Second-Order Schemes for Fokker-Planck Equations with Discontinuous Drift

Second-order finite-difference schemes are developed to solve the corresponding Fokker-Planck equation of Brownian motion with dry friction, which is one of the simplest model of stochastic piecewise-smooth systems. For the Fokker-Planck equation with a discontinuous drift, both explicit and implicit second order schemes are derived by finite volume method. The proposed schemes are proved to be stable both for the one-variable (related to the velocity only) and two-variable (related to the velocity and displacement) cases. Numerical experiments are implemented for both the two cases. Some known analytical results of the considered model are used to confirm the effectiveness and desired accuracy of the schemes.