## 一种维持Saint-Venant方程组移动稳态解的中心格式

1 武汉大学数学与统计学院 武汉 430072

2 国防科技大学文理学院 长沙 410073

## Moving-Water Equilibria Preserving Central Scheme for the Saint-Venant System

Luo Yiming,1, Li Dingfang,1, Liu Min,1, Dong Jian,2

1 School of Mathematics and Statistics, Wuhan University, Wuhan 430072

2 College of Liberal Arts and Sciences, National University of Defense Technology, Changsha 410073

 基金资助: 国家重点研发计划项目.  2017YFC0405901

 Fund supported: the National Key Research and Development Project.  2017YFC0405901

Abstract

In this paper, we propose a second-order unstaggered central finite volume scheme for the Saint-Venant system. Classical central scheme can preserve still-water steady state solution by reconstructing conservative variables and the water level, but generates enormous numerical oscillation when considering moving-water steady state. We choose to reconstruct conservative variables and the energy, and design a new discretization of the source term to preserve moving-water equilibria and capture its small perturbations. In the end, several classical numerical experiments are performed to verify the proposed scheme which is convergent, well-balanced and robust.

Keywords： Saint-Venant system ; Unstaggered central scheme ; Moving-water equilibria

Luo Yiming, Li Dingfang, Liu Min, Dong Jian. Moving-Water Equilibria Preserving Central Scheme for the Saint-Venant System. Acta Mathematica Scientia[J], 2022, 42(3): 891-903 doi:

## 1 引言

Saint-Venant方程组由Saint-Venant于1871年提出[1], 至今仍在浅水流、涌浪传播、水库和明渠流等问题的数值求解中发挥着重要作用. 在一维情形下, 其向量形式可表示为

$$${{{U}}_t} + {{F}}{({{U}})_x} = {{S}}({{U}}),$$$

$$${{U}} = \left( \begin{array}{c} h\\ q \end{array} \right), {\quad} {{F}}\left( {{U}} \right) = \left( \begin{array}{c} q\\ { } h{u^2} + \frac{g}{2}{h^2} \end{array} \right), {\quad} {{S}}\left( {{U}} \right) = \left( \begin{array}{c} 0\\ - gh{B_x} \end{array} \right).{\nonumber}$$$

$h\left( {x, t} \right)$表示水的深度, $u\left( {x, t} \right)$表示速度, $q\left( {x, t} \right) = h\left( {x, t} \right)u\left( {x, t} \right)$表示流量, $g$是重力加速度常数, $B\left( x \right)$表示底部地形. 初值条件$h\left( {x, 0} \right)$, $q\left( {x, 0} \right)$已知, 若求解区域并非整个实数域, 还需添加适当的边值条件.

$$$q \equiv const, {\quad} E: = \frac{{{u^2}}}{2} + g\left( {h + B} \right) \equiv const.{\nonumber}$$$

Nessyahu和Tadmor于1990年提出了用于求解双曲守恒律问题的二阶交错中心格式[2], 相较于迎风格式, 其优势在于避免了求解黎曼问题, 但于相邻时间层, 数值解会在交错单元与非交错单元之间来回更替, 导致了复杂的边界条件设置. 为此, Touma提出了稳健的二阶非交错中心格式[3], 并将其应用于Saint-Venant方程组, 提出了保正的能维持静稳态解的格式[4], 但在干湿界面处, “保正性”与“和谐性”无法同时得到满足. Dong和Li[5]通过重构守恒变量及水位值, 且在干湿界面处构造关于水位的映射, 使格式在干湿界面处经过“保正性”处理后仍可维持静稳态解. 然而在用中心格式求解移动稳态问题时, 在底部地形变化处(${B_x} \ne 0$) 流量与能量会产生较为明显的虚假振荡.

### 2.1 经典的非交错中心格式

(1) 向前投影

(2) 预测步

$\begin{eqnarray} && \int_{{x_i}}^{{x_{i + 1}}} {{{U}}\left( {x, {t^{n + 1}}} \right)} {\rm d}x - \int_{{x_i}}^{{x_{i + 1}}} {{{U}}\left( {x, {t^n}} \right)} {\rm d}x + \int_{{t^n}}^{{t^{n + 1}}} {{{F}}\left( {{{U}}\left( {{x_{i + 1}}, t} \right)} \right)} {\rm d}t {}\\ & &- \int_{{t^n}}^{{t^{n + 1}}} {{{F}}\left( {{{U}}\left( {{x_i}, t} \right)} \right)} {\rm d}t = \int_{{x_i}}^{{x_{i + 1}}} {\int_{{t^n}}^{{t^{n + 1}}} {{{S}}\left( {{U}} \right)} } {\rm d}t{\rm d}x. \end{eqnarray}$

${t^{n + 1}}$时刻${{U}}\left( {x, {t^{n + 1}}} \right)$也通过分片线性函数来近似

$$$\overline {{U}} _{i + \frac{1}{2}}^{n + 1} = \overline {{U}} _{i + \frac{1}{2}}^n - \frac{{\Delta t}}{{\Delta x}}\left( {{{F}}\left( {\overline {{U}} _{i + 1}^{n + \frac{1}{2}}} \right) - {{F}}\left( {\overline {{U}} _i^{n + \frac{1}{2}}} \right)} \right) + \frac{1}{{\Delta x}}\int_{{x_i}}^{{x_{i + 1}}} {\int_{{t^n}}^{{t^{n + 1}}} {{{S}}\left( {{U}} \right)} } {\rm d}t{\rm d}x.$$$

$$$\overline {{U}} _i^{n + \frac{1}{2}} = \overline {{U}} _i^n + \frac{{\Delta t}}{2}\left( { - {{\left( {{{{F}}_i}} \right)}^\prime } + {{S}}_i^n} \right).$$$

${\left( {{{{F}}_i}} \right)^\prime }$表示对通量${{F}}\left( {{U}} \right)$在点$x = {x_i}$的导数的逼近, ${{S}}_i^n \approx {{S}}\left( {{x_i}, \overline {{U}} _i^n} \right)$表示源项的逼近. 为使格式具有稳定性, 在时间步长的选取上应满足CFL条件

(3) 向后投影

### 2.2 新的非交错中心格式

(1) 底部函数处理

$B\left( x \right) $$x = {x_{i + \frac{1}{2}}} 处连续, 则有 {B_{i + \frac{1}{2}}} = B\left( {{x_{i + \frac{1}{2}}}} \right) . 再将 \widetilde B\left( x \right) 在非交错单元 {C_i} 的积分平均值定义为 {B_i} (2) 向前投影 为维持移动稳态解, 首先计算每个非交错单元 {C_i} 的能量 E : {{V}} = {\left( {h, q, E} \right)^{\rm T}} , 而非仅对 {{U}} 做分片线性重构, 得到 {h_{i + \frac{1}{2}}} , {q_{i + \frac{1}{2}}}$$ {E_{i + \frac{1}{2}}}$, 并定义${\widehat E_{i + \frac{1}{2}}}$如下

$$${\widehat E_{i + \frac{1}{2}}}: = \frac{{{u_{i + \frac{1}{2}}}^2}}{2} + g\left( {{h_{i + \frac{1}{2}}} + {{\overline B }_{i + \frac{1}{2}}}} \right),$$$

$$${\widehat E_i}: = \frac{{{u_i}^2}}{2} + g\left( {{h_i} + {B_i}} \right).$$$

$$$\frac{{{q_i}^2}}{{2{h^2}}} + g\left( {h + {B_i}} \right) - {E_i} = 0.$$$

(4) 非线性方程求解

(Ⅰ) 如果$q = 0$, 方程退化为线性方程, $h = \frac{E}{g} - B$.

(Ⅱ) 如果$q \ne 0$, 注意到$\varphi \left( h \right)$为凸函数且在${h_0} = \sqrt[3]{{\frac{{{q^2}}}{g}}}$处达到极小值. 如果$\varphi \left( {{h_0}} \right) > 0$, 方程无正根, 此时无需改变重构值(此情形在数值算例中很少出现); 如果$\varphi \left( {{h_0}} \right) = 0$, 方程只有一个正根: $h = {h_0}$; 如果$\varphi \left( {{h_0}} \right) < 0$, 通过公式法得到方程的三个根

### 2.3 源项的"和谐性"离散

(1) 源项离散

$$${\left( {{{{F}}_i}} \right)^\prime }: = \left( \begin{array}{cc} {\left( {{q_i}} \right)^\prime } \\ {h_i}{\left( {{e_i}} \right)^\prime } + {u_i}{\left( {{q_i}} \right)^\prime } \\ \end{array} \right),$$$

$$${{{S}}_i}: = {{{S}}_{i, L}} + {{{S}}_{i, R}} + {{{S}}_{i, C}}.$$$

$$${{S}}\left( {{{\overline {{U}} }_i}, {{\overline {{U}} }_{i + 1}}} \right) = \left( \begin{array}{cc} 0 \\ { } - g \cdot \frac{{{h_i} + {h_{i + 1}}}}{2} \cdot \frac{{{B_{i + 1}} - {B_i}}}{{\Delta x}} + \frac{{{h_{i + 1}} - {h_i}}}{{4\Delta x}} \cdot {\left( {{u_{i + 1}} - {u_i}} \right)^2} \\ \end{array} \right).$$$

(2) "和谐性"的证明

$$${q_i} \equiv \widehat q, {E_i} = \widehat E, \forall i.$$$

首先来证明(Ⅰ), 不妨设${\left( {{e_i}} \right)^\prime } = \theta \frac{{{e_{i + 1}} - {e_i}}}{{\Delta x}}$, 由条件(2.11), ${\left( {{q_i}} \right)^\prime } = 0$, ${\left( {{E_i}} \right)^\prime } = 0$, 代入(2.8)和(2.9)式

$$${\left( {{{{F}}_i}} \right)^\prime } = \left( \begin{array}{cc} 0 \\ { } {h_i}\theta \frac{{{e_{i + 1}} - {e_i}}}{{\Delta x}} \\ \end{array} \right), {{{S}}_i} = {{{S}}_{i, R}} = \left( \begin{array}{cc} 0 \\ { } - g{h_i}\theta \frac{{{B_{i + 1}} - {B_i}}}{{\Delta x}} \\ \end{array} \right).$$$

${\left( {{e_i}} \right)^\prime }$取其他值时同理可证.

$$${{F}}\left( {{{\overline {{U}} }_{i + 1}}} \right) - {{F}}\left( {{{\overline {{U}} }_i}} \right) = \left( \begin{array}{cc} 0 \\ { } {\widehat q^2}\left( {\frac{1}{{{h_{i + 1}}}} - \frac{1}{{{h_i}}}} \right) + \frac{1}{2}g\left( {{h_{i + 1}} - {h_i}} \right)\left( {{h_{i + 1}} + {h_i}} \right) \\ \end{array} \right).$$$

$$$g\left( {{h_{i + 1}} - {h_i}} \right) = \frac{{{{\widehat q}^2}}}{2}\left( {\frac{1}{{{h_i}^2}} - \frac{1}{{{h_{i + 1}}^2}}} \right) - g\left( {{B_{i + 1}} - {B_i}} \right).$$$

$\overline {{U}} _{i + \frac{1}{2}}^{n + 1} = \overline {{U}} _{i + \frac{1}{2}}^n$, 结论得证.

(a) 超临界流:

(b) 次临界流:

(c) 跨临界流:

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