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

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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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Acknowledgments

Liu was supported by the National Science Foundation under Grant DMS1312636 and by NSF Grant RNMS (Ki-Net) 1107291.

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Authors and Affiliations

  1. Mathematics Department, Iowa State University, Ames, IA, 50011, USA

    Hailiang Liu

  2. Department of Mathematics and Statistics, Florida International University, Miami, FL, 33199, USA

    Zhongming Wang

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  1. Hailiang Liu
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  2. Zhongming Wang
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Correspondence to Hailiang Liu.

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Liu, H., Wang, Z. An Entropy Satisfying Discontinuous Galerkin Method for Nonlinear Fokker–Planck Equations. J Sci Comput 68, 1217–1240 (2016). https://doi.org/10.1007/s10915-016-0174-0

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  • DOI: https://doi.org/10.1007/s10915-016-0174-0

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