An Efficient Modified "Walk On Spheres" Algorithm for the Linearized Poisson-Boltzmann Equation

TL;DR

This paper introduces a new grid-free Monte Carlo method based on a modified Walk On Spheres algorithm to efficiently solve the linearized Poisson-Boltzmann equation, demonstrating high accuracy across test problems.

Contribution

It presents a novel, efficient grid-free random walk algorithm for the LPBE, improving upon previous grid-based methods.

Findings

High accuracy in solving test problems

Excellent agreement with analytical solutions

Efficient computation without grid discretization

Abstract

A discrete random walk method on grids was proposed and used to solve the linearized Poisson-Boltzmann equation (LPBE) \cite{Rammile}. Here, we present a new and efficient grid-free random walk method. Based on a modified `` Walk On Spheres" (WOS) algorithm \cite{Elepov-Mihailov1973} for the LPBE, this Monte Carlo algorithm uses a survival probability distribution function for the random walker in a continuous and free diffusion region. The new simulation method is illustrated by computing four analytically solvable problems. In all cases, excellent agreement is observed.