杜知方， 李杰权^*

（  北京师范大学数学科学学院,北京　100875；  ）

摘要： 本文作者新近提出了对守恒律方程组的两步四阶时间离散方法,并构造了两步四阶广义黎曼问题（Generalized Riemann Problem, GRP）格式。新的格式易于实现,相比其它高阶格式计算开销小,在计算中表现出了对流场的高分辨率。欧拉方程组的二维黎曼问题的研究是流体力学的研究中的一个基础性课题,其中蕴含了很多流体现象,包括激波反射、涡的生成和滑移面的不稳定性等。本文将首先介绍两步四阶 GRP 格式的构造,随后通过对典型二维黎曼问题的计算并与二阶 GRP 格式对比,展示新的格式在二维可压缩流计算方面的性能。结果表明两步四阶格式不仅能清晰分辨解的基本结构,同时与二阶格式相比,分辨滑移面上的小尺度结构具有明显优势。
关键词： 计算数学；二维黎曼问题；两步四阶格式；广义黎曼问题解法器；Lax—Wendroff 方法

Du Zhifang， Li Jiequan^*

（  School of Mathematical Sciences, Beijing Normal University, Beijing 100875；  ）

Abstract： The authors of the present paper have proposed a newly developed two-stage fourth-order time discretization for the hyperbolic conservation laws. And a two-stage fourth-order Generalized Riemann Problem (GRP) scheme has been constructed based on it. The newly developed scheme is easy to implement and requires less computational cost compared with other high-order schemes. It shows excellent resolution in the numerical computations. Two-dimensional Riemann problems of the Euler equations are fundamental in the study of the fluid dynamics. Their solutions reveal a lot of fluid phenomenon such as Mach reflections, spiral formations, interface instabilities and so on. In this paper, we will firstly review the construction of the two-stage fourth-order GRP scheme in detail. Then we will show the performance of the new scheme to resolve the two-dimensional compressible flow by taking it to the simulation of the several classical two-dimensional Riemann problems and compare it with the classical second-order GRP scheme. The results indicate that the new two-stage fourth-order scheme can provide a excellent resolution for the basic structure of the solution. Compared with the second-order scheme, it can also has a great advantage to capture many small-scaled structures along the slip lines.
Keywords： computational mathematics; two-dimensional Riemann problem; two-stage fourth-order scheme; generalized Riemann problem solver; Lax—Wendroff method

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