The weighted particle method for convection-diffusion equations. I. The case of an isotropic viscosity

Authors:
P. Degond and S. Mas-Gallic

Journal:
Math. Comp. **53** (1989), 485-507

MSC:
Primary 65M99

DOI:
https://doi.org/10.1090/S0025-5718-1989-0983559-9

MathSciNet review:
983559

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Abstract: The aim of this paper is to present and study a particle method for convection-diffusion equations based on the approximation of diffusion operators by integral operators and the use of a particle method to solve integro-differential equations described previously by the second author. The first part of the paper is concerned with isotropic diffusion operators, whereas the second part will consider the general case of a nonconstant matrix of diffusion. In the former case, the approximation of the diffusion operator is much simpler than in the general case. Furthermore, we get two possibilities of approximations, depending on whether or not the integral operator is positive.

**[1]**S. Mas-Gallic, "A deterministic particle method for the linearized Boltzmann equation,"*Trans. Theory Stat. Phys.*, v. 16, 1987, pp. 855-887. MR**906929 (88k:82078)****[2]**A. Leonard, "Vortex methods for flow simulations,"*J. Comput. Phys.*, v. 37, 1980, pp. 289-335. MR**588256 (81i:76016)****[3]**A. Chorin, "Numerical study of slightly viscous flow",*J. Fluid Mech.*, v. 57, 1973, pp. 785-796. MR**0395483 (52:16280)****[4]**S. Roberts, "Accuracy of the random vortex method for a problem with a nonsmooth initial condition,"*J. Comput. Phys.*, v. 58, 1985, pp. 29-43. MR**789554 (86f:76022)****[5]**P. R. Spalart,*Numerical Simulation of Separated Flows*, NASA technical memorandum 84328, 1984.**[6]**J. J. Duderstadt & M.-R. Martin,*Transport Theory*, Wiley, New York, 1979. MR**551868 (84k:82099)****[7]**G. H. Cottet & S. Gallic, "A particle method to solve transport-diffusion equations-- Part 1: The linear case,"*Numer. Math.*(To appear.)**[8]**G. H. Cottet & S. Mas-Gallic, "A particle method to solve transport-diffusion equations --Part II: The Navier-Stokes equation," submitted to*Numer. Math.***[9]**S. Huberson,*Modélisation Asymptotique et Numérique de Noyaux Tourbillonaires Enroulés*, Thèse d'état, Université Pierre et Marie Curie, 1986.**[10]**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)****[11]**R. A. Gingold & J. J. Monaghan, "Shock simulation by the particle method S.P.H.,"*J. Comput. Phys.*, v. 52, 1983, pp. 374-389.**[12]**S. Mas-Gallic & P. A. Raviart,*Particle Approximation of Convection-Diffusion Problems*, Pub. Labo. Anal. Num., Univ. Pierre et Marie Curie, 1985.**[13]**S. Mas-Gallic,*C. R. Acad. Sci. Paris*, v. 305, 1987, pp. 431-434 and Thèse d'Etat, Université Pierre et Marie Curie, 1988. MR**916346 (89b:35062)****[14]**J. P. Choquin & B. Lucquin-Desreux, "Accuracy of a deterministic particle method for Navier-Stokes equations," submitted to*Internat. J. Numer. Methods Fluids*.**[15]**J. T. Beale,*On the Accuracy of Vortex Methods at Large Times*, Proceedings of the Workshop on Computational Fluid Dynamics and Reacting Gas Flows, Institute of Applied Mathematics, Minneapolis, MN, 1986.**[16]**J. P. Choquin & S. Huberson,*Application de la Méthode Particulaire aux Écoulements à Grand Nombre de Reynolds*, 18è Congrès National d'Analyse Numérique, Puy St Vincent, 1985.**[17]**B. Lucquin-Desreux, "Approximation particulaire des équations de Navier-Stokes bidimensionnelles,"*Rech. Aérospat.*, v. 4, 1987, pp. 1-12.**[18]**A. Leonard & G. Winckelmans,*Improved Vortex Methods for Three-Dimensional Flows with Application to the Interactions of Two Vortex Rings*, Proceedings of the Workshop on Vortex Dynamics, Xerox training center, Leesburg, 1988. MR**1001786****[19]**S. Mas-Gallic & P. A. Raviart, "A particle method for first order symmetric systems,"*Numer. Math.*, v. 51, 1987, pp. 323-352. MR**895090 (88d:65132)****[20]**P. A. Raviart, "An analysis of particle methods," in*Numerical Methods in Fluid Dynamics*(F. Brezzi, ed.), Lecture Notes in Math., vol. 1127, Springer-Verlag, Berlin, 1985, pp. 243-324. MR**802214 (87h:76010)****[21]**P. Degold & F. J. Mustieles,*A Deterministic Approximation of Diffusion Equations Using Particles*, Rapport Interne no 167, C.M.A.P. Ecole Polytechnique, Oct. 87;*SIAM J. Sci. Statist. Comput.*(To appear.) MR**1037516 (91a:65010)**

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DOI:
https://doi.org/10.1090/S0025-5718-1989-0983559-9

Article copyright:
© Copyright 1989
American Mathematical Society