We consider the compressible Euler equations of gas dynamics with isentropic equation of state. Standard numerical schemes for the Euler equations suffer from stability and accuracy issues in the low Mach regime. These failures are attributed to the transitional behaviour of the governing equations from compressible to incompressible solution in the limit of vanishing Mach number. In this paper we introduce an extra flux term to the momentum flux. This extra term is recognised by looking at the constraints of the incompressible limit system. As a consequence the flux terms enable us to get a suitable splitting, so that an additive IMEX-RK scheme could be applied. Using an elliptic reformulation the scheme boils down to just solving a linear elliptic problem for the density and then explicit updates for the momentum. The IMEX schemes developed are shown to be formally asymptotically consistent with the low Mach number limit of the Euler equations and are shown to be linearly $L^2$ stable. A second order space time fully discrete scheme is obtained in the finite volume framework using a combination of Rusanov flux for the explicit part and simple central differences for the implicit part. Results of numerical case studies are reported which elucidate the theoretical assertions regarding the scheme and its robustness.

翻译：本文研究具有等熵状态方程的气体动力学可压缩欧拉方程。标准欧拉方程数值格式在低马赫数区域存在稳定性与精度问题，这些失效源于控制方程在趋近于零马赫数极限时从可压缩解向不可压缩解的过渡行为。本文通过在动量通量中引入附加通量项来改进数值方法，该附加项是通过分析不可压缩极限系统的约束条件而识别得到的。由此得到的通量项使我们能够实现恰当的算子分裂，从而可应用加性IMEX-RK格式。通过椭圆重构，该格式最终简化为仅需对密度求解线性椭圆问题，随后对动量进行显式更新。所构建的IMEX格式在形式上被证明与欧拉方程的低马赫数极限渐近相容，并具有线性$L^2$稳定性。在有限体积框架下，结合显式部分的Rusanov通量与隐式部分的简单中心差分，获得了二阶时空全离散格式。数值算例的结果验证了该格式的理论性质及其鲁棒性。