Efficient two-step Runge-Kutta methods for fluid dynamics simulations

Explicit two-step Runge-Kutta (TSRK) methods offer an efficient alternative to traditional explicit Low-Storage Runge-Kutta (LSRK) schemes for solving the Navier-Stokes equations. A special class of TSRK methods that reduce requirement compared to previous TSRK schemes are derived. Schemes of fourth, fifth and sixth order are implemented and tested. The new schemes are evaluated with two common test cases, a 2D cylinder and a 3D Taylor-Green vortex. The results are compared with classical time discretization strategies. Timings obtained in three different hardware configurations show that the new TSRK methods of order four are 25% faster than LSRK schemes of the same order. Fifth and sixth order TSRK methods are tested with the same 3D test case and the results are compared to LSRK algorithms. Results show TSRK schemes of fifth and sixth order are competitive compared to LSRK methods of the same orders, as LSRK methods are of second order for non linear differential systems.