We present a numerical approximation of the solutions of the Euler equations with a gravitational source term. On the basis of a Suliciu type relaxation model with two relaxation speeds, we construct an approximate Riemann solver, which is used in a first order Godunov-type finite volume scheme. This scheme can preserve both stationary solutions and the low Mach limit to the corresponding incompressible equations. In addition, we prove that our scheme preserves the positivity of density and internal energy, that it is entropy satisfying and also guarantees not to give rise to numerical checkerboard modes in the incompressible limit. Later we give an extension to second order that preserves positivity, asymptotic-preserving and well-balancing properties. Finally, the theoretical properties are investigated in numerical experiments.

翻译：提出了一种带有重力源项的欧拉方程解的数值近似方法。基于具有两个松弛速度的Suliciu型松弛模型，构造了一个近似黎曼求解器，并将其应用于一阶Godunov型有限体积格式。该格式既能保持稳态解，又能保持对应不可压缩方程的低马赫数极限。此外，我们证明了该格式能保持密度和内能的正性，满足熵条件，且保证在不可压缩极限下不会产生数值棋盘格模式。随后给出了保持正性、渐近保持和良好平衡性质的二阶扩展格式。最后，通过数值实验验证了上述理论性质。