We introduce a semi-implicit two-speed relaxation scheme to solve the compressible Euler equations in the low Mach regime. The scheme involves a relaxation system with two speeds, already introduced by Bouchut et al. (Numer Math, 2020. https://doi.org/10.1007/s00211-020-01111-5) in the barotropic case. It is entropy satisfying and has a numerical viscosity well-adapted to low Mach flows. This relaxation system is solved via a dynamical Mach number dependent splitting, similar to the one proposed by Iampietro et al. (J Comput Appl Math 340:122–150, 2018). Stability conditions are derived, they limit the range of admissible relaxation and splitting parameters. We resolve separately the advection part of the splitting by an explicit method, and the acoustic part by an implicit method. The relaxation speeds are chosen so that the implicit system fully linearizes the acoustics and requires just to invert an elliptic operator with constant coefficients. The scheme is shown to well capture with low cost the incompressible slow scale dynamics with a timestep adapted to the velocity field scale, and rather well the fast acoustic waves.

我们介绍了一种半隐式两速松弛方案，以解决低马赫状态下的可压缩欧拉方程。该方案涉及一种由Bouchut等人介绍的具有两种速度的松弛系统。（Numer Math，2020. https://doi.org/10.1007/s00211-020-01111-5）在正压情况下。它的熵令人满意，并且具有非常适合低马赫数流量的数值粘度。这种松弛系统是通过动态马赫数相关的分裂来解决的，类似于Iampietro等人提出的分裂。（J Comput Appl Math 340：122–150，2018年）。导出了稳定性条件，它们限制了允许的松弛和分裂参数的范围。我们通过显式方法分别解决分裂的对流部分，并通过隐式方法解决声学部分。选择弛豫速度是为了使隐式系统完全线性化声学，并且只需要反转具有恒定系数的椭圆算子即可。该方案显示以低成本很好地捕获了不可压缩的慢尺度动力学，其动态步长适应于速度场尺度，并且很好地捕获了快速声波。