MiS Preprint Repository

Time-stepping procedures for the solution of evolution equations can be performed on parallel architecture by parallelizing the space computation at each time step. This, however, requires heavy communication between processors and becomes inefficient when many time-steps are to be computed and many processors are available. In such cases parallelization in time is advantageous.

In this paper we present a method for parallelization in time of linear multistep discretizations of linear evolution problems; we consider a model parabolic and a model hyperbolic problem, and their, respectively, $A(\theta )$-stable and $A$-stable linear multistep discretizations. The method consists of a discrete decoupling procedure, whereby $N+1$ decoupled Helmholtz problems with complex frequencies are obtained; $N$ being the number of time steps computed in parallel. The usefulness of the method rests on our ability to solve these Helmholtz problems efficiently. We discuss the theory and give numerical examples for multigrid preconditioned iterative solvers of relevant complex frequency Helmholtz problems. The parallel approach can easily be combined with a time-stepping procedure, thereby obtaining a block time-stepping method where each block of steps is computed in parallel. In this way we are able to optimize the algorithm with respect to the number of processors available, the difficulty of solving the Helmholtz problems, and the possibility of both time and space adaptivity. Extensions to other linear evolution problems and to Runge-Kutta time disretizations are briefly mentioned.