Structure-Preserving Numerical Methods for Nonlinear Fokker--Planck Equations with Nonlocal Interactions by an Energetic Variational Approach

Convergence analysis of structure‐preserving numerical methods for nonlinear Fokker–Planck equations with nonlocal interactions

“…The detailed proof has been provided in Duan et all. 12 ; here, we only deduce the trajectory equation briefly.…”

“…A structure-preserving finite difference scheme has been proposed in Duan et al, 12 for the trajectory Equation (2.4) using the convex splitting method, which was first exploited by Eyre 38 to craft energy stable numerical schemes for the Allen-Cahn and Cahn-Hilliard equations. The basic idea is to treat the convex part implicitly and the concave part explicitly.…”

“…Also, we recall the Newton's iteration method that has been developed in 12 to solve the nonlinear difference Equations 3.7 and (3.8).…”

“…[1][2][3][4][5][6][7][8][9] There are many important equations can be regarded as special cases of the FP equations, such as the porous medium equation and aggregation-diffusion equations. Interested readers are referred to the works [10][11][12][13][14] and references therein.…”

“…In a more recent work, 12 the authors constructed a structure‐preserving numerical scheme by an energetic variational approach, which is uniquely solvable, naturally respects mass conservation and positivity, and preserves steady states theoretically. Numerical simulations have revealed not only the preservation of physical structures, such as positivity, mass conservation, discrete energy dissipation, and steady states, but also the capability of solving degenerate cases effectively and robustly.…”