Monte Carlo Method to Solve Diffusion Equation and Error Analysis

Abstract

Three different mathematical approaches for the evolution of diffusion equation are presented. The evolution process of the diffusion equation is explained by principle of conservation law, probability distribution procedure, and finally though stochastic differential equation (SDE) driven by Brownian motion. The Monte Carlo method is discussed to solve the diffusion equation by generating the normally distributed random numbers and the root mean square error is derived for the Monte Carlo method. The numerical solutions are computed for 1-dimensional diffusion equation and results are compared with exact solution. Finally, theoretical root mean square error is compared with the maximum error and the L₂-error by increasing the number of simulated points.