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

Authors:
S. Ogawa;C. Lécot
Affiliations:
Laboratory of Applied Mathematics and Stochastics, Faculty of Technology, Kanazawa University, 2-40-20 Kodatsuno, Kanazawa 920-8667, Japan;Laboratoire de Mathématiques, Université de Savoie, Campus scientifique, 73376 Le Bourget-du-Lac Cedex, France
Venue:
Mathematics and Computers in Simulation - Special issue: 3rd IMACS seminar on Monte Carlo methods - MCM 2001
Year:
2003

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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.