A conservative particle approximation for a boundary advection-diffusion problem. (English) Zbl 0766.65095

The aim of this paper is to present and analyze a conservative two- dimensional extension of the deterministic method to the case of Dirichlet boundary conditions. The basic idea of the method is to add to the usual vorticity an extra term with support in a neighbourhood of the boundary. The boundary condition effects, as well as the diffusion effects, are taken into account by a modification of the weights of the particles. A boundary integral equation formulation is used to construct the method. The order of convergence of the method is of the same kind as in the caes of the whole space.
Reviewer: K.Zlateva (Russe)


65N38 Boundary element methods for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
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[1] C. ANDERSON, Observations on Vorticity Creation Boundary Conditions in Mathematical Aspects of Vortex Dynamics (R. E. Caflish ed.), S.I.A.M., Philadelphia 1988. Zbl0671.76030 MR1001797 · Zbl 0671.76030
[2] J. P. CHOQUIN, S. HUBERSON, Application de la méthode particulaire aux écoulements à grand nombre de Reynolds, 18e Congrès National d’Analyse Numérique, Puy St-Vincent (1985).
[3] J. P. CHOQUIN, B. LUCQUIN-DESREUX, Accuracy of a deterministic particle method for Navier-Stokes equations, Internat. J. Numer. Methods Fluids, 8 (1988), 1439-1458. Zbl0664.76029 · Zbl 0664.76029
[4] A. CHORIN, Numerical study of slightly viscous flow, J. Fluid Mech., 57 (1973), p. 785. MR395483
[5] G. H. COTTET, Boundary conditions and the deterministic vortex methods for the Navier-Stokes equations in Mathematical Aspects of Vortex Dynamics (R. E. Caflish ed.), S.I.A.M., Philadelphia 1988. Zbl0671.76047 MR1001796 · Zbl 0671.76047
[6] G. H. COTTET, S. GALLIC, A particle method to solve transport - diffusion equations - Part I : the linear case, Internal report n^\circ 115, C.M.A.P., École Polytechnique, Palaiseau, France. · Zbl 0678.35077
[7] [7] G. H. COTTET, S. MAS-GALLIC, A particle method to solve the Navier-Stokes system, Numer. Math. 57 (1990), 1-23. Zbl0707.76029 MR1065526 · Zbl 0707.76029
[8] P. DEGOND, S. MAS-GALLIC, The weighted particle method for convection - diffusion equations, part I : the case of an isotropic viscosity, part II : the anisotropic case, Math. Comput. 53 (1989), 485-526. Zbl0676.65121 MR983559 · Zbl 0676.65121
[9] J. GOODMAN, Convergence of the random vortex method, Comm. Pure Appl. Math., 40 (1987), 189-220. Zbl0635.35077 MR872384 · Zbl 0635.35077
[10] S. HUBERSON, Modélisation asymptotique et numérique de noyaux tourbillonaires enroulés, Thèse d’état (1986), Université Pierre et Marie Curie.
[11] S. HUBERSON, A. JOLLES, C. R. Acad. Sci. Paris 309, Série II, 445-448, Paris, 1989, and A Jollès, Résolution des équations de Navier-Stokes par des méthodes particules maillage, Thèse de Doctorat de l’Université, Université Pierre et Marie Curie, février 1989. Zbl0668.76121 MR1022287 · Zbl 0668.76121
[12] A. LEONARD, G. WINCKELMANS, Improved vortex methods for three-dimensional flows with application to the interactions of two vortex rings in Mathematical Aspects of Vortex Dynamics (R. E. Caflish ed.), S.I.A.M., Philadelphia 1988. Zbl0671.76025 MR1001786 · Zbl 0671.76025
[13] B. LUCQUIN-DESREUX, Particle approximation of the two dimensional Navier-Stokes equations, Rech. Aérospat. 4 (1987), 1-12. Zbl0619.76031 · Zbl 0619.76031
[14] B. LUCQUIN-DESREUX, Méthode particulaire conservative avec condition à la limite en dimension 1, internal report, Lab. Anal. Num. 1990.
[15] S. MAS-GALLIC, Thèse d’État de l’Université Pierre et Marie Curie, 1987 and C. R. Acad. Sci. Paris 305, série I, p. 431-434, 1987. Zbl0632.76104 MR916346 · Zbl 0632.76104
[16] S. MAS-GALLIC, C. R. Acad. Sci. Paris 310, série I, p. 465-468, 1990 and Une méthode particulaire déterministe incluant diffusion et conditions aux limites, internal report 90003, Lab. Anal. Num. 1990. Zbl0695.65069 MR1046534 · Zbl 0695.65069
[17] [17] S. MAS-GALLIC, P. A. RAVIART, A particle method for first order symmetric systems, Numer. Math. 51 (1987), 323-352. Zbl0625.65084 MR895090 · Zbl 0625.65084
[18] F. PEPIN, Simulation of the flow past an impulsively started cylinder using a discrete vortex method, Thesis, California Institute of Technology, Pasadena, California (1990).
[19] 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. Zbl0598.76003 MR802214 · Zbl 0598.76003
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