Explicit two-step Runge-Kutta methods for computational fluid dynamics solvers

Computational fluid dynamics (CFD) has emerged as a successful tool for industry applications and basic science during the last decades. However, accurate solutions involving vortex propagation, and separated and turbulent flows, are still associated with high computing costs. In particular, large eddy simulations (LES) of complex geometries, such as a complete automobile, require several days on thousands of cores in order to obtain solutions with statistically relevant information. With an increase in the number of available cores, the number of degrees of freedom (DOF) per core can be reduced accordingly. When the number of DOF per core is below a certain threshold the total simulation time is not bounded by floating point operations (FLOPS), but by the time spend on communication between cores. To overcome this impediment we have identified and tested a class of two-step Runge-Kutta (TSRK) methods of high order with low number of stages, for time discretization of differential systems resulting from space discretization of weakly compressible Navier-Stokes equations. These methods have not been used before in CFD simulations. The advantage of using these methods is reduction in communication times between cores. The numerical experiments indicate that the gains in computational performance of this new class of TSRK methods, as compared with classical Runge-Kutta (RK) methods or low storage Runge-Kutta (LSRK) schemes, are of the order of 25%, with no loss in accuracy.