High Order Sign Preserving Entropy Stable Schemes

Zheng Supei,, Xu Xia,, Feng Jianhu,, Jia Dou,

College of Science, Chang'an University, Xi'an 710064

 基金资助: 国家自然科学基金.  11971075陕西省自然科学基金.  2020JQ-338陕西省自然科学基金.  2019JM-243

 Fund supported: the NSFC.  11971075the NSF of Shaanxi Province.  2020JQ-338the NSF of Shaanxi Province.  2019JM-243

Abstract

Ensuring the sign preserving on entropy variable after the high-order reconstruction is fundamental in constructing the high-order entropy stable schemes. In this paper, we construct the 3rd order compact CWENO-type entropy stable schemes (Fjordholm's the 3rd order entropy conservative schemes with sign preserving compact CWENO reconstruction for the entropy variable) and prove the sign preserving on entropy variable with the 3rd order compact CWENO reconstruction. Numerical results show that the schemes can achieve third-order accuracy, and have high resolution, robustness and non-oscillation.

Keywords： Sign preserving ; Compact CWENO reconstruction ; Entropy stable schemes

Zheng Supei, Xu Xia, Feng Jianhu, Jia Dou. High Order Sign Preserving Entropy Stable Schemes. Acta Mathematica Scientia[J], 2021, 41(5): 1296-1310 doi:

1 引言

2 预备知识

$$$\begin{array}{ll} {\bf u}_t+{\bf f(u)}_x = 0, & {\forall}\ (x, t)\in {{\Bbb R}} \times {{\Bbb R}} _+, \\ {\bf u}(x, 0) = {\bf u}_0(x), & {\forall}\ x\in {{\Bbb R}} , \end{array}$$$

$$$\eta({\bf u})_t+q({\bf u})_x \le 0,$$$

$$$\frac{\rm d}{{\rm d}t}{\bf u}_i (t) = -\frac{1}{h_i}(\hat{{\bf f}}_{i+\frac{1}{2}}-\hat{{\bf f}}_{i-\frac{1}{2}}),$$$

$$$\frac{\rm d}{{\rm d}t}\eta({\bf u}_i(t))+\frac{1}{h_i}(\hat{q}_{i+\frac{1}{2}}-\hat{q}_{i-\frac{1}{2}}) = 0,$$$

$$$\hat{q}_{i+\frac{1}{2}} = \hat{q}({\bf u}_{i-p+1}, \cdots, {\bf u}_{i+p}), \ \ \hat{q}({\bf u}, {\bf u}, \cdots, {\bf u}) = q({\bf u}).$$$

2.1 熵守恒通量

$$$[{\bf v}]_{i+\frac{1}{2}}\cdot\hat{{\bf f}}_{i+\frac{1}{2}} = [\psi]_{i+\frac{1}{2}},$$$

$$$q_{i+\frac{1}{2}} = \bar{{\bf v}}_{i+\frac{1}{2}}\cdot\hat{{\bf f}}_{i+\frac{1}{2}}-\bar{\psi}_{i+\frac{1}{2}}.$$$

4 数值算例

$$$\begin{array}{ll} u^{(1)} = u^n+\Delta t L(u^n), \\ { } u^{(2)} = \frac{3}{4}u^n+\frac{1}{4}u^{(1)}+\frac{1}{4}\Delta t L(u^{(1)}), \\ { }u^{n+1} = \frac{1}{3}u^n+\frac{2}{3}u^{(2)}+\frac{2}{3}\Delta t L(u^{(2)}), \end{array}$$$

4.1 线性对流方程

$$$u_t+u_x = 0.$$$

$(1)$光滑初始条件

$$$u(x, 0) = \sin(\pi x), \ \ |x|\leq1.$$$

$(2)$间断初始条件

$$$u(x, 0) = \left\{\begin{array}{ll} 1, {\quad} &{ } |x|\leqslant \frac{1}{3}, \\ 0, &{ } \frac{1}{3}\leqslant |x|\leqslant 1.\end{array}\right.$$$

 N L1 error Rate L∞ error Rate 40 0.001979313043305 3.629516311932702e-04 80 2.726027476503843e-04 2.86 7.092855086397350e-05 2.36 160 3.522356279884232e-05 2.95 9.975669895746602e-06 2.83 320 4.405706993699641e-06 3.00 1.260137438287312e-06 2.99 640 5.373835462491922e-07 3.01 1.576597490888024e-07 3.00

4.2 Burgers方程

$$$u_t+\bigg(\frac{u^2}{2}\bigg)_x = 0.$$$

$(1)$光滑初始条件

$$$u(x, 0) = u_0-u_1\sin(\pi x), \ \ |x|\leq1.$$$

$(2)$间断初始条件

$$$u(x, 0) = \left\{\begin{array}{ll} 1, &{ } |x|\leqslant \frac{1}{3}, \\ -1, {\quad} &{ } \frac{1}{3}\leqslant |x|\leqslant 1.\end{array}\right.$$$

$$$u(x, 0) = 1+\frac{1}{2}\sin(\pi x), \ \ |x|\leq1.$$$

Burgers方程是一类非线性标量问题, 取熵函数$\eta(u) = \frac{u^2}{2}$, 则熵通量函数为$q(u) = \frac{u^3}{3}$, 可得二阶熵守恒通量

$$$\tilde{f}(u_i, u_{i+1}) = \frac{u_i^2+u_iu_{i+1}+u_{i+1}^2}{6}.$$$

$$${\bf D}_{i+\frac{1}{2}} = |\bar{u}_{i+\frac{1}{2}}|+\frac{1}{6}|[u]_{i+\frac{1}{2}}|.$$$

 N L1 error Rate L∞ error Rate 40 1.827054747602715e-04 1.448773504797440e-04 80 2.851024314352313e-05 2.68 1.785661053588239e-05 3.02 160 3.845862448196772e-06 2.89 2.226299234296333e-06 3.00 320 4.929538860536765e-07 2.96 2.782109209450295e-07 3.00 640 6.139221843903657e-08 3.01 3.478015911254420e-08 3.00

4.3 Euler方程

$$${\bf u}_t+{\bf f}({\bf u})_x = 0,$$$

$$${\bf u} = \left( \begin{array}{c} \rho\\ \rho u \\ E \end{array} \right), \ {\bf f}({\bf u}) = \left( \begin{array}{c} \rho u\\ \rho u^2+p \\ \rho uH \end{array} \right).$$$

$$$\eta({\bf u}) = \frac{-\rho s}{\gamma-1}, \ q({\bf u}) = \frac{-\rho us}{\gamma-1},$$$

Ismail和Roe[9]为Euler方程构造了${\bf v}^{\rm T}\tilde{{\bf f}} = [\psi]$的显式解. 定义参数向量${\bf z}$

$$${\bf z} = \left( \begin{array}{c} z_1\\ z_2 \\ z_3 \end{array} \right) = \sqrt{\frac{\rho}{p}}\left( \begin{array}{c} 1\\ u \\ p \end{array} \right),$$$

$$$\tilde{{\bf f}}_{i+\frac{1}{2}} = \left( \begin{array}{c} \hat{\rho}\hat{u}\\ \hat{\rho}\hat{u}^2+\hat{p}_1 \\ \hat{\rho}\hat{u}\hat{H} \end{array} \right)_{i+\frac{1}{2}},$$$

$$$\left(\ast\right)_{i+\frac{1}{2}}^{\ln} = \frac{[\ast]_{i+\frac{1}{2}}}{[\ln(\ast)]_{i+\frac{1}{2}}} .$$$

(1) Sod激波管问题，初始条件为

$$$\left\{ \begin{array}{ll} \left(\rho_l, u_l, p_l\right) = \left(1, 0, 1\right), \ &x\leq0.5, \\ \left(\rho_r, u_r, p_r\right) = \left(0.125, 0, 0.1\right), \ &x>0.5. \end{array} \right.$$$

Sod激波管问题的初始间断包含左稀疏波, 接触间断和右激波. 本文计算了$T = 0.16$时刻$[0, 1]$区间上的解. 即扰动到达计算区域边界之前的解. 从图 4可以清晰地看到: 与ES格式相比, CCWENO3格式能更好地捕捉激波, 稀疏波和接触间断, 分辨率更高, 鲁棒性更好, 且不产生伪振荡.

图 4

(2) Lax激波管问题，初始条件为

$$$\left\{\begin{array}{ll}\left(\rho_l, u_l, p_l\right) = \left(0.445, 0.698, 3.528\right), \ &x\leq0.5, \\\left(\rho_r, u_r, p_r\right) = \left(0.5, 0, 0.571\right), \ &x>0.5.\end{array} \right.$$$

Lax激波管问题包含了真正的非线性场(稀疏波, 激波)和线性退化场(接触间断). 其计算区间均为$[0, 1]$, 终止时刻为$T = 0.16,$CFL条件为0.25. 图 5表明: 与ES格式相比, CCWENO3格式可以避免过度抹平现象, 提高了数值结果的分辨率, 具有更好的鲁棒性.

图 5

(3) Shu-Osher问题, 初始条件为

$$$\left\{ \begin{array}{ll} \left(\rho_l, u_l, p_l\right) = \left(3.857143, 2.629369, 10.3333\right), \ &x\leq0.1, \\ \left(\rho_r, u_r, p_r\right) = \left((1+0.2\sin(50x)), 0, 1\right), \ &x>0.1. \end{array} \right.$$$

4.4 浅水波方程

$$${\bf u}_t+{\bf f}({\bf u})_x = 0,$$$

$$${\bf u} = \left(\begin{array}{c}h \\ hu\end{array}\right), \; \; \{\bf f}({\bf u}) = \left(\begin{array}{c}hu \\{ } hu^2+\frac{1}{2}gh^2\end{array}\right),$$$

$$$\eta({\bf u}) = \frac{hu^2+gh^2}{2}, \q({\bf u}) = \frac{hu^3}{2}+guh^2.$$$

$$${\bf v} = \left(\begin{array}{c}{ } gh-\frac{u^2}{2} \\ u\end{array}\right), \\psi({\bf u}) = \frac{1}{2}guh^2.$$$

$$$\tilde{{\bf f}}_{i+\frac{1}{2}} = \left(\begin{array}{c}\bar{h}\bar{u}\\{ } \bar{h}\bar{u}^2+\frac{g}{2}\bar{h^2}\end{array}\right)_{i+\frac{1}{2}},$$$

$$${\bf R}_{i+\frac{1}{2}} = \frac{1}{\sqrt{2g}}\left(\begin{array}{cc}1{\quad} & 1 \\\bar{u}-\sqrt{g\bar{h}}{\quad} & \bar{u}+\sqrt{g\bar{h}}\end{array}\right)_{i+\frac{1}{2}},$$$

$$${\bf \tilde{\Lambda}}_{i+\frac{1}{2}} = \left(\begin{array}{cc}{ }|\bar{u}-\sqrt{g\bar{h}}|+\frac{1}{6}[u-\sqrt{gh}]{\quad} & 0 \\0{\quad} &{ } |\bar{u}+\sqrt{g\bar{h}}|+\frac{1}{6}[u+\sqrt{gh}]\end{array}\right)_{i+\frac{1}{2}}.$$$

(1) 溃坝问题, 初始条件为

$$$h(x, 0) = \left\{\begin{array}{ll}2, \ \ &x<0 \\1, \ \ &x>0\end{array} \right. , \ \ \u(x, 0)\equiv0.$$$

图 7

(2) 大型溃坝问题, 初始条件为

$$$h(x, 0) = \left\{\begin{array}{ll}15, \ \ &x<0 \\1, \ \ &x>0\end{array}\right. , \ \ \u(x, 0)\equiv0.$$$

5 总结

(1) 对于双曲守恒律方程组的光滑解可以达到任意阶精度.

(2) 满足离散熵不等式, 是严格熵稳定的.

(3) 每个单元交界面上重构的跳跃与相应单元格值的跳跃具有相同的符号.

(4) 保留紧致CWENO格式的优点, 重构跳跃值具有对称性且计算经济, 即在相同精度要求下所用节点少.

(5) 精准捕捉解的结构, 有效避免非物理现象, 间断附近基本无振荡, 高分辨率, 且具有良好的鲁棒性等.

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