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Overlapping domain decomposition based exponential time differencing methods for semilinear parabolic equations. (English) Zbl 1460.65115

Summary: The localized exponential time differencing method based on overlapping domain decomposition has been recently introduced and successfully applied to parallel computations for extreme-scale numerical simulations of coarsening dynamics based on phase field models. In this paper, we focus on numerical solutions of a class of semilinear parabolic equations with the well-known Allen-Cahn equation as a special case. We first study the semi-discrete system under the standard central difference spatial discretization and prove the equivalence between the monodomain problem and the corresponding multidomain problem obtained by the Schwarz waveform relaxation iteration. Then we develop the fully discrete localized exponential time differencing schemes and, by establishing the maximum bound principle, prove the convergence of the fully discrete localized solutions to the exact semi-discrete solution and the convergence of the iterative solutions. Numerical experiments are carried out to verify the theoretical results in one-dimensional space and test the convergence and accuracy of the proposed algorithms with different numbers of subdomains in two-dimensional space.

MSC:

65M55 Multigrid methods; domain decomposition for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65R20 Numerical methods for integral equations
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35K58 Semilinear parabolic equations
65Y05 Parallel numerical computation
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs

Software:

SERK2
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Full Text: DOI

References:

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