Volume 55, Number 6, November-December 2021
|Page(s)||2567 - 2608|
|Published online||15 November 2021|
A new entropy-variable-based discretization method for minimum entropy moment approximations of linear kinetic equations⋆
Fachbereich Mathematik und Informatik, WWU Münster, Einsteinstrasse 62, 48149 Münster, Germany
* Corresponding author: firstname.lastname@example.org; email@example.com
Accepted: 4 October 2021
In this contribution we derive and analyze a new numerical method for kinetic equations based on a variable transformation of the moment approximation. Classical minimum-entropy moment closures are a class of reduced models for kinetic equations that conserve many of the fundamental physical properties of solutions. However, their practical use is limited by their high computational cost, as an optimization problem has to be solved for every cell in the space-time grid. In addition, implementation of numerical solvers for these models is hampered by the fact that the optimization problems are only well-defined if the moment vectors stay within the realizable set. For the same reason, further reducing these models by, e.g., reduced-basis methods is not a simple task. Our new method overcomes these disadvantages of classical approaches. The transformation is performed on the semi-discretized level which makes them applicable to a wide range of kinetic schemes and replaces the nonlinear optimization problems by inversion of the positive-definite Hessian matrix. As a result, the new scheme gets rid of the realizability-related problems. Moreover, a discrete entropy law can be enforced by modifying the time stepping scheme. Our numerical experiments demonstrate that our new method is often several times faster than the standard optimization-based scheme.
Mathematics Subject Classification: 35L40 / 35F61 / 65M08 / 65M70 / 82C70
Key words: Moment models / minimum entropy / kinetic transport equation / model order reduction / realizability
Supplementary Online Material is only available in electronic form at https://doi.org/10.1051/m2an/2021065/olm.
© The authors. Published by EDP Sciences, SMAI 2021
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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