Time-Parallel Multigrid in an Extrapolation Method for Time-Dependent Partial Differential Equations

We consider the parallel solution of unsteady partial differential equations with the two-dimensional heat equation as a model problem. Conventional implicit integration methods for the solution of this type of equation proceed by solving a sequence of problems iteratively. It is shown that despite the sequential nature of this process, several processors may be employed to solve at several time-steps simultaneously. The accuracy of an integration method such as backward Euler may be enhanced by embedding it in an extrapolation method which itself contains algorithmic parallelism. A solution procedure based on the multigrid method is presented which utilizes both kinds of parallelism. The efficiencies obtained on a message-passing multiprocessor prove the suitability of the method for this type of problem.