We present an implicit-explicit well-balanced finite volume scheme for the Euler equations with a gravitational source term which is able to deal also with low Mach flows. To visualize the different scales we use the non-dimensionalized equations on which we apply a pressure splitting and a Suliciu relaxation. On the resulting model, we apply a splitting of the flux into a linear implicit and an non-linear explicit part that leads to a scale independent time-step. The explicit step consists of a Godunov type method based on an approximative Riemann solver where the source term is included in the flux formulation. We develop the method for a first order scheme and give an extension to second order. Both schemes are designed to be well-balanced, preserve the positivity of density and internal energy and have a scale independent diffusion. We give the low Mach limit equations for well-prepared data and show that the scheme is asymptotic preserving. These properties are numerically validated by various test cases.

我们为带有引力源项的Euler方程提供了一个隐式-显式的均衡有限体积方案，该方案也能够处理低马赫数流量。为了可视化不同的比例，我们使用了无量纲的方程式，在该方程式上应用了压力分裂和Suliciu松弛。在生成的模型上，我们将磁通量分为线性隐式和非线性显式部分，这导致了与比例无关的时间步长。显式步骤由基于近似Riemann求解器的Godunov型方法组成，其中源项包含在通量公式中。我们开发了用于一阶方案的方法，并扩展了二阶方案。两种方案均设计为平衡良好，保持密度和内部能量的正值，并具有与尺度无关的扩散。我们给出了准备好的数据的低马赫数极限方程，并证明了该方案是渐近保持的。这些属性通过各种测试案例进行了数值验证。