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

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.