Convergence of a random walk method for the Burgers equation

Author:
Stephen Roberts

Journal:
Math. Comp. **52** (1989), 647-673

MSC:
Primary 65M10; Secondary 65U05, 76-08

DOI:
https://doi.org/10.1090/S0025-5718-1989-0955753-4

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.

**[1]**J. Thomas Beale and Andrew Majda,*Rates of convergence for viscous splitting of the Navier-Stokes equations*, Math. Comp.**37**(1981), no. 156, 243–259. MR**628693**, https://doi.org/10.1090/S0025-5718-1981-0628693-0**[2]**Yann Brenier,*Averaged multivalued solutions for scalar conservation laws*, SIAM J. Numer. Anal.**21**(1984), no. 6, 1013–1037. MR**765504**, https://doi.org/10.1137/0721063**[3]**Y. Brenier,*A Particle Method for One Dimensional Non-Linear Reaction Advection Diffusion Equations*, Comunicaciones Técnicas, Serie Naranja: Investigaciones, No. 351, 1983, Instituto de Investigaciones en Matematicas Aplicadas y en Sistemas, Universidad Nacional Autonoma de Mexico.**[4]**J. M. Burgers,*A mathematical model illustrating the theory of turbulence*, Advances in Applied Mechanics, Academic Press, Inc., New York, N. Y., 1948, pp. 171–199. edited by Richard von Mises and Theodore von Kármán,. MR**0027195****[5]**Paul R. Chernoff,*Product formulas, nonlinear semigroups, and addition of unbounded operators*, American Mathematical Society, Providence, R. I., 1974. Memoirs of the American Mathematical Society, No. 140. MR**0417851****[6]**Alexandre Joel Chorin,*Numerical study of slightly viscous flow*, J. Fluid Mech.**57**(1973), no. 4, 785–796. MR**0395483**, https://doi.org/10.1017/S0022112073002016**[7]**A. J. Chorin, "Vortex sheet approximation to boundary layers,"*J. Comput. Phys.*, v. 27, 1978, pp. 428-442.**[8]**Alexandre Joel Chorin,*Numerical methods for use in combustion modeling*, Computing methods in applied sciences and engineering (Proc. Fourth Internat. Sympos., Versailles, 1979) North-Holland, Amsterdam-New York, 1980, pp. 229–236. MR**584038****[9]**A. J. Chorin and J. E. Marsden,*A mathematical introduction to fluid mechanics*, Springer-Verlag, New York-Heidelberg, 1979. MR**551053****[10]**Alexandre J. Chorin, Marjorie F. McCracken, Thomas J. R. Hughes, and Jerrold E. Marsden,*Product formulas and numerical algorithms*, Comm. Pure Appl. Math.**31**(1978), no. 2, 205–256. MR**0488713**, https://doi.org/10.1002/cpa.3160310205**[11]**Kai Lai Chung,*A course in probability theory*, 2nd ed., Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1974. Probability and Mathematical Statistics, Vol. 21. MR**0346858****[12]**Julian D. Cole,*On a quasi-linear parabolic equation occurring in aerodynamics*, Quart. Appl. Math.**9**(1951), 225–236. MR**0042889**, https://doi.org/10.1090/S0033-569X-1951-42889-X**[13]**Albert Einstein,*Investigations on the theory of the Brownian movement*, Dover Publications, Inc., New York, 1956. Edited with notes by R. Fürth; Translated by A. D. Cowper. MR**0077443****[14]**William Feller,*An introduction to probability theory and its applications. Vol. I*, Third edition, John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR**0228020****[15]**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****[16]**Avner Friedman,*Partial differential equations of parabolic type*, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. MR**0181836****[17]**A. F. Ghoniem, A. J. Chorin & A. K. Oppenheim, "Numerical modeling of turbulent flow in a combustion tunnel,"*Philos. Trans. Roy. Soc. London Ser. A*, v. 304, 1982, pp. 303-325.**[18]**Ahmed F. Ghoniem and Frederick S. Sherman,*Grid-free simulation of diffusion using random walk methods*, J. Comput. Phys.**61**(1985), no. 1, 1–37. MR**811559**, https://doi.org/10.1016/0021-9991(85)90058-0**[19]**Jonathan Goodman,*Convergence of the random vortex method*, Comm. Pure Appl. Math.**40**(1987), no. 2, 189–220. MR**872384**, https://doi.org/10.1002/cpa.3160400204**[20]**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**, https://doi.org/10.1137/0902007**[21]**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**, https://doi.org/10.1137/0907091**[22]**Wassily Hoeffding,*Probability inequalities for sums of bounded random variables*, J. Amer. Statist. Assoc.**58**(1963), 13–30. MR**0144363****[23]**Eberhard Hopf,*The partial differential equation 𝑢_{𝑡}+𝑢𝑢ₓ=𝜇𝑢ₓₓ*, Comm. Pure Appl. Math.**3**(1950), 201–230. MR**0047234**, https://doi.org/10.1002/cpa.3160030302**[24]**Fritz John,*Partial differential equations*, 4th ed., Applied Mathematical Sciences, vol. 1, Springer-Verlag, New York, 1982. MR**831655****[25]**S. N. Kružkov, "First order quasilinear equations with several space variables,"*Math. USSR-Sb.*, v. 10, 1970, pp. 217-243.**[26]**J. A. Laitone, "A numerical solution for gas particle flows at high Reynolds numbers,"*J. Appl. Mech.*, v. 48, 1981, pp. 465-471.**[27]**C. Marchioro and M. Pulvirenti,*Hydrodynamics in two dimensions and vortex theory*, Comm. Math. Phys.**84**(1982), no. 4, 483–503. MR**667756****[28]**F. J. McGrath,*Nonstationary plane flow of viscous and ideal fluids*, Arch. Rational Mech. Anal.**27**(1967), 329–348. MR**0221818**, https://doi.org/10.1007/BF00251436**[29]**A. K. Oppenheim & A. Ghoniem,*Application of the Random Element Method to One Dimensional Flame Propagation Problems*, AIAA-83-0600, AIAA 21st Aerospace Sciences Meeting, Reno, Nevada, 1983.**[30]**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****[31]**S. G. Roberts,*Convergence of a Random Walk Method for the Burgers Equation*, Ph.D. Thesis, University of California, Berkeley, 1985.**[32]**P. K. Stansby & A. G. Dixon, "Simulation of flows around cylinders by a Lagrangian vortex scheme,"*Appl. Ocean. Res. Ser. 3*, v. 5, 1984, pp. 167-178.**[33]**N. W. Sung, J. A. Laitone & D. J. Pattinson, "Angled jet flow model for a diesel engine intake process--random vortex method,"*Internat. J. Numer. Methods Fluids*, v. 3, 1983, pp. 283-293.**[34]**Zhen Huan Teng,*Elliptic-vortex method for incompressible flow at high Reynolds number*, J. Comput. Phys.**46**(1982), no. 1, 54–68. MR**665805**, https://doi.org/10.1016/0021-9991(82)90005-5**[35]**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.**[36]**G. B. Whitham,*Linear and nonlinear waves*, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. Pure and Applied Mathematics. MR**0483954**

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1989-0955753-4

Article copyright:
© Copyright 1989
American Mathematical Society