Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

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

 
 

 

Convergence of a random particle method to solutions of the Kolmogorov equation $ u\sb t=\nu u\sb {xx}+u(1-u)$


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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • [1] C. Anderson & C. Greengard, "On vortex methods," SIAM J. Numer. Anal., v. 22, 1985, pp. 413-440. MR 787568 (86j:76016)
  • [2] J. T. Beale & A. Majda, "Rates of convergence for viscous splitting of the Navier-Stokes equations," Math. Comp., v. 37, 1981, pp. 243-259. MR 628693 (82i:65056)
  • [3] J. T. Beale & A. Majda, "Vortex methods I: Convergence in three dimensions," Math. Comp., v. 39, 1982, pp. 1-27. MR 658212 (83i:65069a)
  • [4] J. T. Beale & A. Majda, "Vortex methods II: Higher order accuracy in two and three dimensions," Math. Comp., v. 39, 1982, pp. 29-52. MR 658213 (83i:65069b)
  • [5] M. Bramson, "Convergence of solutions of the Kolmogorov equation to travelling waves," Mem. Amer. Math. Soc., no. 285, 1983. MR 705746 (84m:60098)
  • [6] Y. Brenier, A Particle Method for One Dimensional Non-Linear Reaction Advection Diffusion Equations, Technical Report No. 351, Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas, Universidad Nacional Autonoma de Mexico, 1983.
  • [7] A. J. Chorin, "Numerical study of slightly viscous flow," J. Fluid Mech., v. 57, 1973, pp. 785-796. MR 0395483 (52:16280)
  • [8] A. J. Chorin, "Vortex sheet approximation of boundary layers," J. Comput. Phys., v. 27, 1978, pp. 428-442.
  • [9] A. J. Chorin, Numerical Methods For Use in Combustion Modeling. Proc. Internat. Conf. Numer. Methods in Science and Engineering, Versailles, 1979. MR 584038 (82b:80016)
  • [10] K. L. Chung, A Course in Probability Theory, Harcourt, Brace & World Inc., 1974. MR 0346858 (49:11579)
  • [11] S. D. Conte & C. de Boor, Elementary Numerical Analysis, 3rd ed., McGraw-Hill, New York, 1980.
  • [12] G. H. Cottet, Méthodes Particulaires pour l'Equation d'Euler dans le Plan, Thèse de 3ème cycle, Université Pierre et Marie Curie, Paris, 1982.
  • [13] G. H. Cottet & S. Gallic, A Particle Method to Solve Transport-Diffusion Equations, Part I: The Linear Case, Rapport Interne No: 115, Centre de Mathématiques Appliquées, Ecole Polytechnique, Paris, 1984.
  • [14] G. B. Folland, Introduction to Partial Differential Equations, Princeton Univ. Press, 1976. MR 0599578 (58:29031)
  • [15] A. F. Ghoniem & F. S. Sherman, "Grid free simulation of diffusion using random walk methods," J. Comput. Phys., v. 61, 1985, pp. 1-37. MR 811559 (87h:65019)
  • [16] J. Goodman, "Convergence of the random vortex method," Comm. Pure Appl. Math., v. 40, 1987, pp. 189-220. MR 872384 (88d:35159)
  • [17] C. Greengard, "Convergence of the vortex filament method," Math. Comp., v. 47, 1986, pp. 387-398. MR 856692 (88f:65151)
  • [18] O. Hald, Private communication.
  • [19] O. Hald & V. M Del Prete, "Convergence of vortex methods for Euler's equations," Math. Comp., v. 32, 1978, pp. 791-809. MR 492039 (81b:76015a)
  • [20] O. H. Hald, "The convergence of vortex methods for Euler's equations II," SIAM J. Numer. Anal., v. 16, 1979, pp. 726-755. MR 543965 (81b:76015b)
  • [21] O. H. Hald, "Convergence of random methods for a reaction-diffusion equation," SIAM J. Sci. Statist. Comput., v. 2, 1981, pp. 85-94. MR 618634 (83c:65210)
  • [22] O. H. Hald, "Convergence of a random method with creation of vorticity," SIAM J. Sci. Statist. Comput., v. 7, 1986, pp. 1373-1386. MR 857800 (88a:65013)
  • [23] O. H. Hald, "Convergence of vortex methods for Euler's equations, III," SIAM J. Numer. Anal., v. 24, 1987, pp. 538-582. MR 888750 (88i:76003)
  • [24] W. Hoeffding, "Probability Inequalities for Sums of Bounded Random Variables," Amer. Statist. Assoc. J., v. 58, 1963, pp. 13-30. MR 0144363 (26:1908)
  • [25] M. Loève, Probability Theory, Springer-Verlag, Berlin and New York, 1977.
  • [26] D. G. Long, Convergence of the Random Vortex Method in One and Two Dimensions, Ph.D. Thesis, Univ. of California, Berkeley, 1986.
  • [27] E. G. Puckett, "A study of the vortex sheet method and its rate of convergence," SIAM J. Sci. Statist. Comput., v. 10, 1989, pp. 298-327. MR 982225 (90c:76016)
  • [28] P. A. Raviart, "An analysis of particle methods," CIME Course on Numerical Methods in Fluid Dynamics, Publications du Laboratoire d'Analyse numérique, Université Pierre et Marie Curie, Paris, 1983.
  • [29] P. A. Raviart, Particle Approximation of Linear Hyperbolic Equations of the First Order, Publications du Laboratoire d' Analyse numérique, Université Pierre et Marie Curie, Paris, 1983. MR 760461 (86k:65095)
  • [30] S. G. Roberts, "Accuracy of the random vortex method for a problem with non-smooth initial conditions," J. Comput. Phys., v. 58, 1985, pp. 29-43. MR 789554 (86f:76022)
  • [31] S. Roberts, "Convergence of a random walk method for the Burgers equation," this issue, pp. 647-673. MR 955753 (89i:65090)
  • [32] G. Rosen, "Brownian-motion correspondence method for obtaining approximate solutions to nonlinear reaction-diffusion equations," Phys. Rev. Lett., v. 53, 1984, pp. 307-310. MR 766364 (85k:82020)
  • [33] A. S. Sherman & C. S Peskin, "A Monte Carlo method for scalar reaction diffusion equations," SIAM J. Sci. Statist. Comput., v. 7, 1986, pp. 1360-1372. MR 857799
  • [34] A. S. Sherman & C. S. Peskin, "Solving the Hodgkin-Huxley equations by a random walk method," SIAM J. Sci. Statist. Comput., v. 9, 1988, pp. 170-190. MR 922871 (88m:92023)
  • [35] J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer-Verlag, New York, 1983. MR 688146 (84d:35002)
  • [36] G. Strang, "On the construction and comparison of difference schemes," SIAM J. Numer. Anal., v. 5, 1968, pp. 506-517. MR 0235754 (38:4057)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65C05, 65M99

Retrieve articles in all journals with MSC: 65C05, 65M99


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

American Mathematical Society