In the first part of this work, we introduce a new relaxation system in order to approximate the solutions to the barotropic Euler equations. We show that the solutions to this two-speed relaxation model can be understood as viscous approximations of the solutions to the barotropic Euler equations under appropriate sub-characteristic conditions. Our relaxation system is a generalization of the well-known Suliciu relaxation system, and it is entropy satisfying. A Godunov-type finite volume scheme based on the exact resolution of the Riemann problem associated with the relaxation system is deduced, as well as its stability properties. In the second part of this work, we show how the new relaxation approach can be successfully applied to the numerical approximation of low Mach number flows. We prove that the underlying scheme satisfies the well-known asymptotic-preserving property in the sense that it is uniformly (first-order) accurate with respect to the Mach number, and at the same time it satisfies a fully discrete entropy inequality. This discrete entropy inequality allows us to prove strong stability properties in the low Mach regime. At last, numerical experiments are given to illustrate the behaviour of our scheme.

中文翻译：

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满足正压欧拉方程的两速弛豫系统的熵：在低马赫数流的数值近似中的应用

在这项工作的第一部分，我们引入了一个新的松弛系统来逼近正压欧拉方程的解。我们表明，这个双速松弛模型的解可以理解为在适当的子特征条件下正压欧拉方程解的粘性近似。我们的松弛系统是著名的 Suliciu 松弛系统的推广，它是熵满足的。推导出基于与松弛系统相关的黎曼问题的精确分辨率的 Godunov 型有限体积方案及其稳定性属性。在这项工作的第二部分，我们展示了新的松弛方法如何成功地应用于低马赫数流的数值近似。我们证明了基本方案满足众所周知的渐近保持性质，因为它对于马赫数是一致（一阶）准确的，同时它满足完全离散的熵不等式。这种离散熵不等式使我们能够在低马赫数状态下证明强大的稳定性。最后，给出了数值实验来说明我们方案的行为。