zbMATH — the first resource for mathematics

Particle methods for dispersive equations. (English) Zbl 0991.65008
Summary: We introduce a new dispersion-velocity particle method for approximating solutions of linear and nonlinear dispersive equations. This is the first time in which particle methods are being used for solving such equations. Our method is based on an extension of the diffusion-velocity method of P. Degond and F.-J. Mustieles [SIAM J. Sci. Stat. Comput. 11, No. 2, 293-310 (1990; Zbl 0713.65090)] to the dispersive framework. The main analytical result we provide is the short time existence and uniqueness of a solution to the resulting dispersion-velocity transport equation. We numerically test our new method for a variety of linear and nonlinear problems. In particular, we are interested in nonlinear equations which generate structures that have nonsmooth fronts. Our simulations show that this particle method is capable of capturing the nonlinear regime of a compacton-compacton type interaction.

65C35 Stochastic particle methods
65Z05 Applications to the sciences
35Q35 PDEs in connection with fluid mechanics
35K55 Nonlinear parabolic equations
76R50 Diffusion
PDF BibTeX Cite
Full Text: DOI
[1] Anderson, C.; Greengard, C., On vortex method, SIAM J. numer. anal., 22, 413, (1985) · Zbl 0578.65121
[2] Beale, J.T.; Majda, A., Vortex methods I: convergence in three dimensions, Math. comput., 39, 1, (1982) · Zbl 0488.76024
[3] Beale, J.T.; Majda, A., Vortex methods II: high order accuracy in two and three dimensions, Math. comput., 39, 29, (1982)
[4] Beale, J.T.; Majda, A., High order accurate vortex methods with explicit velocity kernels, J. comput. phys., 58, 188, (1985) · Zbl 0588.76037
[5] Carrier, J.; Greengard, L.; Rokhlin, V., A fast adaptive multipole algorithm for particle simulations, SIAM J. sci. stat. comput., 9, 669, (1988) · Zbl 0656.65004
[6] Chorin, A.J., Numerical study of slightly viscous flow, J. fluid mech., 57, 785, (1973)
[7] Chorin, A.J.; Hughes, T.J.R.; McCracken, M.; Marsden, J., Product formulas and numerical algorithms, Comm. pure appl. math., 31, 205, (1978) · Zbl 0358.65082
[8] Cottet, G.-H.; Koumoutsakos, P.D., Vortex methods, (2000), Cambridge Univ. Press Cambridge
[9] Cottet, G.-H.; Mas-Gallic, S., A particle method to solve the navier – stokes system, Numer. math., 57, 805, (1990) · Zbl 0707.76029
[10] Degond, P.; Mass-Gallic, S., The weighted particle method for convection-diffusion equations. part 1 and part 2, Math. comput., 53, 485, (1991)
[11] Degond, P.; Mustieles, F.J., A deterministic approximation of diffusion equations using particles, SIAM J. sci. stat. comput., 11, 293, (1990) · Zbl 0713.65090
[12] Evans, L.C., Partial differential equations, (1998), Am. Math. Soc Providence
[13] Fishelov, D., A new vortex scheme for viscous flows, J. comput. phys., 86, 211, (1990) · Zbl 0681.76036
[14] de Frutos, J.; López-Marcos, M.A.; Sanz-Serna, J.M., A finite difference scheme for the K(2, 2) compacton equation, J. comput. phys., 120, 248, (1995) · Zbl 0840.65090
[15] Goodman, J., Convergence of the random vortex method, Comm. pure appl. math., 40, 189, (1987) · Zbl 0635.35077
[16] Greengard, L.; Rokhlin, V., A fast algorithm for particle simulations, J. comput. phys., 73, 325, (1987) · Zbl 0629.65005
[17] Greengard, L.; Strain, J., The fast Gauss transform, SIAM J. sci. stat. comput., 12, 19, (1991)
[18] K. Gustafson and J. Sethian, Eds., Vortex Methods and Vortex Motion SIAM, Philadelphia, 1991.
[19] Hald, O.H., Convergence of vortex methods for Euler’s equations II, SIAM J. numer. anal., 16, 726, (1979) · Zbl 0427.76024
[20] O. H. Hald, Convergence of vortex methods, in Vortex Methods and Vortex Motion, edited by K. Gustafson and J. SethianSIAM, Philadelphia, 1991, pp. 33-58.
[21] O. H. Hald, private communication, 2000.
[22] Ismail, M.S.; Taha, T.R., A numerical study of compactons, Math. comput. sim., 47, 519, (1998) · Zbl 0932.65096
[23] John, F., Partial differential equations, (1982), Springer-Verlag New York
[24] G. Lacombe, Analyse d’une équation de vitesse de diffusion, preprint. · Zbl 0938.35004
[25] Lacombe, G.; Mas-Gallic, S., Presentation and analysis of a diffusion-velocity method, ESAIM: proc., 7, 225, (1999) · Zbl 0946.76079
[26] Li, Y.A.; Olver, P.J., Convergence of solitary-wave solutions in a perturbed Bihamiltonian dynamical system: I. compactons and peakons, Discrete cont. dynam. systems, 3, 419, (1997) · Zbl 0949.35118
[27] Li, Y.A.; Olver, P.J., Convergence of solitary-wave solutions in a perturbed Bihamiltonian dynamical system. II. complex analytic behavior and convergence to non-analytic solutions, Discrete cont. dynam. system, 4, 159, (1998) · Zbl 0959.35157
[28] K. Lindsay, and, R. Krasny, A Particle Method and Adaptive Treecode for Vortex Sheet Motion in 3-D Flow, preprint. · Zbl 1002.76093
[29] Long, D.-G., Convergence of the random vortex method in two dimensions, J. amer. math. soc., 1, 779, (1988) · Zbl 0664.76024
[30] S. Mas-Gallic, and, P. A. Raviart, Particle Approximation of Convection-Diffusion Equations, International Report 86013, Analyse Numérique, Université Pierre et Marie Curie, 1986.
[31] Nordmark, H.O., Rezoning of higher order vortex methods, J. comput. phys., 97, 366, (1991) · Zbl 0737.76058
[32] Puckett, E.G., (), 335-407
[33] P. A. Raviart, An analysis of particle methods, in, Numerical Methods in Fluid Dynamics, Lecture Notes in Mathematics, edited by, F. Brezzi, Springer-Verlag, Berlin, New York, 1983, Vol, 1127.
[34] Rosenau, P.; Hyman, J.M., Compactons: solitons with finite wavelength, Phys. rev. lett., 70, 564, (1993) · Zbl 0952.35502
[35] Rosenau, P.; Levy, D., Compactons in a class of nonlinearly quintic equations, Phys. lett. A, 252, 297, (1999) · Zbl 0938.35160
[36] Rosenau, P., Compact and noncompact dispersive patterns, Phys. lett. A, 275, 193, (2000) · Zbl 1115.35365
[37] W. Rudin, Functional Analysis, McGraw-Hill, New York, 1991. · Zbl 0867.46001
[38] Russo, G., Deterministic diffusion of particles, Comm. pure appl. math., 43, 697, (1990) · Zbl 0713.65089
[39] Russo, G., A particle method for collisional kinetic equations. I. basic theory and one-dimensional results, J. comput. phys., 87, 270, (1990) · Zbl 0709.65094
[40] Russo, G., A deterministic vortex method for the navier – stokes equations, J. comput. phys., 108, 84, (1993) · Zbl 0778.76076
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.