Embedded paired explicit Runge-Kutta schemes

Abstract

Recently, Paired Explicit Runge-Kutta (P-ERK) schemes were introduced to accelerate the solution of locally-stiff systems of equations. P-ERK schemes use different numbers of active stages based on local stiffness criteria, significantly reducing computational cost relative to classical explicit Runge-Kutta schemes. In the current work we present a framework for incorporating an embedded pair within the original P-ERK formulation. By design, the embedded scheme has a slightly smaller stability limit than its corresponding base scheme. When combined with a suitable controller, the difference between the base and embedded solution approximations can be used to adapt the time step size to be as close as possible to, without exceeding, the stability limit of the base scheme. Numerical results using the high-order flux reconstruction approach for spatial discretization of the compressible Navier-Stokes equations are presented for flow over a circular cylinder, a plunging and pitching NACA 0012 airfoil, and a stalled NACA 0020 airfoil. It is demonstrated that adaptive time stepping using embedded P-ERK schemes yields excellent agreement with reference data, while being up to seven times less computationally expensive than classical embedded pairs for all cases.