基于WENO重构的真正多维黎曼求解器  

A genuinely multidimensional Riemann solver based on WENO reconstruction

作　　者：胡立军 谭诗德 袁海专 HU Li-jun;TAN Shi-de;YUAN Hai-zhuan(College of Mathematics and Statistics,Hengyang Normal University,Hengyang 421002,China;School of Mathematics and Computational Science,Xiangtan University,Xiangtan 411105,China)

机构地区：[1]衡阳师范学院数学与统计学院,衡阳421002 [2]湘潭大学数学与计算科学学院,湘潭411105

出　　处：《计算力学学报》2023年第6期1036-1043,共8页Chinese Journal of Computational Mechanics

基　　金：湖南省自然科学基金(2021JJ40009);湖南省教育厅优秀青年项目(21B0626);湖南省“双一流”应用特色学科项目(湘教通[2018]469号);湖南省重点实验室项目(2016TP1020)资助。

摘　　要：具有良好守恒性与网格适应性的有限体积格式在流体力学的数值计算中占有重要地位。其中,求解数值流通量是实施有限体积法的关键步骤。一维情形下,通过求解局部黎曼问题来获得数值流通量的相关理论已经比较成熟。但是在计算多维问题时,传统的维度分裂方法仅考虑沿界面法向传播的信息,这不仅影响格式的精度,还可能会造成数值不稳定性从而诱发非物理现象。本文基于对流-压力通量分裂方法来构造真正多维的黎曼求解器,通过求解网格顶点处的多维黎曼问题来实现格式的多维特性。采用五阶WENO重构方法来获得空间的高阶精度,时间离散采用三阶TVD龙格-库塔格式。一系列数值实验的结果表明,真正多维的黎曼求解器不仅具有更高的分辨率还能有效克服多维强激波模拟中的数值不稳定性。The finite volume scheme plays an important role in numerical computation of fluid dynamics due to its good conservation and mesh adaptability.Numerical determination of fluxes is the key to implement the finite volume method.In one-dimensional cases,the theory of obtaining fluxes by solving the local Riemann problem has been relatively mature.However,when solving multidimensional problems,the traditional dimension splitting method only considers the information propagating along the direction normal to the interface,which not only affects the accuracy but also may cause numerical instability and thus induce the unphysical phenomenon.A genuinely multidimensional Riemann solver based on the convection-pressure flux splitting method is constructed,and the multidimensional property is achieved by solving multidimensional Riemann problems at vertices of the grid.The high-order accuracy in space is achieved by implementing the fifth-order WENO reconstruction and the third-order TVD Runge-Kutta scheme for time discretization.Results of a series of numerical experiments show that the genuinely multidimensional Riemann solver not only has a higher resolution but also can effectively overcome the numerical instability in calculations of multidimensional strong shock waves.

关 键 词：黎曼求解器 真正多维 WENO重构 通量分裂 激波不稳定性 

分 类 号：O354[理学—流体力学] O241.82[理学—力学]