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

We consider the compressible Euler equations of gas dynamics with isentropic equation of state. In the low Mach number regime i.e. when the fluid velocity is very very small in comparison to the sound speed in the medium, the solution of the compressible system converges to the solution of its incompressible counter part. Standard numerical schemes fail to respect this transition property and hence are plagued with inaccuracies as well as instabilities. In this paper we introduce an extra flux term to the momentum flux. This extra term is brought to fore by looking at the incompressibility constraints of the asymptotic limit system. This extra flux term enables us to get a suitable flux 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. 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. Numerical results are reported which elucidate the theoretical assertions regarding the scheme and its robustness.