

Preprint 26/2009
Parallel multistep methods for linear evolution problems
Lehel Banjai and Daniel Peterseim
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Submission date: 24. Jun. 2009 (revised version: February 2010)
Pages: 27
published in: IMA journal of numerical analysis, 32 (2012) 3, p. 1217-1240
DOI number (of the published article): 10.1093/imanum/drq040
Bibtex
MSC-Numbers: 65M20, 65Y05, 35L05, 35K05
Keywords and phrases: wave equation, heat equation, parallel computation
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Abstract:
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,
-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.