Structure Preserving Schemes for Fokker–Planck Equations of Irreversible Processes

In this paper, we construct structure preserving schemes for solving Fokker–Planck equations associated with irreversible processes. The proposed method is first order in time. We consider two structure-preserving spatial discretizations, which are second order and fourth order accurate finite difference schemes. They are derived via finite difference implementation of the classical Q^k ( k=1,2 ) finite element methods on uniform meshes. Under mild mesh conditions and practical time step constraints, the schemes are proved monotone, thus are positivity-preserving and energy dissipative. In particular, our scheme is suitable for capturing steady state solutions in large final time simulations.