Third-order Paired Explicit Runge-Kutta schemes for stiff systems of equations

The ability to advance locally-stiff systems of equations in time depends on accurate and efficient temporal schemes. Recently, a new family of Paired Explicit Runge-Kutta (P-ERK) methods has been proposed. This approach allows different Runge-Kutta schemes with different numbers of active stages to be assigned based on local stiffness criteria. Whereas the original P-ERK formulation was only second-order accurate, in this paper we propose a new third-order family. We present the general formulation for these schemes, and then optimize them for the high-order discontinuous Galerkin method recovered via the flux reconstruction approach. We then verify that these schemes obtain their designed third-order accuracy for non-linear systems of equations. Performance results show that they achieve speedup factors up to four relative to a classical third-order Runge-Kutta methods for laminar and turbulent flow over an SD7003 airfoil, and turbulent flow over a tandem sphere configuration. Based on these results, this new family of third-order P-ERK schemes provides an appealing approach for accurate and efficient solution of locally-stiff systems of equations. •We propose novel third-order Paired Explicit Runge-Kutta (P-ERK) schemes.•These are suitable for explicit solution of locally-stiff systems of equations.•These are then optimized for flux reconstruction and discontinuous Galerkin discretizations.•Verification shows all schemes achieve their designed order of accuracy in time.•Speedup factors of over four observed relative to classical RK schemes, at equivalent accuracy.