A Deterministic Particle Method for One-Dimensional Reaction-Diffusion Equations

Name:    Prof. Michael Mascagni         
Address: Department of Computer Science and
         School of Computational Science
         Florida State University
         Tallahassee, FL  32306-4530  USA
Offices: 498 Dirac Science Library/172 Love Building
Phone:   +1.850.644.3290
FAX:     +1.850.644.0098
e-mail:  mascagni@fsu.edu

Title: A Deterministic Particle Method for One-Dimensional Reaction-Diffusion Equations

Abstract:

We derive a deterministic particle method for the solution of nonlinear reaction-diffusion equations in one spatial dimension. This deterministic method is an analog of a Monte Carlo method for these problems previously investigated by the author. The deterministic method uses a uniform discretization of a dependent variable (the solution) instead of an independent variable (such as space). This discretization leads to the consideration of a system of nonlinear ordinary differential equations for the positions of particles defined by their solution value. These ODEs are nonlinear, with their nonlinearity corresponding to the reaction-diffusion equation's linear diffusion term, and are linear corresponding to the reaction-diffusion equation's nonlinear term. We then study explicit and implicit methods to solve numerically these equations. These methods are especially attractive when the solution has a traveling wave solution, as the particles concentrate themselves where the gradient of the solution is steepest. Thus these methods are naturally adaptive. Nagumo's equation is known to have traveling wave solutions, and so we compute the solution to this initial value problem using the new method. We then compute the wave speed of the numerical solution and compare it to the wave speed of exact solution. This is done analytically and is confirmed numerically. We then discuss future research into adapting these methods to solve non-monotonic problems and systems. The eventual goal of this research is the rapid solution of nerve equations for neural simulation.