Temporal accuracy of FastRK3

Abstract

FastRK3 by Aithal and Ferrante (2020) [1] is an explicit, three-stage, third-order Runge-Kutta (RK3) based projection-method for solving the incompressible Navier-Stokes equations written in curvilinear coordinates using the orthogonal formulation. FastRK3 requires solving the Poisson equation for pressure only once per time step versus three times of standard RK3 methodology. In this work, we report the results of a theoretical and numerical study which is focused on the temporal accuracy of FastRK3. For free-shear flows, we derive theoretically and show numerically that FastRK3 is third-order accurate in time for velocity and second-order accurate in time for pressure when the RK3 coefficients and the pressure extrapolation scheme satisfy specific conditions herein derived. For ‘stiff’ wall-bounded flows, we show numerically that the temporal accuracy of velocity and pressure is the same, i.e., first- or second-order depending on the CFL number, for both FastRK3 and standard RK3. In summary, we derive theoretically and verify numerically the conditions that the RK3 coefficients and the pressure extrapolation scheme need to satisfy in order for FastRK3 to preserve the temporal accuracy of the underlying RK3 methodology.