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.

- Sylvie Mas-Gallic,
*A deterministic particle method for the linearized Boltzmann equation*, Proceedings of the conference on mathematical methods applied to kinetic equations (Paris, 1985), 1987, pp. 855–887. MR**906929**, DOI https://doi.org/10.1080/00411458708204318 - A. Leonard,
*Vortex methods for flow simulation*, J. Comput. Phys.**37**(1980), no. 3, 289–335. MR**588256**, DOI https://doi.org/10.1016/0021-9991%2880%2990040-6 - Alexandre Joel Chorin,
*Numerical study of slightly viscous flow*, J. Fluid Mech.**57**(1973), no. 4, 785–796. MR**395483**, DOI https://doi.org/10.1017/S0022112073002016 - 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**, DOI https://doi.org/10.1016/0021-9991%2885%2990154-8
P. R. Spalart, - James J. Duderstadt and William R. Martin,
*Transport theory*, John Wiley & Sons, New York-Chichester-Brisbane, 1979. A Wiley-Interscience Publication. MR**551868**
G. H. Cottet & S. Gallic, "A particle method to solve transport-diffusion equations— Part 1: The linear case," - 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**, DOI https://doi.org/10.1090/S0025-5718-1981-0628693-0
R. A. Gingold & J. J. Monaghan, "Shock simulation by the particle method S.P.H.," - Sylvie Mas-Gallic,
*Méthode particulaire pour une équation de convection-diffusion*, C. R. Acad. Sci. Paris Sér. I Math.**305**(1987), no. 10, 431–434 (French, with English summary). MR**916346**
J. P. Choquin & B. Lucquin-Desreux, "Accuracy of a deterministic particle method for Navier-Stokes equations," submitted to - G. Winckelmans and A. Leonard,
*Improved vortex methods for three-dimensional flows*, Mathematical aspects of vortex dynamics (Leesburg, VA, 1988) SIAM, Philadelphia, PA, 1989, pp. 25–35. MR**1001786** - S. Mas-Gallic and P.-A. Raviart,
*A particle method for first-order symmetric systems*, Numer. Math.**51**(1987), no. 3, 323–352. MR**895090**, DOI https://doi.org/10.1007/BF01400118 - P.-A. Raviart,
*An analysis of particle methods*, Numerical methods in fluid dynamics (Como, 1983) Lecture Notes in Math., vol. 1127, Springer, Berlin, 1985, pp. 243–324. MR**802214**, DOI https://doi.org/10.1007/BFb0074532 - Pierre Degond and Francisco-José Mustieles,
*A deterministic approximation of diffusion equations using particles*, SIAM J. Sci. Statist. Comput.**11**(1990), no. 2, 293–310. MR**1037516**, DOI https://doi.org/10.1137/0911018

*Numerical Simulation of Separated Flows*, NASA technical memorandum 84328, 1984.

*Numer. Math.*(To appear.) G. H. Cottet & S. Mas-Gallic, "A particle method to solve transport-diffusion equations —Part II: The Navier-Stokes equation," submitted to

*Numer. Math.*S. Huberson,

*Modélisation Asymptotique et Numérique de Noyaux Tourbillonaires Enroulés*, Thèse d’état, Université Pierre et Marie Curie, 1986.

*J. Comput. Phys.*, v. 52, 1983, pp. 374-389. S. Mas-Gallic & P. A. Raviart,

*Particle Approximation of Convection-Diffusion Problems*, Pub. Labo. Anal. Num., Univ. Pierre et Marie Curie, 1985.

*Internat. J. Numer. Methods Fluids*. 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. 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. B. Lucquin-Desreux, "Approximation particulaire des équations de Navier-Stokes bidimensionnelles,"

*Rech. Aérospat.*, v. 4, 1987, pp. 1-12.

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Article copyright:
© Copyright 1989
American Mathematical Society