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Journal of Scientific Computing

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08-02-2016

An Entropy Satisfying Discontinuous Galerkin Method for Nonlinear Fokker–Planck Equations

Authors: Hailiang Liu, Zhongming Wang

Published in: Journal of Scientific Computing | Issue 3/2016

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Abstract

We propose a high order discontinuous Galerkin method for solving nonlinear Fokker–Planck equations with a gradient flow structure. For some of these models it is known that the transient solutions converge to steady-states when time tends to infinity. The scheme is shown to satisfy a discrete version of the entropy dissipation law and preserve steady-states, therefore providing numerical solutions with satisfying long-time behavior. The positivity of numerical solutions is enforced through a reconstruction algorithm, based on positive cell averages. For the model with trivial potential, a parameter range sufficient for positivity preservation is rigorously established. For other cases, cell averages can be made positive at each time step by tuning the numerical flux parameters. A selected set of numerical examples is presented to confirm both the high-order accuracy and the efficiency to capture the large-time asymptotic.

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Title: An Entropy Satisfying Discontinuous Galerkin Method for Nonlinear Fokker–Planck Equations
Authors: Hailiang Liu
Zhongming Wang
Publication date: 08-02-2016
Publisher: Springer US
Published in: Journal of Scientific Computing / Issue 3/2016
Print ISSN: 0885-7474
Electronic ISSN: 1573-7691
DOI: https://doi.org/10.1007/s10915-016-0174-0

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