Convergence of a random particle method to solutions of the Kolmogorov equation

Author:
Elbridge Gerry Puckett

Journal:
Math. Comp. **52** (1989), 615-645

MSC:
Primary 65C05; Secondary 65M99

DOI:
https://doi.org/10.1090/S0025-5718-1989-0964006-X

MathSciNet review:
964006

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Abstract: We study a random particle method for solving the reaction-diffusion equation which is a one-dimensional analogue of the random vortex method. It is a fractional step method in which is solved by random walking the particles while is solved with a numerical ordinary differential equation solver such as Euler's method. We prove that the method converges when , i.e. the Kolmogorov equation, and that when the time step is the rate of convergence is like where *N* denotes the number of particles. Furthermore, we show that this rate of convergence is uniform as the diffusion coefficient tends to 0. Thus, travelling waves with arbitrarily steep wavefronts may be modeled without an increase in the computational cost. We also present the results of numerical experiments including the use of second-order time discretization and second-order operator splitting and use these results to estimate the expected value and standard deviation of the error.

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1989-0964006-X

Keywords:
Kolmogorov equation,
particle method,
random vortex method,
random walk

Article copyright:
© Copyright 1989
American Mathematical Society