Convergence of a random walk method

for a partial differential equation

Author:
Weidong Lu

Journal:
Math. Comp. **67** (1998), 593-602

MSC (1991):
Primary 65C05, 65M99

MathSciNet review:
1443122

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Abstract | References | Similar Articles | Additional Information

Abstract: A Cauchy problem for a one-dimensional diffusion-reaction equation is solved on a grid by a random walk method, in which the diffusion part is solved by random walk of particles, and the (nonlinear) reaction part is solved via Euler's polygonal arc method. Unlike in the literature, we do not assume monotonicity for the initial condition. It is proved that the algorithm converges and the rate of convergence is of order , where is the spatial mesh length.

**1.**Alexandre Joel Chorin,*Numerical study of slightly viscous flow*, J. Fluid Mech.**57**(1973), no. 4, 785–796. MR**0395483****2.**A. J. Chorin,*Vortex sheet approximation of boundary layer,*27(1978), 428-442.*J. Comp. Phys,***3.**A. J. Chorin and J. E. Marsden,*A mathematical introduction to fluid mechanics*, Springer-Verlag, New York-Heidelberg, 1979. MR**551053****4.**William Feller,*An introduction to probability theory and its applications. Vol. II.*, Second edition, John Wiley & Sons, Inc., New York-London-Sydney, 1971. MR**0270403****5.**Jonathan Goodman,*Convergence of the random vortex method*, Comm. Pure Appl. Math.**40**(1987), no. 2, 189–220. MR**872384**, 10.1002/cpa.3160400204**6.**Ole H. Hald,*Convergence of random methods for a reaction-diffusion equation*, SIAM J. Sci. Statist. Comput.**2**(1981), no. 1, 85–94. MR**618634**, 10.1137/0902007**7.**Ole H. Hald,*Convergence of a random method with creation of vorticity*, SIAM J. Sci. Statist. Comput.**7**(1986), no. 4, 1373–1386. MR**857800**, 10.1137/0907091**8.**D. G. Long,*Convergence of the random vortex method in one and two dimensions,*1986.*Ph.D. Thesis, Univ. of California, Berkeley,***9.**Robert D. Richtmyer and K. W. Morton,*Difference methods for initial-value problems*, Second edition. Interscience Tracts in Pure and Applied Mathematics, No. 4, Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney, 1967. MR**0220455****10.**Elbridge Gerry Puckett,*Convergence of a random particle method to solutions of the Kolmogorov equation 𝑢_{𝑡}=𝜈𝑢ₓₓ+𝑢(1-𝑢)*, Math. Comp.**52**(1989), no. 186, 615–645. MR**964006**, 10.1090/S0025-5718-1989-0964006-X**11.**Stephen Roberts,*Accuracy of the random vortex method for a problem with nonsmooth initial conditions*, J. Comput. Phys.**58**(1985), no. 1, 29–43. MR**789554**, 10.1016/0021-9991(85)90154-8**12.**Stephen Roberts,*Convergence of a random walk method for the Burgers equation*, Math. Comp.**52**(1989), no. 186, 647–673. MR**955753**, 10.1090/S0025-5718-1989-0955753-4

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Additional Information

**Weidong Lu**

Affiliation:
Department of Mathematics, Fudan University, Shanghai, 200433, China

DOI:
https://doi.org/10.1090/S0025-5718-98-00917-X

Keywords:
Random walk method,
partial differential equation,
Cauchy problem.

Received by editor(s):
July 20, 1995

Received by editor(s) in revised form:
December 11, 1996

Additional Notes:
This work is partially supported by the Chinese State Education Commission Natural Science Foundation.

Article copyright:
© Copyright 1998
American Mathematical Society