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Partitioned Adaptive Parallel Integrators for Coupled Stiff Systems

Banff International Research Station for Mathematical Innovation and Discovery

The efficient numerical simulation of stiff multiphysics systems remains a core challenge in scientific computing. Examples are fluid structure interaction, earth system models or turbulent flames. We consider problems with the following characteristics: They are large scale, all components are stiff, possibly on different time scales and there are codes for the subproblems available. Thus, we want a partitioned numerical method, meaning that reuse of the existing codes is possible. Thereby, we assume that while we have access to the source codes, we want to edit that code as little as possible. In particular we assume that we can repeat a time step. We are then looking for numerical methods that are implicit and at least order two, time adaptive, allow the subsolvers to run in parallel and allow for different time steps in the different models. We are not aware of a method that fulfills all of these properties and suggest two methods of our own for the case of two systems being coupled. The core idea is the following: We have a time integration method of at least order two for each subproblem and assume that we can restart these with new initial data and that during time integration, information for the other solver at all times can be provided using interpolation. This continuous representation of the numerical solution is updated after each local time step. Then the solvers run in parallel over a macro time window and are free to choose their own timesteps in an adaptive way without outside interference. At the end of the macrostep, it is checked if the coupled system is fulfilled up to a tolerance, if not, the time window is repeated. Crucial questions are order of the time integration method and convergence of the time window iteration, also called waveform relaxation. This is shown numerically for representative test cases. For the specific case of two linear heat equations with different material properties coupled across an interface, we suggest to do the waveform relaxation in the form of a Neumann-Neumann coupling, known from domain decomposition. There, the choice of the relaxation parameter is crucial and previous analysis by Gander and Kwok for the semidiscrete case does not apply. We thus perform a fully discrete analysis for the case of fixed but different time steps for the subproblems. Numerical results show that this can be used for the time adaptive case as well.

Mathematics; Numerical analysis; Partial differential equations; Scientific computing

video/mp4

Author affiliation: Lund University

2019-06-02

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/70452

Unreviewed

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