2016/01/11 by Hailiang Liu, Liu, Hailiang, Zhongming Wang +1 · 1 citation
Engineering · Physics and Astronomy · #35B40 #65M60 #92D15 #Computational Fluid Dynamics and Aerodynamics #FOS: Mathematics #Fluid Dynamics and Turbulent Flows #Numerical Analysis (math.NA) #Statistical Mechanics and Entropy
paper · pdf · doi:10.48550/arxiv.1601.02547
openalex publication_date 2016/01/11 · openalex created_date 2019/06/27 · openalex updated_date 2026/07/28
We propose a high order discontinuous Galerkin (DG) 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.