Volume 5, Issue 1
An Improved WPE Method for Solving Discontinuous Fokker-Planck Equations

Y. Saka, M. Gunzburger & J. Burkardt

Int. J. Numer. Anal. Mod., 5 (2008), pp. 1-23

Published online: 2008-05

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  • 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 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.

  • Keywords

Fokker-Planck equation detailed balance numerical solutions

  • AMS Subject Headings

65Mxx 82Cxx

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COPYRIGHT: © Global Science Press

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@Article{IJNAM-5-1, author = {}, title = {An Improved WPE Method for Solving Discontinuous Fokker-Planck Equations}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2008}, volume = {5}, number = {1}, pages = {1--23}, 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 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. }, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/794.html} }
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.
Y. Saka, M. Gunzburger & J. Burkardt. (1970). An Improved WPE Method for Solving Discontinuous Fokker-Planck Equations. International Journal of Numerical Analysis and Modeling. 5 (1). 1-23. doi:
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