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Telescopic Projective Integration for Linear Kinetic Equations with Multiple Relaxation Times

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Abstract

We study a general, high-order, fully explicit numerical method for simulating kinetic equations with a BGK-type collision model with multiple relaxation times. In that case, the problem is stiff and its spectrum consists of multiple separated eigenvalue clusters. Projective integration methods are explicit integration schemes that first take a few small (inner) steps with a simple, explicit method, after which the solution is extrapolated forward in time over a large (outer) time step. These are very efficient schemes, provided there are only two clusters of eigenvalues, one corresponding to a single fast relaxation time scale, and one corresponding to the slow macroscopic dynamics. Here, we show how telescopic projective integration can be used to efficiently integrate kinetic equations with multiple relaxation times. Telescopic projective integration generalizes the idea of projective integration by constructing a hierarchy of projective levels. The main idea is to adjust the size of the inner time step at each level to one of the relaxation time scales. We show that the size of the outer time step, as well as the required number of inner steps at each level, does not depend on the stiffness of the problem. The computational cost of the method depends on the stiffness of the problem only via the number of projective levels. For problems with a fixed number of well-separated spectral clusters, the number of projective levels is independent of the stiffness, and the computational cost of telescopic projective integration is independent of the stiffness. For problems with a time-varying spectrum, the number of projective levels grows logarithmically with the stiffness. We illustrate numerically that, also in that case, the resulting computational cost is acceptable.

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Acknowledgements

We would like to thank Thomas Rey (Laboratoire Paul Painlevé, Université de Lille) for providing us with clear background material on Boltzmann and BGK kinetic equations and his assistance with their linearization, which supplemented the motivation of this work.

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  1. Department of Computer Science, K.U. Leuven, Celestijnenlaan 200A, 3001, Louvain, Belgium

    Ward Melis & Giovanni Samaey

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  1. Ward Melis
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Correspondence to Ward Melis.

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Melis, W., Samaey, G. Telescopic Projective Integration for Linear Kinetic Equations with Multiple Relaxation Times. J Sci Comput 76, 697–726 (2018). https://doi.org/10.1007/s10915-017-0635-0

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