samples/texts/1513018/page_24.md

where

$${\mathcal{F}}_{i,j}^{\ast }=F(U_{i,j}^{\ast },U_{j,i}^{\ast },{\mathbf{n}}_{i,j}),{\mathcal{S}}_{i,j}^{\ast }=S(U_i,U_{i,j}^{\ast },{\mathbf{n}}_{i,j})=\left(\begin{array}{c}0\\ \frac{g}{2}{l}_1(h_{i,j}^{\ast 2}-h_i^2){\mathbf{n}}_{i,j}\\ :\\ \frac{g}{2}{l}_N(h_{i,j}^{\ast 2}-h_i^2){\mathbf{n}}_{i,j}\end{array}\right),(93)\backslash mathcal\{F\}\_\{i,j\}^*\; =\; F(U\_\{i,j\}^*,\; U\_\{j,i\}^*,\; \backslash mathbf\{n\}\_\{i,j\}),\; \backslash quad\; \backslash mathcal\{S\}\_\{i,j\}^*\; =\; S(U\_i,\; U\_\{i,j\}^*,\; \backslash mathbf\{n\}\_\{i,j\})\; =\; \backslash begin\{pmatrix\}\; 0\; \backslash \backslash \; \backslash frac\{g\}\{2\}\; l\_1\; (h\_\{i,j\}^\{*2\}\; -\; h\_i^2)\; \backslash mathbf\{n\}\_\{i,j\}\; \backslash \backslash \; :\; \backslash \backslash \; \backslash frac\{g\}\{2\}\; l\_N\; (h\_\{i,j\}^\{*2\}\; -\; h\_i^2)\; \backslash mathbf\{n\}\_\{i,j\}\; \backslash end\{pmatrix\},\; \backslash quad\; (93)$$

with

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

$$f_{\alpha ,i}^{n+1-}=f_{\alpha ,i}^{n+1\mathrm{/}2-}+\mathrm{\Delta }{t}^n(N_{\alpha +1\mathrm{/}2,i}^{\ast ,n+1-}-N_{\alpha -1\mathrm{/}2,i}^{\ast ,n+1-}),(96)f\_\{\backslash alpha,i\}^\{n+1-\}\; =\; f\_\{\backslash alpha,i\}^\{n+1/2-\}\; +\; \backslash Delta\; t^n\; (N\_\{\backslash alpha+1/2,i\}^\{*,n+1-\}\; -\; N\_\{\backslash alpha-1/2,i\}^\{*,n+1-\}),\; \backslash quad\; (96)$$

where

$${\theta }_{\alpha ,i,j}=\left(\begin{array}{c}\xi -u_{\alpha ,i}\\ \gamma -v_{\alpha ,i}\end{array}\right)\mathrm{.}{n}_{i,j}\mathrm{.}\backslash theta\_\{\backslash alpha,i,j\}\; =\; \backslash begin\{pmatrix\}\; \backslash xi\; -\; u\_\{\backslash alpha,i\}\; \backslash \backslash \; \backslash gamma\; -\; v\_\{\backslash alpha,i\}\; \backslash end\{pmatrix\}\; .\; n\_\{i,j\}.$$

For the exchange terms, by analogy with (89) we define

$$N_{\alpha +1\mathrm{/}2,i}^{\ast ,n+1-}(\xi ,\gamma )=\frac{G_{\alpha +1\mathrm{/}2,i}^{\ast }}{h_i}{f}_{\alpha +1\mathrm{/}2,i}^{n+1-},(97)N\_\{\backslash alpha+1/2,\; i\}^\{*,n+1-\}(\backslash xi,\; \backslash gamma)\; =\; \backslash frac\{G\_\{\backslash alpha+1/2,\; i\}^\{*\}\}\{h\_i\}\; f\_\{\backslash alpha+1/2,\; i\}^\{n+1-\},\; \backslash quad\; (97)$$

and using (65) we get

$$G_{\alpha +1\mathrm{/}2,i}^{\ast }=-\frac{1}{\mathrm{\mid }{C}_i\mathrm{\mid }}\sum _{k=1}^N(\sum _{p=1}^{\alpha }{l}_p-1_{k\le \alpha })\sum _{j\in K_i}{L}_{i,j}{\int }_{{\mathbb{R}}^2}(M_{k,i,j}^{\ast }{\zeta }_{i,j}{1}_{{\zeta }_{i,j}\ge 0}+M_{k,i,j}^{\ast }{\zeta }_{i,j}{1}_{{\zeta }_{i,j}\le 0})d\xi d\gamma \mathrm{.}G\_\{\backslash alpha+1/2,\; i\}^*\; =\; -\; \backslash frac\{1\}\{|C\_i|\}\; \backslash sum\_\{k=1\}^\{N\}\; \backslash left(\; \backslash sum\_\{p=1\}^\{\backslash alpha\}\; l\_p\; -\; 1\_\{k\; \backslash le\; \backslash alpha\}\; \backslash right)\; \backslash sum\_\{j\; \backslash in\; K\_i\}\; L\_\{i,j\}\; \backslash int\_\{\backslash mathbb\{R\}^2\}\; \backslash left(\; M\_\{k,i,j\}^*\; \backslash zeta\_\{i,j\}\; 1\_\{\backslash zeta\_\{i,j\}\; \backslash ge\; 0\}\; +\; M\_\{k,i,j\}^*\; \backslash zeta\_\{i,j\}\; 1\_\{\backslash zeta\_\{i,j\}\; \backslash le\; 0\}\; \backslash right)\; d\backslash xi\; d\backslash gamma.$$

It is easy to see that in the previous formula, we have the moment relations

$${\int }_{{\mathbb{R}}^2}(M_{\alpha ,i}-M_{\alpha ,i,j}^{\ast }){\theta }_{\alpha ,i,j}d\xi d\gamma =0,(98)\backslash int\_\{\backslash mathbb\{R\}^2\}\; (M\_\{\backslash alpha,i\}\; -\; M\_\{\backslash alpha,i,j\}^*)\; \backslash theta\_\{\backslash alpha,i,j\}\; d\backslash xi\; d\backslash gamma\; =\; 0,\; \backslash quad\; (98)$$

$${\int }_{{\mathbb{R}}^2}\left(\genfrac{}{}{0px}{}{\xi }{\gamma }\right)(M_{\alpha ,i}-M_{\alpha ,i,j}^{\ast }){\theta }_{\alpha ,i,j}d\xi d\gamma =\frac{g}{2}{l}_{\alpha }(h_{i,j}^{\ast 2}-h_i^2)n_{i,j},(99)\backslash int\_\{\backslash mathbb\{R\}^2\}\; \backslash binom\{\backslash xi\}\{\backslash gamma\}\; (M\_\{\backslash alpha,i\}-M\_\{\backslash alpha,i,j\}^*)\; \backslash theta\_\{\backslash alpha,i,j\}\; d\backslash xi\; d\backslash gamma\; =\; \backslash frac\{g\}\{2\}\; l\_\backslash alpha\; (h\_\{i,j\}^\{*2\}\; -\; h\_i^2)\; n\_\{i,j\},\; \backslash quad\; (99)$$

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 $\alpha, i$,

$$\mathrm{\mid }\xi \mathrm{\mid }\ge v_m\text{ or }\mathrm{\mid }\gamma \mathrm{\mid }\ge v_m\Rightarrow M(U_{\alpha ,i},\xi ,\gamma )=0.(100)|\backslash xi|\; \backslash geq\; v\_m\; \backslash text\{\; or\; \}\; |\backslash gamma|\; \backslash geq\; v\_m\; \backslash Rightarrow\; M(U\_\{\backslash alpha,i\},\; \backslash xi,\; \backslash gamma)\; =\; 0.\; \backslash quad\; (100)$$