Parallel methods for ordinary differential equations. (English) Zbl 0675.65068
The author gives a survey on the methods for parallel integration of ordinary differential equations. Parallelism across space and parallelism across time are considered. Parallelism across time (or parallelism across the method) appears to be more appropriate for small-scale parallelism and it is provided by the use of block methods and frontal methods. Block methods generate a set of new values in a single integration step and include explicit and implicit Runge-Kutta, multistep and predictor-corrector approaches. Frontal methods are based on modified predictor-corrector method.
Large scale parallelism across time is appropriate for some special cases such as linear problems and nearly linear problems and the best results are obtained for the solution of stiff equations with O(log N) parallel time required. Parallelism across space which is most effective for the systems having a regular structure is briefly discussed. The ways in which parallelism must be applied effectively in future investigations are also considered.
Large scale parallelism across time is appropriate for some special cases such as linear problems and nearly linear problems and the best results are obtained for the solution of stiff equations with O(log N) parallel time required. Parallelism across space which is most effective for the systems having a regular structure is briefly discussed. The ways in which parallelism must be applied effectively in future investigations are also considered.
Reviewer: Yu.V.Rogovchenko
MSC:
| 65L05 | Numerical methods for initial value problems involving ordinary differential equations |
| 65Y05 | Parallel numerical computation |
| 34A34 | Nonlinear ordinary differential equations and systems |
Keywords:
Runge-Kutta method; multistep method; predictor-corrector method; Parallelism across space; parallelism across time; block methods; frontal methods; stiff equationsReferences:
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