A regularized entropy-based moment method for kinetic equations

Graham W. Alldredge; Martin Frank; Cory D. Hauck

doi:10.1137/18M1181201

A regularized entropy-based moment method for kinetic equations

Graham W. Alldredge, Martin Frank, Cory D. Hauck

Multiscale Methods

Research output: Contribution to journal › Article › peer-review

15 Scopus citations

Abstract

We present a new entropy-based moment method for the velocity discretization of kinetic equations. This method is based on a regularization of the optimization problem defining the original entropy-based moment method, and this gives the new method the advantage that the moment vectors of the solution do not have to take on realizable values. We show that this equation still retains many of the properties of the original equations, including hyperbolicity, an entropy-dissipation law, and rotational invariance. The cost of the regularization is mismatch between the moment vector of the solution and that of the ansatz returned by the regularized optimization problem. However, we show how to control this error using the parameter defining the regularization. This suggests that with proper choice of the regularization parameter, the new method can be used to generate accurate solutions of the original entropy-based moment method, and we confirm this with numerical simulations.

Original language	English
Pages (from-to)	1627-1653
Number of pages	27
Journal	SIAM Journal on Applied Mathematics
Volume	79
Issue number	5
DOIs	https://doi.org/10.1137/18M1181201
State	Published - 2019

Bibliographical note

Publisher Copyright:
© 2019 Society for Industrial and Applied Mathematics

Funding

\ast Received by the editors April 16, 2018; accepted for publication (in revised form) June 18, 2019; published electronically September 5, 2019. https://doi.org/10.1137/18M1181201 Funding: The work of the first author was supported by the Deutsche Forschungsgemeinschaft through project ID AL 2030/1-1. This work was supported by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing, and performed at Oak Ridge National Laboratory (ORNL), managed by UT-Battelle, LLC, for the U.S. Department of Energy under contract DE-AC05-00OR22725. \dagger Department of Mathematics and Computer Science, Freie Universit\a"t Berlin, 14195 Berlin, Germany ([email protected]). \ddagger Department of Mathematics, Karlsruhe Institute of Technology, D-76128 Karlsruhe, Germany ([email protected]). \S Computational Mathematics Group, Computer Science and Mathematics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831 ([email protected]).

Funders	Funder number
UT-Battelle	DE-AC05-00OR22725
U.S. Department of Energy
Office of Science
Advanced Scientific Computing Research
Oak Ridge National Laboratory	ORNL
Deutsche Forschungsgemeinschaft	ID AL 2030/1-1

Keywords

Entropy
Moment methods
Numerical methods for kinetic equations

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10.1137/18M1181201

Cite this

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AB - We present a new entropy-based moment method for the velocity discretization of kinetic equations. This method is based on a regularization of the optimization problem defining the original entropy-based moment method, and this gives the new method the advantage that the moment vectors of the solution do not have to take on realizable values. We show that this equation still retains many of the properties of the original equations, including hyperbolicity, an entropy-dissipation law, and rotational invariance. The cost of the regularization is mismatch between the moment vector of the solution and that of the ansatz returned by the regularized optimization problem. However, we show how to control this error using the parameter defining the regularization. This suggests that with proper choice of the regularization parameter, the new method can be used to generate accurate solutions of the original entropy-based moment method, and we confirm this with numerical simulations.

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A regularized entropy-based moment method for kinetic equations

Abstract

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