Accelerated implicit-explicit Runge-Kutta schemes for locally stiff systems

Abstract

In this paper we introduce a family of accelerated implicit-explicit (AIMEX) schemes for the solution of stiff systems of equations. Similar to conventional IMEX schemes, AIMEX schemes allow a problem to be split into its stiff and non-stiff constituent parts. These are then advanced in time using an implicit and explicit scheme, respectively. By design, AIMEX schemes have an arbitrarily large number of stages. This allows for optimization of the explicit part to improve its stability properties, increasing the allowable time step size. Importantly, only two implicit stages are required regardless of the total number of stages, meaning a larger global time step can be taken with significantly fewer implicit stages for a given simulation time. Numerical results demonstrate that AIMEX schemes achieve their designed order of accuracy for linear and non-linear problems involving mesh induced stiffness. Simulations of unsteady flow over an SD7003 airfoil using the compressible Navier-Stokes equations demonstrate that AIMEX schemes can significantly outperform classical explicit Runge-Kutta schemes by over a factor of 20, and conventional IMEX schemes by over a factor of two, with negligible impact on quantitative results. Finally, a demonstration case of Implicit Large Eddy Simulation (ILES) of flow over a stalled NACA0020 airfoil shows the utility of AIMEX schemes for wall-bounded turbulent flows.