Paired explicit Runge-Kutta schemes for stiff systems of equations

Abstract

In this paper we introduce a family of explicit Runge-Kutta methods, referred to as Paired Explicit Runge-Kutta (P-ERK) schemes, that are suitable for the solution of stiff systems of equations. The P-ERK approach allows Runge-Kutta schemes with a large number of derivative evaluations and large region of absolute stability to be used in the stiff parts of a domain, while schemes with relatively few derivative evaluations are used in non-stiff parts to reduce computational cost. Importantly, different P-ERK schemes with different numbers of derivative evaluations can be chosen based on local stiffness requirements and seamlessly paired with one another. We then verify that P-ERK schemes obtain their designed order of accuracy using the Euler equations with arbitrary combinations of schemes. We then demonstrate that P-ERK schemes can achieve speedup factors of approximately five for simulations using the Navier-Stokes equations including laminar and turbulent flow over an SD7003 airfoil. These results demonstrate that P-ERK schemes can significantly accelerate the solution of stiff systems of equations when using an explicit approach, and that they maintain accuracy with respect to conventional Runge-Kutta methods and available reference data.