Optimal explicit Runge-Kutta methods for compressible Navier-Stokes equations

We focus our attention on the numerical simulations of compressible flows obtained by using Finite Difference in time /Finite Element in space approximation. In particular, we determine optimal explicit Runge-Kutta methods capable to maximize the stability features of the resulting numerical scheme. Two different regimes characterized by low and moderate Mach numbers have been taken into account. In the former regime, we have determined an explicit Runge-Kutta method of fourth order that is approximately 15% more efficient than classical ERK(4,4) schemes. For moderate Mach numbers, Ma=0.4, and transitional Reynolds numbers we have determined ERK schemes that outperform classic ERK(3,3) or ERK(4,4). Optimal ERK have a reduced CFL approximatively four or five times larger than classical ones. These optimized ERK schemes are then promising for the study of transitional flows for global stability or transient growth analyses.