Low-Mach correction for Lagrangian acoustic Riemann solvers on unstructured meshes

We propose a low-Mach correction for cell-centered schemes in Lagrangian frame. After transposing some classical results in Eulerian frame to Lagrangian frame, we show why classical cell-centered schemes in Lagrangian frame are not able to capture the low-Mach regime except by using unreasonably fine meshes. Consequently, we propose a slight modification of the original scheme, which is easy to implement in any scheme using a acoustic Godunov Riemann solver on unstructured mesh, and is costless in term of CPU time. We demonstrate that this modification cures this flaw. The properties of the original semi-discrete scheme (consistence, conservation) are preserved. Particular attention is paid to the entropy condition, proving its compatibility with the modification proposed. We assess this new scheme on several low and high-Mach problems, to demonstrate its good behaviour in all regimes. Last test problem is devoted to the study of the growth rate of instability in convergent configuration. It shows that even if the problem is globally very compressible, the low-Mach correction can have a significant impact on the solution.