Speaker: Professor Cheng Wang

Time: 09:00-10:00, 26 May 2026 (Tue.)

Online: Tencent meeting 767-953-4082

Abstract

Afinite difference numerical scheme is proposed and analyzed for the Poisson-Nernst-Planckequation (PNP) system. To understand the energy structure of the PNP model, we make use of theEnergetic Variational Approach, so that the PNP system could be reformulated as a non-constantmobility, conserved gradient flow, with singular logarithmic energy potentials involved. To ensurethe unique solvability and energy stability, the mobility function is explicitly treated, while both thelogarithmic and the electric potential diffusion terms are treated implicitly, due to the convex natureof these two energy functional parts. The positivity-preserving property for both concentrations isestablished at a theoretical level. This is based on the subtle fact that the singular nature of thelogarithmic term around the value of 0 prevents the numerical solution reaching the singular value,so that the numerical scheme is always well-defined. In addition, an optimal rate convergenceanalysis is provided in this work, in which many highly non-standard estimates have to be involved,due to the nonlinear parabolic coefficients. The higher order asymptotic expansion (up to third ordertemporal accuracy and fourth order spatial accuracy), the rough error estimate (to establish thediscrete maximum norm bound), and the refined error estimate have to be carried out to accomplishsuch a convergence result. In our knowledge. this work will be the first to combine the followingthree theoretical properties for a numerical scheme for the PNP system: (i) unique solvability andpositivity, (ii) energy stability, and (iii) optimal rate convergence. A few numerical results are alsopresented in this talk, which demonstrates the robustness of the proposed numerical scheme.

About the Speaker

Dr. Cheng Wang is a professor in Department of Mathematics at the University of MassachusettsDartmouth (UMassD). He obtained his PhD degree from Temple University, under the supervisionof Prof. Jian-Guo Liu. Prior to joining UMassD in 2008 as an assistant professor, he was a Zornpostdoc at Indiana University from 2000 to 2003, under the supervision of Roger Temam andShouhong Wang, and he worked as an assistant professor at University of Tennessee at Knoxvillefrom 2003 to 2008. Dr. Wang's research interests include development of stable, accuratenumerical algorithms for partial differential equations and numerical analysis. He has publishedmore than 140 papers with more than 9000 citations. He also serves in the Editorial Board of"Numerical Mathematics: Theory, Methods and Applications".