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.
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Keywords: Moment models, minimum entropy, kinetic transport equation, model order reduction, realizability
@article{M2AN_2021__55_6_2567_0,
author = {Leibner, Tobias and Ohlberger, Mario},
title = {A new entropy-variable-based discretization method for minimum entropy moment approximations of linear kinetic equations},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {2567--2608},
year = {2021},
publisher = {EDP-Sciences},
volume = {55},
number = {6},
doi = {10.1051/m2an/2021065},
mrnumber = {4338789},
language = {en},
url = {https://www.numdam.org/articles/10.1051/m2an/2021065/}
}
TY - JOUR AU - Leibner, Tobias AU - Ohlberger, Mario TI - A new entropy-variable-based discretization method for minimum entropy moment approximations of linear kinetic equations JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2021 SP - 2567 EP - 2608 VL - 55 IS - 6 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/m2an/2021065/ DO - 10.1051/m2an/2021065 LA - en ID - M2AN_2021__55_6_2567_0 ER -
%0 Journal Article %A Leibner, Tobias %A Ohlberger, Mario %T A new entropy-variable-based discretization method for minimum entropy moment approximations of linear kinetic equations %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2021 %P 2567-2608 %V 55 %N 6 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/m2an/2021065/ %R 10.1051/m2an/2021065 %G en %F M2AN_2021__55_6_2567_0
Leibner, Tobias; Ohlberger, Mario. A new entropy-variable-based discretization method for minimum entropy moment approximations of linear kinetic equations. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 55 (2021) no. 6, pp. 2567-2608. doi: 10.1051/m2an/2021065
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