Immersed Interface Method for Fokker-Planck Equation with Discontinuous Drift

The Immersed Interface Method (IIM) is derived to solve the corresponding Fokker-Planck equation of Brownian motion with pure dry friction, which is one of the simplest models of piecewise-smooth stochastic systems. The IIM is capable of treating a discontinuity in the drift of Fokker-Planck equation and it is readily extended to the dry and viscous friction model. Analytic results of the considered model are used to confirm the effectiveness and design accuracy of the scheme.


Introduction
In recent years, piecewise-smooth stochastic systems (governed by piecewise-smooth stochastic differential equations) are usually used to describe biological and physical systems.Although for some simple piecewise-linear stochastic differential equations, analytical solutions of the transition probability distribution can be obtained [1] [2], it is difficult to attain analytical expressions for many other cases.Hence, we need to develop some effective numerical methods to deal with the difficulty in order to know more dynamical behaviors of the systems.
In this paper, we attempt to solve numerically a Fokker-Planck equation with discontinuous drift, which results from a so-called Brownian motion with pure dry friction [3].This dry friction model can be described as the following piecewise linear Langevin equation ( ) Here ( ) ( ) The corresponding initial condition is Since Equation (1.2) has a discontinuous drift ( ) v σ , we must deal with it carefully.The IIM is a sharp interface method which can accurately capture discontinuities in the solution and the flux.This method has been used for many problems, such as elliptic interface problems [6], parabolic interface problems [7], moving interface problems [8] and many other applications [9] [10] [11] (see [12] [13] for excellent reviews).To the best of our knowledge, there is no literature about the IIM for solving Fokker-Planck equations with discontinuous drift so far.Hence, our goal is to solve it.The rest of this paper is organized as follows.In Section 2, we derive the IIM for the Fokker-Planck Equation (1.2).The numerical results are compared with the analytical solutions in Section 3. In addition, the accuracy of the scheme is also obtained.Finally, conclusions are made in Section 4. where + and − stand for the limiting values from the right-and left-hand sides of 0 v = .Integrating (1.2) across the discontinuity, we find

The Scheme
and then Therefore, the grid points can be expressed as ( 1) + with the discontinuous point being between j v and We hope to develop finite difference scheme of the form where τ is the time-step size.This means that we need to determine the coeffi- cients γ and the correction term n i C so that ( ) .
At a regular grid point i v , , 1 i j j ≠ + , the coefficients γ in the explicit dif- ference scheme (2.7) are obtained by the standard approximation as follows and the correction term 0.

i j =
In a similar way, we can compute the coefficients at the irregular grid point and the correction term 1 0.

Numerical results
For the Fokker-Planck Equation (1.2), using spectral decomposition method, one can get the transition probability distribution in closed analytic form [1] [14]: where ( ) is the transition probability distribution in non-dimensional units and To see the accuracy of the scheme numerically, we consider the 2 L and L ∞ errors between the numerical solutions and the exact solutions defined by where i p is the numerical solution and i p is the exact solution.Then we cal- culate the order of accuracy.A small time-step chosen and the problem is recalculated from time 0.01 t = to 1 t = .As illustrated in Table 1, the scheme is approximated second order in the velocity direction.

We set 1 D
= for convenience.At the discontinuous point 0 v = , we have the matching condition for the solution,

1 D = , 0 2 v
function.In addition, when t → ∞ the Fokker-Planck Equation (1.2) admits a steady stationary state = and the computing interval be [ 10,10] condition for computing.Figure1shows the comparison of numerical and analytical results of the probability distribution 0 ( , | , 0) p v t v at different times.It can be seen that the numerical solutions (points) coincide with the exact solutions (solid lines), indicating the effectiveness of the Scheme (2.7).

Figure 1 .
Figure 1.Transition probability distribution The notation i stands for the average overall possible realizations of the noise, and δ is the Dirac delta func- tion.The transition probability distribution

Table 1 .
Accuracy test in the velocity direction for 1 t = and