The efficient numerical solution of compressible Euler equations of gas dynamics poses several computational challenges. Severe time restrictions are required by standard explicit time discretization techniques, in particular for flow regimes characterized by low Mach number values. We analyze schemes based on a general Implicit-Explicit (IMEX) Runge-Kutta (RK) time discretization for the compressible Euler equations of gas dynamics. We prove the asymptotic-preserving (AP) property of the numerical scheme in the low Mach number limit for both a single asymptotic length scale and two length scales. The analysis is carried out considering a general equation of state (EOS) and therefore it is not restricted to the ideal gas law as done for standard AP schemes. We couple implicitly the energy equation to the momentum one, while treating the continuity equation in an explicit fashion.

We also present an alternative strategy, which consists of employing a semi-implicit temporal integrator based on IMEX-RK methods (SI-IMEX-RK). The stiff dependence is carefully analyzed, so as to avoid the solution of a nonlinear equation for the pressure also for equations of state (EOS) of non-ideal gases.

The asymptotic-preserving (AP) and the asymptotically-accurate (AA) properties of the two approaches are assessed on a number of classical benchmarks for ideal gases and on their extension to non-ideal gases. Particular attention is devoted to the robustness of the numerical methods with respect to the boundary conditions and to non well-prepared initial conditions.

This work is based on a collaboration with Luca Bonaventura (Politecnico di Milano), Sebastiano Boscarino and Giovanni Russo (Università di Catania).