Optimised Runge-Kutta schemes applied to the spectral difference method

The discontinuous spectral methods (Discontinuous Galerkin, Spectral Difference, Spec- tral Volume, Flux Reconstruction) offer high order accuracy, with spectral properties in agreement with the requirements for Large Eddy Simulations and Direct Numerical Simu- lations. Today, the standard way to integrate the compressible Navier-Stokes equations to deal with LES and DNS follows the Runge-Kutta time integration procedure. Even though this technique is one of the oldest ones, it is still applied successfully to the discontinuous spectral methods, but with a strong Courant Friedrichs Lewy (CFL) constraint to main- tain stability. In this paper, we propose new sets of coefficients for the Runge-Kutta time integration scheme, dedicated to the spectral difference method, that gives a higher stable CFL number compared to the standard Runge-Kutta schemes.