AN IMPROVED WPE METHOD FOR SOLVING DISCONTINUOUS FOKKER-PLANCK EQUATIONS

Abstract. In mathematical studies of molecular motors, the stochastic motor motion is modeled using the Langevin equation. If we consider an ensemble of motors, the probability density is governed by the corre-sponding Fokker-Planck equation. Average quantities, such as, average velocity, effective diffusion and randomness parameter, can be calcu-lated from the probability density. The WPE method was previously developed to solve Fokker-Planck equations (H. Wang, C. Peskin and T. Elston, J. Theo. Biol., Vol. 221, 491-511, 2003). The WPE method has the advantage of preserving detailed balance, which ensures that the numerical method still works even when the potential is discontinu-ous. Unfortunately, the accuracy of the WPE method drops to first order when the potential is discontinuous. Here we propose an improved ver-sion of the WPE method. The improved WPE method a) maintains the second order accuracy even when the potential is discontinuous, b) has got rid of a numerical singularity in the WPE method, and c) is as sim-ple and easy to implement as the WPE method. Numerical examples are shown to demonstrate the robust performance of the improved WPE method. Key Words. Fokker-Planck equation, detailed balance, numerical solu-tions