Coarse-grid operator optimization in multigrid reduction in time for time-dependent Stokes and Oseen problems

Abstract

Multigrid reduction in time (MGRIT), one of the most popular parallel-in-time approaches, extracts temporal parallelism by constructing coarse grids in the time direction. The coarse-grid operator optimization method for MGRIT has achieved high convergence for one of the hyperbolic problems that had poor convergence performance: the one-dimensional linear advection problems with constant coefficients. This paper applies this optimization method to two-dimensional linear time-dependent Stokes and Oseen problems using the pressure projection and the staggered grid discretization methods. Although the time-stepping operator involves the projection operator, the commutativity in the periodic boundary conditions allows a similar adaptation of the coarse-grid operator optimization for scalar equations. This method can also be applied to Dirichlet boundary problems by modifying the operator obtained based on the assumption of periodic boundary conditions. We demonstrate that MGRIT can achieve reasonable convergence rates for these problems with a practical number of non-zero elements by using the optimization method. Numerical experiments show convergence estimates for periodic boundary problems, applications to Dirichlet boundary problems, and parallel results compared to the sequential time-stepping method.