Asymptotic Preserving Linearly Implicit Additive IMEX-RK Finite Volume Schemes for Low Mach Number Isentropic Euler Equations

Abstract: 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.