Temporal accuracy of FastRK3

FastRK3 is an explicit, three-stage, third-order Runge-Kutta (RK3) based projection-method which requires solving the Poisson equation for pressure only once per time step versus three times of standard RK3 methodology (Aithal & Ferrante, J. Comp. Phys., 2020). The present work is focused on the temporal accuracy of FastRK3 in comparison with the underlying standard RK3 methodology. We derive theoretically and show numerically that, for free-shear flows, FastRK3 is third-order and second-order accurate in time for velocity and pressure, respectively, when the RK3 coefficients and the pressure extrapolation scheme satisfy specific conditions herein derived. For wall-bounded flows, specifically the lid-driven polar cavity, we show numerically that the temporal accuracy of velocity and pressure is second-order 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.