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Using time-parallel methods for the simulation of multi-domain parabolic equations

Banff International Research Station for Mathematical Innovation and Discovery

We consider parabolic partial differential equations defined on multiple domains. These domains are coupled at the boundary by Robin boundary conditions where the region of overlap is time-dependent. The rate of change of the geometry is much faster than the time scale of heat conduction. We apply the spectral deferred correction approach as well as a splitting in slow and fast components to this type of problem. We use the PFASST concept to obtain a parallel implementation of these concepts. One basic ingredient of PFASST are the underlying spectral deferred correction methods. Spectral deferred correction (SDC) methods start from a provisional solution. Using a simpler time integrator, this provisional solution will be iteratively improved. There are several adaptions of SDC for multi-rate problems. MISDC methods treat every scale independently in the sweeper and allows to construct high-order multi-rate methods. Typically, slow processes are treated explicitly and fast processes implicitly. We adapt the idea of the MISDC methods for the coupled heat equation and treat the diffusion part implicitly, but the fast sources in an explicit manner to avoid implicit solves for the geometry. The method will be analyzed with respect to order and stability. Finally, we present numerical results.

Mathematics; Numerical analysis; Partial differential equations

video/mp4

Author affiliation: Technische Universität Dresden

2017-05-15

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/61657

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/