• 论文 •

### 粘性Burgers方程的高阶隐式WCNS格式

1. 西南科技大学理学院模型与算法研究所, 绵阳 621010
• 收稿日期:2020-12-28 发布日期:2022-06-10
• 通讯作者: 蒋艳群,Email:jyq2005@mail.ustc.edu.cn.
• 基金资助:
国家数值风洞工程项目(NNW2018-ZT4A08)和国家自然科学基金(11872323)资助

Zhang Xu, Jiang Yanqun, Chen Xun, Hu Yinggang. HIGH-ORDER IMPLICIT WCNS SCHEME FOR VISCOUS BURGERS’ EQUATIONS[J]. Journal on Numerica Methods and Computer Applications, 2022, 43(2): 188-201.

### HIGH-ORDER IMPLICIT WCNS SCHEME FOR VISCOUS BURGERS’ EQUATIONS

Zhang Xu, Jiang Yanqun, Chen Xun, Hu Yinggang

1. Model and Algorithm Research Institute, Department of Mathematics, Southwest University of Science and Technology, Mianyang 621000, China
• Received:2020-12-28 Published:2022-06-10
JFNK (Jacobian-free Newton-Krylov)方法是由外层Newton迭代法和内层Krylov子空间迭代法构成的嵌套迭代方法.本文提出了一种基于JFNK方法的高阶隐式WCNS (weighted compact nonlinear scheme)格式，并用于求解一维、二维粘性Burgers方程.外层迭代法采用含参数的多步Newton迭代法，给出了收敛性分析，内层迭代法采用无矩阵GMRES迭代法.粘性Burgers方程的非线性对流项采用五阶WCNS格式计算.为提高方法精度和计算效率，时间离散采用三阶隐式的DIRK (diagonal implicit Runge-Kutta)方法.数值结果表明基于JFNK方法的隐式WCNS格式在时间上能达到三阶精度，与显式TVD Runge-Kutta WCNS方法相比，计算效率更高.此外，基于JFNK方法的隐式WCNS格式稳定性好，且具有良好的激波捕捉能力.
The JFNK (Jacobian-free Newton-Krylov) method is a nested iteration composed of the outer Newton iteration and the inner Krylov subspace iteration. A high-order JFNKbased implicit WCNS (weighted compact nonlinear scheme) is presented to solve the onedimensional and two-dimensional viscous Burgers' equations. A multi-step Newton iteration method with several parameters as the outer iteration is designed and its convergence analysis is also given. A matrix-free iterative algorithm based on the GMRES method is used as the inner iteration. The fifth-order WCNS is applied to calculate the nonlinear convection terms of the viscous Burgers' equations. In order to improve the accuracy and computational efficiency of the method, the third-order implicit DIRK (diagonal implicit Runge-Kutta)method is used as the time discretization. Numerical results show that the JFNK-based implicit WCNS can achieve third-order accuracy in time and it is more advantageous in computational efficiency than the explicit TVD Runge-Kutta WCNS. In addition, the JFNKbased implicit WCNS has good stability and shock-capturing ability.

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