We propose and analyze a new asymptotic preserving (AP) finite volume scheme for the multidimensional compressible barotropic Euler equations to simulate low Mach number flows. The proposed scheme uses a stabilized upwind numerical flux, with the stabilization term being proportional to the stiff pressure gradient, and we prove its conditional energy stability and consistency. Utilizing the concept of dissipative measure-valued (DMV) solutions, we rigorously illustrate the AP properties of the scheme for well-prepared initial data. In particular, we prove that the numerical solutions will converge weakly to a DMV solution of the compressible Euler equations as the mesh parameter vanishes, while the Mach number is fixed. The DMV solutions then converge to a classical solution of the incompressible Euler system as the Mach number goes to zero. Conversely, we show that if the mesh parameter is kept fixed, we obtain an energy stable and consistent finite-volume scheme approximating the incompressible Euler equations as the Mach number goes to zero. The numerical solutions generated by this scheme then converge weakly to a DMV solution of the incompressible Euler system as the mesh parameter vanishes. Invoking the weak-strong uniqueness principle, we conclude that the DMV solution and classical solution of the incompressible Euler system coincide, proving the AP property of the scheme. We also present an extensive numerical case study in order to illustrate the theoretical convergences, wherein we utilize the techniques of K-convergence.

翻译：本文提出并分析了一种新的渐近保持有限体积格式，用于模拟多维可压缩正压欧拉方程中的低马赫数流动。该格式采用稳定的迎风数值通量，其中稳定项与刚性压力梯度成正比，并证明了其条件能量稳定性与相容性。利用耗散测度值解的概念，我们严格论证了该格式在良好预备初始数据下的渐近保持性质。特别地，我们证明：当网格参数趋于零而马赫数固定时，数值解弱收敛于可压缩欧拉方程的耗散测度值解；随后当马赫数趋于零时，该耗散测度值解收敛于不可压缩欧拉系统的经典解。反之，若固定网格参数，当马赫数趋于零时，我们得到一个逼近不可压缩欧拉方程的能量稳定相容有限体积格式。由此生成的数值解在网格参数消失时弱收敛于不可压缩欧拉系统的耗散测度值解。通过引用弱-强唯一性原理，我们得出不可压缩欧拉系统的耗散测度值解与经典解等价，从而证明了格式的渐近保持性质。最后，我们利用K-收敛技术开展了广泛的数值案例研究以验证理论收敛性。