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

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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. T. Beale & A. Majda, "Rates of convergence for viscous splitting of the Navier-Stokes equation,"*Math. Comp.*, v. 37, 1981, pp. 243-259. MR**628693 (82i:65056)****[2]**Y. Brenier, "Averaged multivalued solutions for scalar conservation laws,"*SIAM J. Numer. Anal.*, v. 21, 1984, pp. 1013-1037. MR**765504 (86b:65099)****[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,"*Adv. in Appl. Mech.*, v. 1, 1948, pp. 171-199. MR**0027195 (10:270b)****[5]**P. R. Chernoff,*Product Formulas, Nonlinear Semigroups and Addition of Unbounded Operators*, Mem. Amer. Math. Soc., No. 140, Amer. Math. Soc., Providence, R. I., 1974. MR**0417851 (54:5899)****[6]**A. J. Chorin, "Numerical study of slightly viscous flows,"*J. Fluid Mech.*, v. 57, 1973, pp. 785-796. MR**0395483 (52:16280)****[7]**A. J. Chorin, "Vortex sheet approximation to boundary layers,"*J. Comput. Phys.*, v. 27, 1978, pp. 428-442.**[8]**A. J. Chorin,*Numerical Methods for Use in Combustion Modelling*, Proc. Internat. Conf. Numer. Methods in Science and Engineering, Versailles, 1979. MR**584038 (82b:80016)****[9]**A. J. Chorin & J. Marsden,*A Mathematical Introduction to Fluid Mechanics*, Springer-Verlag, New York, 1979. MR**551053 (81m:76001)****[10]**A. J. Chorin, T. J. R. Hughes, M. T. McCracken & J. E. Marsden, "Product formulas and numerical algorithms,"*Comm. Pure Appl. Math.*, v. 31, 1978, pp. 205-256. MR**0488713 (58:8230)****[11]**K. L. Chung,*A Course in Probability Theory*, Academic Press, New York, 1974. MR**0346858 (49:11579)****[12]**J. D. Cole, "On a quasi-linear parabolic equation occurring in aerodynamics,"*Quart. Appl. Math.*, v. 9, 1951, pp. 225-236. MR**0042889 (13:178c)****[13]**A. Einstein,*Investigation on the Theory of Brownian Movement*, Translation, Methuen, London, 1956. MR**0077443 (17:1035g)****[14]**W. Feller,*An Introduction to Probability Theory and Its Applications*, Vol. I, Wiley, New York, 1968. MR**0228020 (37:3604)****[15]**W. Feller,*An Introduction to Probability Theory and Its Applications*, Vol. II, Wiley, New York, 1971. MR**0270403 (42:5292)****[16]**A. Friedman,*Partial Differential Equations of Parabolic Type*, Prentice-Hall, N. J., 1964. MR**0181836 (31:6062)****[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]**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)****[19]**J. Goodman, "Convergence of the random vortex method,"*Comm. Pure Appl. Math.*, v. 40, 1987, pp. 189-220. MR**872384 (88d:35159)****[20]**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)****[21]**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)****[22]**W. Hoeffding, "Probability inequalities for sums of bounded random variables,"*J. Amer. Statist. Assoc.*, v. 58, 1963, pp. 13-30. MR**0144363 (26:1908)****[23]**E. Hopf, "The partial differential equation ,"*Comm. Pure Appl. Math.*, v. 3, 1950, pp. 201-230. MR**0047234 (13:846c)****[24]**F. John,*Partial Differential Equations*, 3rd ed., Springer-Verlag, New York, 1978. MR**831655 (87g:35002)****[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 & M. Pulvirenti, "Hydrodynamics in two dimensions and vortex theory,"*Comm. Math. Phys.*, v. 84, 1982, pp. 483-503. MR**667756 (84e:35126)****[28]**F. J. McGrath, "Nonstationary plane flow of viscous and ideal fluids,"*Arch. Rational Mech. Anal.*, v. 27, 1968, pp. 329-348. MR**0221818 (36:4870)****[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]**R. D. Richtmyer & K. W. Morton,*Difference Methods for Initial Value Problems*, 2nd ed., Interscience, New York, 1967. MR**0220455 (36:3515)****[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]**Z. Teng, "Elliptic-vortex method for incompressible flow at high Reynolds number,"*J. Comput. Phys.*, v. 46, 1982, pp. 54-68. MR**665805 (83g:76019)****[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, New York, 1974. MR**0483954 (58:3905)**

Retrieve articles in *Mathematics of Computation*
with MSC:
65M10,
65U05,
76-08

Retrieve articles in all journals with MSC: 65M10, 65U05, 76-08

Additional Information

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

Article copyright:
© Copyright 1989
American Mathematical Society