A quasi-random walk method for one-dimensional reaction–diffusion equations

Abstract

Probabilistic methods are presented to solve one-dimensional nonlinear reaction–diffusion equations. Computational particles are used to approximate the spatial derivative of the solution. The random walk principle is used to model the diffusion term. We investigate the effect of replacing pseudo-random numbers by quasi-random numbers in the random walk steps. This cannot be implemented in a straightforward fashion, because of correlations. If the particles are reordered according to their position at each time step, this has the effect of breaking correlations. For simple demonstration problems, the error is found to be significantly less when quasi-random sequences are used than when a standard random walk calculation is performed using pseudo-random points.