﻿ 基于移动网格的HLL格式求解浅水波方程

# 基于移动网格的HLL格式求解浅水波方程Solving Shallow Water Wave Equation with HLL Scheme Based on Moving Grid

Abstract: In this paper, an HLL scheme based on moving grid is proposed to solve shallow water wave equations. Based on the iterative process, the moving mesh method redistributes the mesh in each iteration by using the equal distribution principle, and then updates its numerical solution by using the conservative interpolation formula, the main idea of this method is to preserve the mass conservation of the numerical solution under each redistribution step. HLL scheme is a numerical flux with good robustness and can eliminate the carbuncle. The time direction is advanced by the third- order strongly stable Runge-Kutta method, and the comparison of numerical results shows that the HLL scheme based on the moving grid has good resolution.

1. 引言

2. 控制方程

${U}_{t}+F{\left(U\right)}_{x}+G{\left(U\right)}_{y}=-S,$ (1)

$U=\left[\begin{array}{c}h\\ hu\\ hv\end{array}\right],F\left(U\right)=\left[\begin{array}{c}hu\\ h{u}^{2}+\frac{1}{2}g{h}^{2}\\ huv\end{array}\right],G\left(U\right)=\left[\begin{array}{c}hv\\ huv\\ h{v}^{2}+\frac{1}{2}g{h}^{2}\end{array}\right],$

$S=\left[\begin{array}{c}0\\ gh\frac{\partial {z}_{b}}{\partial x}+gh{S}_{{f}_{x}}\\ gh\frac{\partial {z}_{b}}{\partial y}+gh{S}_{{f}_{y}}\end{array}\right],$

${z}_{b}$ 表示底部地势函数， $gh\left(\partial {z}_{b}/\partial x\right)$$gh\left(\partial {z}_{b}/\partial y\right)$ 表示水下底部作用力沿x方向和y方向上的分力， $gh{S}_{{f}_{x}}$$gh{S}_{{f}_{y}}$ 为水下底面摩擦力沿x方向和y方向的量， ${S}_{{f}_{x}}$${S}_{{f}_{y}}$ 为x方向和y方向上的摩阻比率，有

${S}_{{f}_{x}}=\frac{u}{{K}^{2}}\frac{{\left({u}^{2}+{v}^{2}\right)}^{1/2}}{{h}^{4/3}},{S}_{{f}_{y}}=\frac{v}{{K}^{2}}\frac{{\left({u}^{2}+{v}^{2}\right)}^{1/2}}{{h}^{4/3}},$

K表示摩擦因数，通常情况取 $K=50$

3. 自适应移动网格法

$\begin{array}{l}{\alpha }_{j+\frac{1}{2},k}\left({\stackrel{\to }{z}}_{j+1,k}^{\left[v\right]}-{\stackrel{\to }{z}}_{j,k}^{\left[v+1\right]}\right)-{\alpha }_{j-\frac{1}{2},k}\left({\stackrel{\to }{z}}_{j,k}^{\left[v+1\right]}-{\stackrel{\to }{z}}_{j-1,k}^{\left[v+1\right]}\right)\\ \text{ }+{\beta }_{j,k+1/2}\left({\stackrel{\to }{z}}_{j,k+1}^{\left[v\right]}-{\stackrel{\to }{z}}_{j-1,k}^{\left[v+1\right]}\right)-{\beta }_{j,k+1/2}\left({\stackrel{\to }{z}}_{j,k}^{\left[v+1\right]}-{\stackrel{\to }{z}}_{j,k-1}^{\left[v+1\right]}\right)=0\end{array}$

${\alpha }_{j±\frac{1}{2},k}=w\left({u}_{j±\frac{1}{2},k}^{\left[v\right]}\right)=w\left(\frac{1}{2}\left({u}_{j±\frac{1}{2},k+\frac{1}{2}}^{\left[v\right]}+{u}_{j±\frac{1}{2},k-\frac{1}{2}}^{\left[v\right]}\right)\right)$

${\beta }_{j,k±\frac{1}{2}}=w\left({u}_{j,k±\frac{1}{2}}^{\left[v\right]}\right)=w\left(\frac{1}{2}\left({u}_{j+\frac{1}{2},k±\frac{1}{2}}^{\left[v\right]}+{u}_{j-\frac{1}{2},k±\frac{1}{2}}^{\left[v\right]}\right)\right)$

2) 新网格上的数值解 $\left\{{u}_{j+1/2,k+1/2}^{\left[v+1\right]}\right\}$ 通过守恒插值进行更新；

3) 重复迭代过程1) 和2) 到指定的迭代步数(3~5步)或直到 $‖{\stackrel{\to }{z}}^{\left[v+1\right]}-{\stackrel{\to }{z}}^{\left[v\right]}‖\le \epsilon$

4. HLL通量格式

${H}^{HLL}=\frac{{S}_{R}^{+}{H}_{k}\left({U}_{L}\right)-{S}_{L}^{-}{H}_{k}\left({U}_{R}\right)}{{S}_{R}^{+}-{S}_{L}^{-}}+\frac{{S}_{R}^{+}{S}_{L}^{-}}{{S}_{R}^{+}-{S}_{L}^{-}}\Delta U,$ (2)

${H}_{k}=\left(F,G\right)\cdot n\left(k=x,y\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}n=\left({n}_{x},{n}_{y}\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}\Delta U={U}_{R}-{U}_{L},$

${S}_{R}^{+}=\mathrm{max}\left(0,{S}_{R}\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}{S}_{L}^{-}=\mathrm{min}\left(0,{S}_{L}\right),$

${S}_{L},{S}_{R}$ 分别表示左波速和右波速，它们的值分别为Jacobi矩阵 ${\stackrel{^}{A}}_{k}=\frac{\partial {H}_{k}}{\partial U}$ 特征值的最小值和最大值，即

${S}_{L}=\mathrm{min}\left({\left({q}_{n}\right)}_{L}-{c}_{L},{\stackrel{^}{q}}_{n}-\stackrel{^}{c}\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}{S}_{R}=\mathrm{max}\left({\left({q}_{n}\right)}_{R}+{c}_{R},{\stackrel{^}{q}}_{n}+\stackrel{^}{c}\right),$

${\left({q}_{n}\right)}_{L,R}={\left(u,v\right)}_{L,R}\cdot n,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\stackrel{^}{q}}_{n}=\left(\stackrel{^}{u},\stackrel{^}{v}\right)\cdot n,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\stackrel{^}{q}}_{t}=\left(\stackrel{^}{u},\stackrel{^}{v}\right)\cdot \left(-{n}_{x},{n}_{y}\right).$

${H}^{\text{HLL}}=\left\{\begin{array}{ll}{F}_{L}\cdot {n}_{x}+{G}_{L}\cdot {n}_{y}\hfill & 0\le {S}_{L},\hfill \\ \frac{{S}_{R}^{+}{H}_{k}\left({U}_{L}\right)-{S}_{L}^{-}{H}_{k}\left({U}_{R}\right)}{{S}_{R}^{+}-{S}_{L}^{-}}+\frac{{S}_{R}^{+}{S}_{L}^{-}}{{S}_{R}^{+}-{S}_{L}^{-}}\Delta U,\hfill & {S}_{L}\le 0\le {S}_{R},\hfill \\ {F}_{R}\cdot {n}_{x}+{G}_{R}\cdot {n}_{y},\hfill & 0\ge {S}_{R}.\hfill \end{array}$ (3)

$\begin{array}{l}{U}^{\left(1\right)}={U}^{n}+\Delta tL\left({U}^{n}\right),\\ {U}^{\left(2\right)}=\frac{3}{4}{U}^{n}+\frac{1}{4}{U}^{\left(1\right)}+\frac{\Delta t}{4}L\left({U}^{\left(1\right)}\right),\\ {U}^{n+1}=\frac{1}{3}{U}^{n}+\frac{2}{3}{U}^{\left(2\right)}+\frac{2}{3}\Delta tL\left({U}^{\left(2\right)}\right).\end{array}$ (4)

5. 数值算例

$\begin{array}{l}h\left(x,y,0\right)=\left\{\begin{array}{l}10,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\sqrt{{\left(x-25\right)}^{2}+{\left(y-25\right)}^{2}}<11\\ 1,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{others}.\end{array}\\ u\left(x,y,0\right)=v\left(x,y,0\right)=0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{z}_{b}\left(x,y\right)=0.\end{array}$

Figure 1. Grid evolution

(a) (b)

Figure 2. The cylindrical dam-break density contour. (a) The fixed grid diagram; (b) The moving grid diagram

$\begin{array}{l}h\left(x,y,0\right)=\left\{\begin{array}{l}2,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\sqrt{{x}^{2}+{y}^{2}}<0.5\\ 1,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{others}.\end{array}\\ u\left(x,y,0\right)=v\left(x,y,0\right)=0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{z}_{b}\left(x,y\right)=0.\end{array}$

Figure 3. Grid evolution

(a) (b)

Figure 4. The circular dam-break density contour. (a) The fixed grid diagram; (b) The moving grid diagram

$h\left(x,y,0\right)={z}_{b}\left(x,y,0\right)+\zeta \left(x,y,0\right),$

$u\left(x,y,0\right)=v\left(x,y,0\right)=0$

${z}_{b}\left(x,y\right)=1+0.01\mathrm{cos}\left(\pi x/2\right)\mathrm{cos}\left(\pi y/2\right)$

$CFL=0.45$，时间 $t=0.3$$g=0.98$，控制参数 $\alpha =0.935$，边界条件采用周期性条件。图5表示的是二维潮汐问题密度等值线图，网格演化效果不明显，通过对比可以发现两种格式没有明显差别，均对称性良好，结构清晰。

(a) (b)

Figure 5. The density contour of tidal. (a) The fixed grid diagram; (b) The moving grid diagram

6. 总结

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