Optimal Runge-Kutta Stability Polynomials for Multidimensional High-Order Methods

In this paper we generate optimized Runge-Kutta stability polynomials for multidimensional discontinuous Galerkin methods recovered using the flux reconstruction approach. Results from linear stability analysis demonstrate that these stability polynomials can yield significantly larger time-step sizes for triangular, quadrilateral, hexahedral, prismatic, and tetrahedral elements with speedup factors of up to 1.97 relative to classical Runge-Kutta methods. Furthermore, performing optimization for multidimensional elements yields modest performance benefits for the triangular, prismatic, and tetrahedral elements. Results from linear advection demonstrate these schemes obtain their designed order of accuracy. Results from Direct Numerical Simulation (DNS) of a Taylor-Green vortex demonstrate the performance benefit of these schemes for unsteady turbulent flows, with negligible impact on accuracy.

Automated summary

This paper presents optimized Runge-Kutta stability polynomials for multidimensional discontinuous Galerkin methods using the flux reconstruction approach. The stability polynomials significantly increase time-step sizes for various elements, with up to a speedup factor of 1.97 compared to classical methods. The optimization also yields modest performance benefits for certain elements and maintains the designed accuracy levels.