Applied Mathematics Seminar

Speaker: Shan Zhao, Department of Mathematics, University of Alabama 

Title: New Developments of Alternating Direction Implicit (ADI) Algorithms for Biomolecular Solvation Analysis 

Abstract: In this talk, I will first present some tailored alternating direction implicit (ADI) algorithms for solving nonlinear PDEs in biomolecular solvation analysis. Based on the variational formulation, we have previously proposed a pseudo-transient continuation model to couple a nonlinear Poisson-Boltzmann (NPB) equation for the electrostatic potential with a geometric flow equation defining the biomolecular surface. To speed up the simulation, we have reformulated the geometric flow equation so that an unconditionally stable ADI algorithm can be realized for molecular surface generation. Meanwhile, to overcome the stability issue associated with the strong nonlinearity, we have introduced an operator splitting ADI method for solving the NPB equation. Motivated by our biological applications, we have also recently carried out some studies on the algorithm development for solving the parabolic interface problem. A novel matched ADI method has been developed to solve a 2D diffusion equation with material interfaces involving complex geometries. For the first time in the literature, the ADI finite difference method is able to deliver a second order of accuracy in space for arbitrarily shaped interfaces and spatial-temporal dependent interface conditions.