Parallel defect control
Enright, W.H. and Higham, D.J. (1991) Parallel defect control. BIT Numerical Mathematics, 31 (4). pp. 647663. ISSN 00063835 (http://dx.doi.org/10.1007/BF01933179)
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How can smallscale parallelism best be exploited in the solution of nonstiff initial value problems? It is generally accepted that only modest gains inefficiency are possible, and it is often the case that ldquofastrdquo parallel algorithms have quite crude error control and stepsize selection components. In this paper we consider the possibility of using parallelism to improvereliability andfunctionality rather than efficiency. We present an algorithm that can be used with any explicit RungeKutta formula. The basic idea is to take several smaller substeps in parallel with the main step. The substeps provide an interpolation facility that is essentially free, and the error control strategy can then be based on a defect (residual) sample. If the number of processors exceeds (p  1)/2, wherep is the order of the RungeKutta formula, then the interpolant and the error control scheme satisfy very strong reliability conditions. Further, for a given orderp, the asymptotically optimal values for the substep lengths are independent of the problem and formula and hence can be computed a priori. Theoretical comparisons between the parallel algorithm and optimal sequential algorithms at various orders are given. We also report on numerical tests of the reliability and efficiency of the new algorithm, and give some parallel timing statistics from a 4processor machine.
ORCID iDs
Enright, W.H. and Higham, D.J. ORCID: https://orcid.org/0000000266353461;

Item type: Article ID code: 205 Dates: DateEventDecember 1991PublishedKeywords: RungeKutta, parallelism, defectinterpolation, numerical mathematics, Mathematics Subjects: Science > Mathematics Department: Faculty of Science > Mathematics and Statistics Depositing user: Ms Sarah Scott Date deposited: 09 Mar 2006 Last modified: 07 Dec 2022 02:46 URI: https://strathprints.strath.ac.uk/id/eprint/205