TY - JOUR
T1 - An Improved WPE Method for Solving Discontinuous Fokker-Planck Equations
JO - International Journal of Numerical Analysis and Modeling
VL - 1
SP - 1
EP - 23
PY - 2008
DA - 2008/05
SN - 5
DO - http://doi.org/
UR - https://global-sci.org/intro/article_detail/ijnam/794.html
KW - Fokker-Planck equation
KW - detailed balance
KW - numerical solutions
AB - 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
corresponding Fokker-Planck equation. Average quantities, such as,
average velocity, effective diffusion and randomness parameter, can be
calculated 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 discontinuous.
Unfortunately, the accuracy of the WPE method drops to first order when
the potential is discontinuous. Here we propose an improved version 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 simple and easy to
implement as the WPE method. Numerical examples are shown to demonstrate
the robust performance of the improved WPE method.