Positivity Limiters for Filtered Spectral Approximations of Linear Kinetic Transport Equations

  • M. Paul Laiu
  • Cory D. Hauck


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.


Kinetic equation Spectral methods Positivity-preserving limiters Filters 

Mathematics Subject Classification

35L02 41A10 41A25 41A36 42B37 65M70 82C70 82D75 


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Copyright information

© This is a U.S. Government work and not under copyright protection in the US; foreign copyright protection may apply  2018

Authors and Affiliations

  1. 1.Computational and Applied Mathematics Group, Computer Science and Mathematics DivisionOak Ridge National LaboratoryOak RidgeUSA
  2. 2.Department of MathematicsUniversity of TennesseeKnoxvilleUSA

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