Optimized explicit runge–kutta schemes for entropy stable discontinuous collocated methods applied to the euler and navier–stokes equations

In this work, we design a new set of optimized explicit Runge–Kutta schemes for the integration of systems of ordinary differential equations arising from the spatial discretization of wave propagation problems with entropy stable collocated discontinuous Galerkin methods. The optimization of the new time integration schemes is based on the spectrum of the discrete spatial operator for the advection equation. To demonstrate the efficiency and accuracy of the new schemes compared to some widely used classic explicit Runge–Kutta methods, we report the wall-clock time versus the error for the simulation of the two-dimensional advection equation and the propagation of an isentropic vortex with the compressible Euler equations. The efficiency and robustness of the proposed optimized schemes for more complex flow problems are presented for the three-dimensional Taylor–Green vortex at a Reynolds number of Re = 1.6 × 103 and Mach number Ma = 0.1, and the flow past two identical spheres in tandem at a Reynolds number of Re = 3.9 × 103 and Mach number Ma = 0.1.