In this paper, we propose a modification of an acoustic-transport operator splitting Lagrange-projection method for simulating compressible flows with gravity. The original method involves two steps that respectively account for acoustic and transport effects. Our work proposes a simple modification of the transport step, and the resulting modified scheme turns out to be a flux-splitting method. This new numerical method is less computationally expensive, more memory efficient, and easier to implement than the original one. We prove stability properties for this new scheme by showing that under classical CFL conditions, the method is positivity preserving for mass, energy and entropy satisfying. The flexible flux-splitting structure of the method enables straightforward extensions of the method to multi-dimensional problems (with respect to space) and high-order discretizations that are presented in this work. We also propose an interpretation of the flux-splitting solver as a relaxation approximation. Both the stability and the accuracy of the new method are tested against one-dimensional and two-dimensional numerical experiments that involve highly compressible flows and low-Mach regimes.

翻译：本文提出对一种声学-输运算子分裂拉格朗日-投影方法进行改进，用于模拟含重力的可压缩流动。原始方法包含两个步骤，分别处理声学效应和输运效应。我们的工作提出了对输运步骤的简单改进，得到的修正格式实质上是一种通量分裂方法。与原始方法相比，这种新数值方法计算成本更低、内存效率更高且更易于实现。我们证明了该新格式的稳定性性质：在经典CFL条件下，该方法能保持质量、能量和熵的正性。该方法的灵活通量分裂结构使其可便捷地扩展至多维问题（空间维度）及高阶离散格式，本文对此进行了论述。我们还提出了将通量分裂求解器解释为松弛近似的方法。通过涉及高可压缩流动和低马赫数工况的一维及二维数值实验，验证了新方法的稳定性和精度。