A Conservative,Free Energy Dissipating,and Positivity Preserving Finite Difference Scheme For Multi-Dimensional Nonlocal Fokker-Planck Equation

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 prop-erties that are possessed by analytical solutions:?)mass conservation,?)free-energy dissipation,and ?)positivity,are proved to be maintained at discrete level.The semi-implicit scheme is further shown to preserve positivity unconditionally,whereas a constraint on a mesh ratio is required for the explicit scheme to ensure positivity.In addition,our estimates on the upper bound of condition numbers indicate that the developed discretization based on harmonic-mean approximations can effectively solve a known issue—a large condition number is often accompanied by the use of non-logarithmic Landau variables.Extensive numerical tests are performed to vali-date aforementioned properties numerically and is second-order accurate in space and first-order accurate in time.