We present an implicit relaxation scheme for the simulation of compressible flows in all Mach number regimes based on a Jin Xin relaxation approach. The main features of the proposed scheme lie in its simplicity and effectiveness. Thanks to the linearity of the flux in the relaxation system, the time-semi discrete scheme can be reformulated in linear decoupled elliptic equations resulting in the same number of unknowns as in the original system. To obtain the correct numerical diffusion in all Mach number regimes, a convex combination of upwind and centred fluxes is applied. The numerical scheme is validated by applying it on a Eulerian model for non-linear elasticity. Simulations of gas and fluid flows, as well as deformations of compressible solids are carried out to assess the performance of the numerical scheme in accurately approximating material waves in different Mach regimes.

翻译：基于金欣松弛方法，我们提出了一种适用于全马赫数范围内可压缩流动模拟的隐式松弛格式。该格式的主要特点在于其简洁性与有效性。借助松弛系统中通量的线性特性，时间半离散格式可重构为线性解耦椭圆方程，其未知量数目与原系统保持一致。为在不同马赫数范围内获得正确的数值耗散，我们采用迎风型通量与中心型通量的凸组合策略。将该数值格式应用于非线性弹性欧拉模型进行验证，通过气体与流体流动模拟以及可压缩固体变形算例，评估了格式在不同马赫数区域准确近似材料波的能力。