Numerical studies of a multigrid version of the parareal algorithm

Abstract

In this work, a parallel-in-time method is combined with a multigrid algorithm and further on with a spatial coarsening strategy. The most famous parallel-in-time method is the parareal algorithm. Depending on two different operators, it enables the parallelism of time-dependent problems. The operator with huge effort is carried out in parallel. But despite parallelization this can lead to long run times for long-term problems. Since the parareal algorithm has a two-level structure and the time-parallel multigrid methods are also widespread in the area of parallel time integration, we combine these approaches. We use the parareal algorithm as a smoothing operator in the basic framework of a geometrical multigrid method, where we apply a coarsening strategy in time. So we get a multigrid in time method which is strongly parallelizable. For partial differential equations we add an extra spatial coarsening strategy to our multigrid parareal version. All in all we get a method, which has a high parallel efficiency and converges fast due to the multigrid framework, which is shown in the numerical studies of this work. So we will get a highly accurate solution and can greatly reduce the parallel complexity, which is especially important for long-term problems with a limited number of processors.