真正多维时空高精度黎曼解法器及其应用

The finite volume scheme and discontinuous finite element method play dominant roles in the scientific computation of compressible fluid flows and related problems. In this context, one of key ingredients is the construction of numerical fluxes, which is quite well understood in one space dimension. However, related constructions have to be often simplified to one dimensional cases due to notoriously complex wave interactions around cell vertices when space dimensions are multiple. On the other hand, very high order accuracy both in space and time is required to deal with many practical problems. This project aims to investigate the construction of genuinely dimensional Riemann solvers of high order accuracy both in space and time, based on the underlying theoretical results that the principal applicant obtained together with his collaborators in past years. The research will start from the multidimensional linear systems of constant coefficients, to compressible Euler equations, to magneto-hydrodynamical systems, to shallow water equations on sphere (earth) and to gas dynamical combustion. It is expected that such a contribution will cure some numerical instabilities such as carbuncle phenomenon in the simulation of strong shock propagation and provide further numerical tools for CFD and related engineering problems.

在可压缩流体及相关问题的科学计算中，有限体积格式（FV）和间断有限元（DG）方法等占有重要地位。其中，数值流通量的构造是至关重要的，它可归结为相关联广义黎曼问题的数值求解。一个空间维数的情形下相关研究比较成熟，而在多个空间维数的情形下，由于网格顶点附近非线性波的复杂相互作用，人们常使用局部一维化的方法计算数值流通量，由此可能导致数值不稳定性或虚假物理特征；另一方面，许多实际问题常需要时空高阶精度的算法。本项目将结合申请者多年来对非线性多维守恒律系统的理论成果，探索时空高精度、真正多维黎曼解法器。研究路线拟从多维常系数线性方程组入手，然后到可压缩欧拉方程，再到更一般的磁流体力学方程组、球面上浅水波方程组及其气体动力燃烧等；主要特色和关键是黎曼解法器真正多维性和时空高精度。期望研究结果能克服诸如强激波模拟中红宝石现象等多维数值不稳定性，为计算流体力学及相关领域的工程问题提供理论支撑。

本项目以流体力学中的可压缩欧拉方程组为背景，发展时空耦合、真正多维的高精度广义黎曼解法器。以此项目为依托，取得了一系列研究成果，特别两步四阶时空耦合的高精度数值方法产生了重要影响，该方法结合广义黎曼解法器和多步Rung-Kutta型方法的优点，规避了它们的劣势。 目前在国际一流刊物上发表论文1 0多篇，其中SIAM J Sci. Comput.1篇， SIAM J. Numer. Anal. 1篇，Journal of Comput. Phys. 5篇，Physics of Fluids 1篇， 其它刊物多篇，另有投稿论文多篇。目前这一领域的研究正在进一步拓展之中，可望在多介质、颗粒流等多方面做出贡献。