An All Speed Second Order IMEX Relaxation Scheme for the Euler Equations (Communications in Computational Physics, 2020)
We present an implicit-explicit finite volume scheme for the Euler equations.We start from the non-dimensionalised Euler equations where we split the pressure ina slow and a fast acoustic part. We use a Suliciu type relaxation model which we splitin an explicit part, solved using a Godunov-type scheme based on an approximateRiemann solver, and an implicit part where we solve an elliptic equation for the fastpressure. The relaxation source terms are treated projecting the solution on the equilibrium manifold. The proposed scheme is positivity preserving with respect to thedensity and internal energy and asymptotic preserving towards the incompressibleEuler equations. For this first order scheme we give a second order extension whichmaintains the positivity property. We perform numerical experiments in 1D and 2D toshow the applicability of the proposed splitting and give convergence results for thesecond order extension.