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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2024 MCQ for Mathematics of Computation is 1.78.

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Convergence of a random particle method to solutions of the Kolmogorov equation $u_ t=\nu u_ {xx}+u(1-u)$
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by Elbridge Gerry Puckett PDF
Math. Comp. 52 (1989), 615-645 Request permission

Abstract:

We study a random particle method for solving the reaction-diffusion equation ${u_t} = \nu {u_{xx}} + f(u)$ which is a one-dimensional analogue of the random vortex method. It is a fractional step method in which ${u_t} = \nu {u_{xx}}$ is solved by random walking the particles while ${u_t} = f(u)$ is solved with a numerical ordinary differential equation solver such as Euler’s method. We prove that the method converges when $f(u) = u(1 - u)$, i.e. the Kolmogorov equation, and that when the time step $\Delta t$ is $O({\sqrt [4]{N}^{ - 1}})$ the rate of convergence is like $\ln N \cdot {\sqrt [4]{N}^{ - 1}}$ where N denotes the number of particles. Furthermore, we show that this rate of convergence is uniform as the diffusion coefficient $\nu$ 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
  • © Copyright 1989 American Mathematical Society
  • 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