Abstract
Periodic solutions of autonomous parabolic partial differential systems can be computed by using a shooting method, applied to the (large) system of ordinary differential equations that results after spatial discretization. Many time integrations of this system may be required in each shooting iteration step. Hence these calculations can be extremely expensive, especially in the case of fine spatial discretizations and/or higher dimensional problems.
This paper deals with techniques to make the shooting method more feasible by reducing the arithmetic complexity of the standard approach and by exploiting parallelism and vectorization. The former is achieved by providing a ‘coarse grid approximation’ to the Jacobian matrix, that arises in the shooting algorithm. This leads to a substantial reduction of the number of required time integrations. The latter is based on the application of the Multigrid Waveform Relaxation algorithm, a recent technique for time-integration of parabolic partial differential equations that is well suited for vector and parallel computers. Results are given for the time-integration and the calculation of periodic solutions of the two-dimensional Brusselator-model on a distributed memory parallel computer.
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Roose, D., Vandewalle, S. (1991). Efficient Parallel Computation of Periodic Solutions of Parabolic Partial Differential Equations. In: Seydel, R., Schneider, F.W., Küpper, T., Troger, H. (eds) Bifurcation and Chaos: Analysis, Algorithms, Applications. International Series of Numerical Mathematics / Internationale Schriftenreihe zur Numerischen Mathematik / Série Internationale d’Analyse Numérique, vol 97. Birkhäuser Basel. https://doi.org/10.1007/978-3-0348-7004-7_40
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DOI: https://doi.org/10.1007/978-3-0348-7004-7_40
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