On Asymptotic Behavior of HLL-Type Schemes at Low Mach Numbers

Hang Yu,Zhengyu Tian,Fan Yang,Hua Li

doi:10.1155/2020/7451240

Hang Yu, Zhengyu Tian + Show 2 more

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https://doi.org/10.1155/2020/7451240

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Abstract

The HLLEM approximate Riemann solver can capture discontinuities sharply, maintain positive definiteness, and satisfy the entropy condition automatically. These attractive properties make the HLLEM scheme widely used in the simulations of many compressible fluid problems. However, in the simulations of low Mach incompressible flow, the accuracy of HLLEM solver cannot be guaranteed. In the current study, a detailed discrete asymptotic analysis is conducted on the HLLEM scheme and the responsible terms for the loss of accuracy are identified. This allows us to develop two modified methods to solve this low Mach number problem. The first method is to add a low Mach number correction term on the basis of the original HLLEM scheme. The second is to simply rescale the responsible terms with a Mach number-based function. The asymptotic analysis of these two low Mach correction methods shows that the difference between the continuous system and the discrete system disappears, which means the resulting LM-HLLEM and LM-HLLEM2 schemes are both capable of obtaining physically correct solutions in low Mach limit. The results obtained from various test cases demonstrate that both these two HLLEM-type schemes can simulate incompressible and compressible fluid problems accurately and robustly.

Highlights

Due to the robustness and clear physical interpretation, Godunov-type Riemann solvers have been widely applied in practical engineering problem calculation and theoretical research
Guillard et al [2, 3] conducted an asymptotic analysis of the Godunov-type schemes at low Mach numbers. ey found the results calculated by the numerical schemes contained pressure fluctuations of order Ma, while the actual physical pressure varied with Ma2. ey applied a preconditioned technique to rescale the Roe scheme [4] and modified the numerical dissipation to solve this problem
A detailed asymptotic analysis on the HLLEM scheme is conducted in this paper to study its low Mach number behavior. rough this analysis, we have identified the responsible terms which result in the loss of accuracy of the HLLEM Riemann solver

Summary

Introduction

Due to the robustness and clear physical interpretation, Godunov-type Riemann solvers have been widely applied in practical engineering problem calculation and theoretical research. Dellacherie et al [13, 14] have done a series of research in order to solve this low Mach limit problem and put forward a corresponding correction method By applying their low Mach correction, a new low Mach Godunov scheme for solving linear wave equations was developed, and they further generalized it to nonlinear Euler cases. Compared with the existing fixes for the HLLEM scheme in low Mach number situations like Park et al.’s preconditioned HLLEM method [32] and Qu et al.’s HLLEMS-AS scheme [34], our newly proposed schemes have much simpler formulas and only the actual responsible terms for the loss of accuracy are rescaled In this sense, the proposed low Mach HLLEM-type schemes are effective for use as reliable tools for all-speed flow computations.

Low Mach Number Behavior of Compressible Euler Equations

Detailed Asymptotic Analysis of the HLLEM Scheme

Conclusions

Findings

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