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.
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X. Li’s work is partially supported by National Natural Science Foundation of China grant 11801024. L. Ju’s work is partially supported by US National Science Foundation grant DMS-1818438 and US Department of Energy grants DE-SC0016540 and DE-SC0020270.
Communicated by Christian Lubich.
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Li, X., Ju, L. & Hoang, T. Overlapping domain decomposition based exponential time differencing methods for semilinear parabolic equations. Bit Numer Math (2020). https://doi.org/10.1007/s10543-020-00817-0
- Semilinear parabolic equation
- Overlapping domain decomposition
- Localized exponential time differencing
- Parallel Schwarz iteration
- Waveform relaxation
- Convergence analysis
Mathematics Subject Classification