Conservative, Positivity Preserving and Energy Dissipative Numerical Methods for the Poisson-Nernst-Planck Equations

Abstract

We design and analyze some numerical methods for solving the Poisson-Nernst-Planck (PNP) equations. The numerical schemes, including finite difference method and discontinuous Galerkin method, respect three desired properties that are possessed by analytical solutions: I) conservation, II) positivity of solution, and III) free-energy dissipation. Advantages of different types of methods are discussed. Numerical experiments are performed to validate the numerical analysis. Modified PNP system that incorporating size and solvation effect is also studied to demonstrate the effectiveness of our schemes in solving realistic problems.

This is joint work with D. Jie, H. Liu, P. Yin, H. Yu and S. Zhou.

Speaker

Zhongming Wang, Ph.D., received his bachelor's degree in Computing Mathematics from City University of Hong Kong in 2003 and doctorate degree in Applied Mathematics from Iowa State University in 2008. He was a Postdoctoral Fellow at the University of California, San Diego from 2008 to 2011. He joined the Department of Mathematics and Statistics, Florida International University in 2011 and was promoted to associate professor in 2016.

Dr. Wang's research interests are in computational and applied mathematics. In particular, he has been working on

1. level set methods for high frequency wave propagations, two-phase flows and the Euler-Poisson equations;

2. direct Discontinuous Galerkin method for nonlinear Fokker-Planck equations

3. conservative, positivity preserving and energy dissipative numerical methods for the Poisson-Nernst-Planck equations.