Novel adaptive IMEX two-step Runge-Kutta temporal discretization methods for unsteady flows

Efficient and accurate simulation of unsteady flow presents a significant challenge that needs to be overcome in computational fluid dynamics. Temporal discretization method plays a crucial role in the simulation of unsteady flows. To enhance computational efficiency, we propose the Implicit-Explicit Two-Step Runge-Kutta (IMEX-TSRK) time-stepping discretization methods for unsteady flows, and develop a novel adaptive algorithm that correctly partitions spatial regions to apply implicit or explicit methods. The novel adaptive IMEX-TSRK schemes effectively handle the numerical stiffness of the small grid size and improve computational efficiency. Compared to implicit and explicit Runge-Kutta (RK) schemes, the IMEX-TSRK methods achieve the same order of accuracy with fewer first derivative calculations. Numerical case tests demonstrate that the IMEX-TSRK methods maintain numerical stability while enhancing computational efficiency. Specifically, in high Reynolds number flows, the computational efficiency of the IMEX-TSRK methods surpasses that of explicit RK schemes by more than one order of magnitude, and that of implicit RK schemes several times over.