Positivity-preserving and energy-dissipating discontinuous Galerkin methods for nonlinear nonlocal Fokker-Planck equations
<|reference_start|>Positivity-preserving and energy-dissipating discontinuous Galerkin methods for nonlinear nonlocal Fokker-Planck equations: 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.<|reference_end|>
@article{carrillo2024positivity-preserving,
    title={Positivity-preserving and energy-dissipating discontinuous Galerkin
  methods for nonlinear nonlocal Fokker-Planck equations},
    author={Jos'e A. Carrillo, Hailiang Liu, Hui Yu},
    journal={arXiv preprint arXiv:2403.15643},
    year={2024},
    archivePrefix={arXiv},
    eprint={2403.15643},
    primaryClass={math.NA cs.NA}
}