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Diffusion in a nonhomogeneous medium: quasi-random walk on a lattice

Author

Listed:
  • El Haddad Rami

    (Département de Mathématiques, Faculté des Sciences, Université Saint-Joseph, BP 11-514 Riad El Solh, Beyrouth 1107 2050, Lebanon. E-mail:)

  • Lécot Christian

    (Laboratoire de Mathématiques, UMR 5127 CNRS and Université de Savoie, Campus scientifique, 73376 Le Bourget-du-Lac Cedex, France. E-mail:)

  • Venkiteswaran Gopalakrishnan

    (Department of Mathematics, Birla Institute of Technology and Science, Vidya Vihar Campus, Pilani, 333 031 Rajasthan, India. E-mail:)

Abstract

We are interested in Monte Carlo (MC) methods for solving the diffusion equation: in the case of a constant diffusion coefficient, the solution is approximated by using particles and in every time step, a constant stepsize is added to or subtracted from the coordinates of each particle with equal probability. For a spatially dependent diffusion coefficient, the naive extension of the previous method using a spatially variable stepsize introduces a systematic error: particles migrate in the directions of decreasing diffusivity. A correction of stepsizes and stepping probabilities has recently been proposed and the numerical tests have given satisfactory results. In this paper, we describe a quasi-Monte Carlo (QMC) method for solving the diffusion equation in a spatially nonhomogeneous medium: we replace the random samples in the corrected MC scheme by low-discrepancy point sets. In order to make a proper use of the better uniformity of these point sets, the particles are reordered according to their successive coordinates at each time step. We illustrate the method with numerical examples: in dimensions 1 and 2, we show that the QMC approach leads to improved accuracy when compared with the original MC method using the same number of particles.

Suggested Citation

  • El Haddad Rami & Lécot Christian & Venkiteswaran Gopalakrishnan, 2010. "Diffusion in a nonhomogeneous medium: quasi-random walk on a lattice," Monte Carlo Methods and Applications, De Gruyter, vol. 16(3-4), pages 211-230, January.
    • Handle: RePEc:bpj:mcmeap:v:16:y:2010:i:3-4:p:211-230:n:2
    DOI: 10.1515/mcma.2010.009
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    References listed on IDEAS

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    1. Venkiteswaran, G. & Junk, M., 2005. "Quasi-Monte Carlo algorithms for diffusion equations in high dimensions," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 68(1), pages 23-41.
    2. Coulibaly, Ibrahim & Lécot, Christian, 1998. "Simulation of diffusion using quasi-random walk methods," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 47(2), pages 153-163.
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