Positivity Limiters for Filtered Spectral Approximations of Linear Kinetic Transport Equations

M. Paul Laiu; Cory D. Hauck

doi:10.1007/s10915-018-0790-y

Positivity Limiters for Filtered Spectral Approximations of Linear Kinetic Transport Equations

M. Paul Laiu, Cory D. Hauck

Multiscale Methods

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

8 Scopus citations

Abstract

We analyze the properties and compare the performance of several positivity limiters for spectral approximations with respect to the angular variable of linear transport equations. It is well-known that spectral methods suffer from the occurrence of (unphysical) negative spatial particle concentrations due to the fact that the underlying polynomial approximations are not always positive at the kinetic level. Positivity limiters address this defect by enforcing positivity of the polynomial approximation on a finite set of preselected points. With a proper PDE solver, they ensure positivity of the particle concentration at each step in a time integration scheme. We review several known positivity limiters proposed in other contexts and also introduce a modification for one of them. We give error estimates for the consistency of the positive approximations produced by these limiters and compare the theoretical estimates to numerical results. We then solve two benchmark problems with these limiters, make qualitative and quantitative observations about the solutions, and then compare the efficiency of the different limiters.

Original language	English
Pages (from-to)	918-950
Number of pages	33
Journal	Journal of Scientific Computing
Volume	78
Issue number	2
DOIs	https://doi.org/10.1007/s10915-018-0790-y
State	Published - Feb 15 2019

Funding

This manuscript has been authored, in part, by UT-Battelle, LLC, under Contract No. DE-AC0500OR22725 with the U.S. Department of Energy. The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this manuscript, or allow others to do so, for the United States Government purposes. The Department of Energy will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan (http://energy.gov/downloads/doe-public-access-plan). M. Paul Laiu: Supported by the U.S. Department of Energy, under the SCGSR program administered by the Oak Ridge Institute for Science and Education under Contract No. DE-AC05-06OR23100. Supported by the U.S. National Science Foundation under Grant No. 1217170. Cory D. Hauck: This author’s research was sponsored by the Office of Advanced Scientific Computing Research and performed at the Oak Ridge National Laboratory, which is managed by UT-Battelle, LLC under Contract No. DE-AC05-00OR22725. We thank Professor Xiangxiong Zhang for pointing out the sweeping limiter in [31] to the authors.

Funders	Funder number
National Science Foundation
U.S. Department of Energy
Directorate for Mathematical and Physical Sciences	1217170
Advanced Scientific Computing Research	DE-AC05-00OR22725
Oak Ridge Institute for Science and Education	DE-AC05-06OR23100
UT-Battelle	DE-AC0500OR22725

Keywords

Filters
Kinetic equation
Positivity-preserving limiters
Spectral methods

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10.1007/s10915-018-0790-y

Cite this

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abstract = "We analyze the properties and compare the performance of several positivity limiters for spectral approximations with respect to the angular variable of linear transport equations. It is well-known that spectral methods suffer from the occurrence of (unphysical) negative spatial particle concentrations due to the fact that the underlying polynomial approximations are not always positive at the kinetic level. Positivity limiters address this defect by enforcing positivity of the polynomial approximation on a finite set of preselected points. With a proper PDE solver, they ensure positivity of the particle concentration at each step in a time integration scheme. We review several known positivity limiters proposed in other contexts and also introduce a modification for one of them. We give error estimates for the consistency of the positive approximations produced by these limiters and compare the theoretical estimates to numerical results. We then solve two benchmark problems with these limiters, make qualitative and quantitative observations about the solutions, and then compare the efficiency of the different limiters.",

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N2 - We analyze the properties and compare the performance of several positivity limiters for spectral approximations with respect to the angular variable of linear transport equations. It is well-known that spectral methods suffer from the occurrence of (unphysical) negative spatial particle concentrations due to the fact that the underlying polynomial approximations are not always positive at the kinetic level. Positivity limiters address this defect by enforcing positivity of the polynomial approximation on a finite set of preselected points. With a proper PDE solver, they ensure positivity of the particle concentration at each step in a time integration scheme. We review several known positivity limiters proposed in other contexts and also introduce a modification for one of them. We give error estimates for the consistency of the positive approximations produced by these limiters and compare the theoretical estimates to numerical results. We then solve two benchmark problems with these limiters, make qualitative and quantitative observations about the solutions, and then compare the efficiency of the different limiters.

AB - We analyze the properties and compare the performance of several positivity limiters for spectral approximations with respect to the angular variable of linear transport equations. It is well-known that spectral methods suffer from the occurrence of (unphysical) negative spatial particle concentrations due to the fact that the underlying polynomial approximations are not always positive at the kinetic level. Positivity limiters address this defect by enforcing positivity of the polynomial approximation on a finite set of preselected points. With a proper PDE solver, they ensure positivity of the particle concentration at each step in a time integration scheme. We review several known positivity limiters proposed in other contexts and also introduce a modification for one of them. We give error estimates for the consistency of the positive approximations produced by these limiters and compare the theoretical estimates to numerical results. We then solve two benchmark problems with these limiters, make qualitative and quantitative observations about the solutions, and then compare the efficiency of the different limiters.

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Positivity Limiters for Filtered Spectral Approximations of Linear Kinetic Transport Equations

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