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An empirical numerical study of the Parareal method applied to the Burgers Equation

Banff International Research Station for Mathematical Innovation and Discovery

Solving industrial sized problems in the field of computational fluid dynamics is a big challenge. Especially in flows with high Reynolds numbers (flows with a high ratio between convection and diffusion) small turbulent structures have to be resolved with a very fine grid in space and time. These fine grids lead, even with spatial parallelization, to long computational times, since also a reasonable amount of physical time has to be simulated. This problem can be partially circumvented by modelling the turbulent structures as it is done with statistical turbulence models. These modelling techniques allow coarser grids and shorter simulation runtimes with the drawback of a less accurate solution. Nevertheless, these simulations can still have a runtime in the range of months. For these problems a parallelization in time in addition to the parallelization in space is very appealing. The parallelization in time can be used to reduce the computational load of the typically longer physical time which is to be simulated. The easiest parallel in time method, the Parareal method, would be a good starting point for the runtime reduction. Unfortunately, it was already shown by multiple publications, that the Parareal method in its original formulation is not suitable for high Reynolds flows [e.g. Kreienbuehl et al., 2015]. The Parareal method has stability problems which occur with high Reynolds number flows. These stability problems can be investigated by applying the Parareal method to the viscous Burgers equation, which is closely related to the Navier-Stokes equations. Therefore, a parameter study was done by varying the Reynolds number of the simulated flow. In addition, it was investigated whether a scale difference of the structures in the flow have an impact on the convergence. These structures which model the turbulent scales are induced according to [Kooij et al., 2015]. Furthermore, the effect of applying different time stepping schemes was studied. The used schemes were of explicit Runge-Kutta and IMEX Runge-Kutta nature. Moreover, it will be presented whether it is possible to reduce the stability problems of applying the Parareal method to high Reynolds number flows with a semi-Lagragian formulation of the examined equations. The work of Andreas Schmitt is supported by the 'Excellence Initiative' of the German Federal and State Governments and the Graduate School of Computational Engineering at Technische Universit\"at Darmstadt.

Mathematics; Numerical analysis; Partial differential equations

video/mp4

Author affiliation: Technische Universität Darmstadt

2017-05-15

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/61654

Unreviewed

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