Nonlinear convergence analysis for the parareal algorithm. (English) Zbl 1140.65336

Gander, Martin J.; Hairer, Ernst

Langer, Ulrich (ed.) et al., Domain decomposition methods in science and engineering XVII. Selected papers based on the presentations at the 17th international conference on domain decomposition methods, St. Wolfgang/Strobl, Austria, July 3–7, 2006. Berlin: Springer (ISBN 978-3-540-75198-4/pbk). Lecture Notes in Computational Science and Engineering 60, 45-56 (2008).

From the text: We show that the parareal algorithm applied to a system of nonlinear ordinary differential equations converges superlinearly on any bounded time interval. We illustrate this result with four nonlinear examples coming from chemical reactions, planetary orbits, weather forecast and fluid flow problems.
These examples show that parallel speedup in time is possible, although not at the same level as in space, where one often asks for perfect speedup, i.e. the computation with one hundred processors should be one hundred times faster. For time parallelization with the parareal algorithm, one has to be satisfied with less, but if this is the only option left to speedup the solution time, it might be worthwhile considering it.
For the entire collection see [Zbl 1130.65004].

MSC:

65L05	Numerical methods for initial value problems
34A34	Nonlinear ordinary differential equations and systems, general theory
65L20	Stability and convergence of numerical methods for ordinary differential equations
65Y05	Parallel numerical computation

Keywords:

convergence; parallel computation; numerical examples; chemical reactions; planetary orbits; weather forecast; fluid flow problems