j\}\; \backslash ge\; 0\})\; \backslash le\; \backslash sigma\_i\; (\backslash max\backslash \{|\backslash xi|,\; |\backslash gamma|\backslash \}\; +\; \backslash max\backslash \{|\backslash xi\; -\; u\_\{\backslash alpha,i\}|,\; |\backslash gamma\; -\; v\_\{\backslash alpha,i\}|\backslash \})\; \backslash le\; 1,$$

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

$$M_{\alpha ,i}=M_{\alpha ,i,j}^{\ast },\mathrm{\forall }\alpha ,i,j\mathrm{.}M\_\{\backslash alpha,i\}\; =\; M\_\{\backslash alpha,i,j\}^*,\; \backslash quad\; \backslash forall\; \backslash alpha,\; i,\; j.$$

From (95)-(96), this gives (102). ■