Multirate timestepping for the incompressible Navier-Stokes equations in overlapping grids

Ketan Mittal, Som Dutta, Paul Fischer

Abstract: Highlights • Multirate timestepping is developed for the incompressible Navier-Stokes equations. • Overlapping grids use timestep size based on local CFL for temporal integration. • Predictor-corrector scheme is used to maintain high-order temporal accuracy. • Method scales to arbitrary number of subdomains with arbitrary timestep ratio. • Method is demonstrated to accurately model complex turbulent flow phenomenon. Abstract We develop a multirate timestepper for semi-implicit solutions of the unsteady incompressible Navier-Stokes equations (INSE) based on a recently-developed multidomain spectral element method (SEM) [1]. For incompressible flows, multirate timestepping (MTS) is particularly challenging because of the tight coupling implied by the incompressibility constraint, which manifests as an elliptic subproblem for the pressure at each timestep. The novelty of our approach stems from the development of a stable overlapping Schwarz method applied directly to the Navier-Stokes equations, rather than to the convective, viscous, and pressure substeps that are at the heart of most INSE solvers. Our MTS approach is based on a predictor-corrector (PC) strategy that preserves the temporal convergence of the underlying semi-implicit timestepper. We present numerical results demonstrating that this approach scales to an arbitrary number of overlapping grids, accurately models complex turbulent flow phenomenon, and improves computational efficiency in comparison to singlerate timestepping-based calculations.