Optimized explicit Runge–Kutta schemes for high-order collocated discontinuous Galerkin methods for compressible fluid dynamics

In compressible computational fluid dynamics, the step size of explicit time integration schemes is often constrained by stability when high-order accurate spatial discretizations are used. We report a set of new optimized explicit Runge–Kutta schemes for the integration of systems of ordinary differential equations arising from the spatial discretization of wave propagation problems with high-order entropy stable collocated discontinuous Galerkin methods. The eigenvalues of the discrete spatial operator for the advection equation and the propagation of an isentropic vortex with the compressible Euler equations for various values of the problems' parameters are used to optimize the stability region of the proposed time integration schemes. To demonstrate the efficiency and the robustness of the methods, we solve the compressible turbulent flow past the Valeo controlled-diffusion airfoil and a delta wing at a Reynolds number of 8.3×10⁵ and 10⁶, respectively. A thorough analysis of the performance of the two families of optimized schemes revealed that methods generated using the spectra of the vortex problem are 6-to-20% faster than methods constructed using the spectra of the advection equation. Compared to widely used explicit Runge–Kutta schemes, the methods designed using the spectra of the vortex problem yield a time-to-solution saving of approximately 6-to-38%. For large-scale time-dependent partial differential equations computations, these gains mean saving hundreds of thousands if not millions of core hours. In addition, the new methods can be effectively and efficiently applied to integrate systems of ordinary differential equations arising from a wide range of spatial discretization, including discontinuous Galerkin spectral element methods, spectral difference methods, and flux reconstruction methods.