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Positivity-preserving and energy-dissipating discontinuous Galerkin methods for nonlinear nonlocal Fokker-Planck equations (2403.15643v1)

Published 22 Mar 2024 in math.NA and cs.NA

Abstract: This paper is concerned with structure-preserving numerical approximations for a class of nonlinear nonlocal Fokker-Planck equations, which admit a gradient flow structure and find application in diverse contexts. The solutions, representing density distributions, must be non-negative and satisfy a specific energy dissipation law. We design an arbitrary high-order discontinuous Galerkin (DG) method tailored for these model problems. Both semi-discrete and fully discrete schemes are shown to admit the energy dissipation law for non-negative numerical solutions. To ensure the preservation of positivity in cell averages at all time steps, we introduce a local flux correction applied to the DDG diffusive flux. Subsequently, a hybrid algorithm is presented, utilizing a positivity-preserving limiter, to generate positive and energy-dissipating solutions. Numerical examples are provided to showcase the high resolution of the numerical solutions and the verified properties of the DG schemes.

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Authors (3)
  1. José A. Carrillo (119 papers)
  2. Hailiang Liu (65 papers)
  3. Hui Yu (119 papers)
Citations (1)
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