A semi-implicit in time, entropy stable finite volume scheme for the compressible barotropic Euler system is designed and analyzed and its weak convergence to a dissipative measure-valued (DMV) solution [E. Feireisl et al., Dissipative measure-valued solutions to the compressible Navier-Stokes system, Calc. Var. Partial Differential Equations, 2016] of the Euler system is shown. The entropy stability is achieved by introducing a shifted velocity in the convective fluxes of the mass and momentum balances, provided some CFL-like condition is satisfied to ensure stability. A consistency analysis is performed in the spirit of the Lax's equivalence theorem under some physically reasonable boundedness assumptions. The concept of K-convergence [E. Feireisl et al., K-convergence as a new tool in numerical analysis, IMA J. Numer. Anal., 2020] is used in order to obtain some strong convergence results, which are then illustrated via rigorous numerical case studies. The convergence of the scheme to a DMV solution, a weak solution and a strong solution of the Euler system using the weak-strong uniqueness principle and relative entropy are presented.

翻译：本文设计并分析了一种时间半隐式、熵稳定的有限体积格式，用于可压缩正压欧拉系统，并证明了该格式弱收敛于欧拉系统的耗散测度值（DMV）解 [E. Feireisl等，可压缩Navier-Stokes系统的耗散测度值解，《Calc. Var. Partial Differential Equations》，2016]。通过在质量和动量守恒方程的对流通量中引入偏移速度，并满足类似CFL的稳定性条件，实现了熵稳定性。在若干物理合理的有限性假设下，遵循Lax等价定理的思想进行了相容性分析。文中采用K-收敛概念 [E. Feireisl等，K-收敛：数值分析的新工具，《IMA J. Numer. Anal.》，2020] 以获取强收敛结果，并通过严谨的数值算例加以验证。基于弱-强唯一性原理和相对熵，展示了该格式收敛于欧拉系统的DMV解、弱解及强解的过程。