Convergence of a random walk method for the Burgers equation
Author:
Stephen Roberts
Journal:
Math. Comp. 52 (1989), 647673
MSC:
Primary 65M10; Secondary 65U05, 7608
MathSciNet review:
955753
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Abstract: We show that the solution of the Burgers equation can be approximated in , to within , by a random walk method generated by particles. The nonlinear advection term of the equation is approximated by advecting the particles in a velocity field induced by the particles. The diffusive term is approximated by adding an appropriate random perturbation to the particle positions. It is also shown that the corresponding viscous splitting algorithm approximates the solution of the Burgers equation in to within when k is the size of the time step. This work provides the first proof of convergence in a strong sense, for a random walk method, in which the related advection equation allows for the formation of shocks.
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 J. T. Beale & A. Majda, "Rates of convergence for viscous splitting of the NavierStokes equation," Math. Comp., v. 37, 1981, pp. 243259. MR 628693 (82i:65056)
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 Y. Brenier, "Averaged multivalued solutions for scalar conservation laws," SIAM J. Numer. Anal., v. 21, 1984, pp. 10131037. MR 765504 (86b:65099)
 [3]
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 J. A. Laitone, "A numerical solution for gas particle flows at high Reynolds numbers," J. Appl. Mech., v. 48, 1981, pp. 465471.
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 N. W. Sung, J. A. Laitone & D. J. Pattinson, "Angled jet flow model for a diesel engine intake processrandom vortex method," Internat. J. Numer. Methods Fluids, v. 3, 1983, pp. 283293.
 [34]
 Z. Teng, "Ellipticvortex method for incompressible flow at high Reynolds number," J. Comput. Phys., v. 46, 1982, pp. 5468. MR 665805 (83g:76019)
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 J. J. W. Van der Vegt & R. H. M. Huijsmans, "Numerical simulation of flow around bluff bodies at high Reynolds numbers," Z 50457, Netherlands Ship Model Basin, ONR Paper, August, 1982.
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DOI:
http://dx.doi.org/10.1090/S00255718198909557534
PII:
S 00255718(1989)09557534
Article copyright:
© Copyright 1989
American Mathematical Society
