Positivity-preserving and energy-dissipative finite difference schemes for the Fokker–Planck and Keller–Segel equations

Abstract In this work we introduce semi-implicit or implicit finite difference schemes for the continuity equation with a gradient flow structure. Examples of such equations include linear Fokker–Planck and Keller–Segel equations. The two proposed are first-order accurate in time, explicitly solvable, second-order fourth-order space, which obtained via implementation classical continuous element method. fully discrete proved to be positivity preserving energy dissipative: scheme can achieve so unconditionally while only requires mild time step mesh size constraint. particular, is first high order spatial discretization that both decay properties, suitable long simulation obtain steady state solutions.