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A low-dissipation central scheme for solving compressible flows

DOI：

 作者 单位 邮编 胡立军* 衡阳师范学院数学与统计学院 421002

含激波的可压缩流动问题广泛存在于科学工程领域，求解这类问题的数值方法大多都是基于黎曼求解器来构造的，这严重依赖于控制方程组的特征结构。此外，许多可以准确捕捉接触间断的黎曼求解器(如工业软件中流行的Roe格式)在捕捉激波时会出现不同形式的数值异常现象，这大大影响了格式在工程实际问题中的应用。本文利用广义黎曼不变量的基本思想结合单元边界变差最小化(Boundary Variation Diminishing, BVD)原理构造了一种不依赖于系统特征结构的中心型格式。该格式形式简单、可以准确捕捉接触间断并且不会出现激波数值异常现象，理论分析和数值实验证明了该格式的优良性能和表现。由于该格式独立于系统的特征结构，因此它可以简单地被推广到其他守恒律系统的数值计算。

Compressible flow problems with shock waves exist widely in the field of science and engineering. Most of the numerical methods for solving such problems are constructed based on Riemann solvers which rely heavily on the eigen-structure of the governing equations. In addition, many Riemann solvers that can accurately capture contact discontinuity (such as the Roe scheme popular in industrial software) will suffer from different forms of numerical anomalies when capturing shock waves, which greatly affects their application in practical engineering problems. In the current work, based on the basic idea of generalized Riemann invariants and the principle of Boundary Variation Diminishing (BVD), a central scheme which is independent of the eigen-structure of the system is constructed. The scheme is simple, accurate for contact discontinuities, and does not induce numerical shock anomalies. Theoretical analyses and numerical experiments demonstrate the good performance of the current central scheme. Since the scheme is independent of the eigen-structure of the system, it can be easily generalized to numerical calculations of other conservation laws.
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