A conservative, free energy dissipating, and positivity preserving finite difference scheme for multi-dimensional nonlocal Fokker–Planck equation

Abstract

In this work, we design and analyze a conservative, positivity preserving, and free energy dissipating finite difference method for the multi-dimensional nonlocal Fokker–Planck (FP) equation. Based on a non-logarithmic Landau transformation, a central-differencing spatial discretization using harmonic-mean approximations is developed. Both forward and backward Euler discretizations in time are employed to derive an explicit scheme and a linearized semi-implicit scheme, respectively. Three desired properties that are possessed by analytical solutions: i) mass conservation, ii) free-energy dissipation, and iii) positivity, are proved to be maintained at discrete level. Remarkably, numerical analysis demonstrates that the semi-implicit time discretization ensures the property of positivity preserving unconditionally. Due to the advantages brought by the harmonic-mean approximations, an estimate on the upper bound of the condition number of the resulting coefficient matrix is further established for the semi-implicit scheme. Extensive numerical tests are performed to validate aforementioned properties numerically.