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Grid-free simulation of diffusion using random wall methods. (English) Zbl 0595.65132
The authors investigate the simulation of diffusion of a set of particles by means of the random walk displacement. They consider a type of statistical approach that differs from classical random walk methods both of the discrete and of the floating type. The method they expose can be called of ”gradient random walk”, where particles following a Brownian motion transport flux elements while their timely displacements are constructed to approximate, in statistical sense, the gradient of the concentration distribution; this latter is then obtained by repeated integration. The method can be considered as originated by the researches of Chorin et al. [see e.g. A. J. Chorin, Commun. Pure Appl. Math. 34, 853-866 (1981; Zbl 0458.76042)]. The paper contains also some examples of applications and some considerations about the advantages that this method presents from various angles in comparison with classical Monte Carlo methods.
Reviewer: M.Cugiani

MSC:
65Z05 Applications to the sciences
65C05 Monte Carlo methods
35K05 Heat equation
60J65 Brownian motion
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References:
[1] Anderson, C., Vortex methods for variable density flows, ()
[2] Aref, H., Ann. rev. fluid mech., 15, 345, (1983)
[3] Ashurst, W.T., (), 402
[4] Bailer, A.J., Finite-element computational fluid mechanics, (1983), McGraw-Hill/Hemisphere New York
[5] Beale, J.T.; Majda, A., Math. comp., 37, No.156, 243, (1981)
[6] Beale, J.T.; Majda, A., Math. comp., 39, No. 159, 28, (1982)
[7] Bramson, M., Mem. amer. math. soc., 44, 285, (1983)
[8] Burliarello, G.; Jackson, E.D.; Burliarello, G.; Jackson, E.D., (), 3, No. 2, 461, (1967)
[9] Carslaw, H.S.; Jaeger, J.C., Conduction of heat in solids, (1949), Oxford Univ. Press London · Zbl 0029.37801
[10] Cebeci, T.; Keller, H.B.; Williams, P.G., J. comput. phys., 31, 363, (1979)
[11] Chandraseilmr, S., Rev. mod. phys., 15, No. 1, 1, (1943)
[12] Cheer, A.Y., Program BOUNDL, ()
[13] Cheer, A.Y., Numerical analysis of time dependent flow structure generated by an impulsively started circular cylinder in a slightly viscous incompressible liquid, (1983), PAM-145, Center Pure Appl. Math., University of California Berkeley · Zbl 0524.76047
[14] Chorin, A.J., J. fluid mech., 57, No. 4, 785, (1973)
[15] Chorin, A.J., J. comput. phys., 27, 423, (1978)
[16] Chorin, A.J., Numerical methods for use in combustion modeling, (), 229
[17] Chorin, A.J., SIAM J. sci. stat. comput., 1, 1, (1980)
[18] Chorin, A.J., Comm. pure appl. math., 34, 843, (1981)
[19] Chorin, A.J.; Hughes, T.J.R.; McCracken, M.T.; Marsden, J.E., Comm. pure applied math., 31, 205, (1978)
[20] Chorin, A.J.; Marsden, J., A mathematical introduction to fluid mechanics, (1979), Springer-Verlag New York · Zbl 0417.76002
[21] Chu, C.K., Adv. appl. mech., 18, 285, (1978)
[22] Courant, R.; Friedrichs, K.; Lewy, H.; Courant, R.; Friedrichs, K.; Lewy, H., IBM j., Math. ann., 100, 32, (1928), 1967
[23] Courant, R.; Hilbert, D., Methods of mathematical physics, I, II, (1951), Interscience New York · Zbl 0729.35001
[24] Crank, J., The mathematics of diffusion, (1956), Oxford Univ. Press London · Zbl 0071.41401
[25] Dawson, J., Rev. mod. phys., 55, 403, (1983)
[26] Doob, J.L., Trans. amer. math. soc., 78, 168, (1955)
[27] Doob, J.L., Trans. amer. math. soc., 80, 216, (1955)
[28] Dwyer, H.A.; Raiszadeh, F.; Otey, G.R., ()
[29] Dwyer, H.A.; Smooke, M.D.; Kee, R.J., ()
[30] Eastwood, J.; Hockney, R.W., Computer simulation using particles, (1981), McGraw-Hill New York · Zbl 0662.76002
[31] Einstein, A.; Einstein, A., Investigation on the theory of the Brownian movement, (1956), McGraw-Hill New York, reprint · JFM 53.0876.09
[32] Emery, J.F.; Carson, W.W., J. heat transfer, 90, 328, (1968)
[33] Fichtl, G.H.; Perimutter, M.; Frost, W., (), 433
[34] Fife, P.C., Mathematical aspects of reacting and diffusing systems, () · Zbl 0403.92004
[35] Fraley, S.K.; Hoffman, T.F.; Stevens, P.N., J. heat transfer, 102, 121, (1980)
[36] Ghoniem, A.F.; Chorin, A.J.; Oppenheim, A.K., Phil. trans. R. soc. London ser. A, 304, 303, (1982) · Zbl 0478.76066
[37] Ghoniem, A.F.; Oppenheim, A.K., (), 224
[38] Ghoniem, A.F.; Oppenheim, A.K., Aiaa j., 22, No. 10, 1429, (1983)
[39] Ghoniem, A.F.; Chen, D.Y.; Oppenheim, A.K., Formation and inflammation of turbulent jets, (1984), Amer. Inst. Aeronautics and Astronautics New York, AIAA-84-0572
[40] Ghoniem, A.F.; Sethian, J.A., Structure of turbulence in a recirculating flow; A computational study, (1985), AIAA-85-0146, Amer. Inst. Aeronautics and Astronautics New York
[41] Gingold, R.A.; Monaghan, J.J., J. comput. phys., 46, 429, (1982)
[42] Haji-sheikh, J.; Sparrow, E.M., J. heat transfer, 122, 121, (1967)
[43] Hald, O., SIAM J. num. anal., 16, 726, (1979)
[44] Hald, O., SIAM J. sci. stat. comput., 2, 85, (1981)
[45] Hald, O., ()
[46] Hammersley, J.M.; Handscomb, D.C., Monte Carlo methods, (1964), Methuen London · Zbl 0121.35503
[47] Hirschfilder, J.O.; Curtiss, C.F.; Bird, R.B., Molecular theory of gases and liquids, (1964), Wiley New York
[48] Hsiao, C.C.; Ghoniem, A.F.; Chorin, A.J.; Oppenheim, A.K., ()
[49] Illingworth, C.R., (), 603, No. 4
[50] Ito, K.; McKean, H.P., Diffusion processes and their sample spcaded, (1974), Springer-Verlag New York/Berlin · Zbl 0285.60063
[51] John, F., Partial differential equations, (1979), Springer-Verlag New York
[52] Kellogg, O.D., Foundations of potential theory, (1953), Dover New York · Zbl 0053.07301
[53] Laitone, J.A., Separation effects in gas-particle flows at high Reynolds numbers, (1979), LBL-9996, Lawrence Berkeley Laboratory U. C. Berkeley · Zbl 0469.76076
[54] Lamb, J., Hydrodynamics, (1945), Dover New York · Zbl 0828.01012
[55] Lamperti, J., Probability theory, (1966), Benjamin New York · Zbl 0147.15502
[56] Landau, L.D.; Lifshitz, E.M., Fluid mechanics, (1975), Pergamon New York, translated from Russian · Zbl 0146.22405
[57] Lighthill, M.J., (), 46
[58] Leonard, A., J. comput. phys., 37, No. 3, 289, (1980)
[59] Leonard, A.; Spalart, P.R., ()
[60] Lin, S.S., Theoretical study of a reaction-diffusion model for flame propagation in a gas, (1979), LBL9487, Lawrence Berkeley Laboratory U. C. Berkeley
[61] Marchioro, C.; Pulvirenti, M., Comm. math. phys., 84, 483, (1982)
[62] McCracken, M.F.; Peskin, C.S., J. comput. phys., 35, 183, (1980)
[63] McKean, H.P., Comm. pure appl. math., 28, 323, (1975)
[64] Milinazzo, F.; Saffman, P.G., J. comput. phys., 23, 380, (1979)
[65] Monaghan, J.J., Why particle methods work, SIAM J. sci. stat. comput., 3, 422, (1982) · Zbl 0498.76010
[66] Morse, P.M.; Feshbach, H., Methods of theoretical physics, (1953), McGraw-Hill New York · Zbl 0051.40603
[67] Nakamura, Y.; Leonard, A.; Spalart, P., ()
[68] Niederreiter, H., Bull. amer. math. soc., 84, 957, (1978)
[69] Peskin, C.S., Ann. rev. fluid mech., 14, 235, (1982)
[70] Potter, D., Computational physics, (1973), Wiley-Interscience New York · Zbl 0314.70008
[71] \(J. Ramos\), Int. J. Num. Meth. Fluid Mech., in press.
[72] Rayleigh, J.W.; Rayleigh, J.W., (), 17, 371, (1889)
[73] Rayleigh, J.W.; Rayleigh, J.W., (), 72, 256, (1905)
[74] Richtmyer, R.D.; Morton, K.W., Difference methods for initial value problems, (1967), Interscience New York · Zbl 0155.47502
[75] Roache, P.J., Computational fluid dynamics, (1972), Hermosa Albuquerque, N.M · Zbl 0251.76002
[76] Roberts, S., Accuracy of the random vortex method for a problem with non-smooth initial conditions, (1983), PAM-162, Center of Pure and Applied Math U. C. Berkeley
[77] Rubinstein, R.Y., Simulation and the Monte Carlo method, (1981), Wiley New York · Zbl 0529.68076
[78] A. H. SHAPIRO, “Vorticity,” motion picture, NSF-Sponsored series on Fluid Mechanics, Nat. Comm. for Fluid Mechanics, Encyclopedia Britannica Films.
[79] Shen, S.F., Adv. appl. mech., 18, 177, (1978)
[80] Sherman, F.S., ()
[81] Shestakov, A.I., A hybrid vortex-AID solution for flows of low viscosity, J. comp. phys., 31, 313, (1979) · Zbl 0442.76036
[82] Shireider, Y.A., The Monte Carlo method, the method of statistical trial, (1966), Pergamon New York, translated from Russian
[83] Smith, J.H., Survey of three-dimensional finite difference forms of heat equation, SC-M-70-83, (1970), Sandia Laboratories Albuquergue, N. M
[84] Sod, G.A., Computational fluid dynamics with stochastic techniques, () · Zbl 0635.65126
[85] Telionis, D.P., Unsteady viscous flows, (1981), Springer-Verlag New York · Zbl 0484.76054
[86] Tend, Z.H., J. comput. phys., 46, 54, (1982)
[87] Von Smoluchowski, M., Drei vortage uber diffusion, brownsche bewegung and koalzulation von kolloidteilchen, Physik. zeits., 17, 557, (1916)
[88] Yanenko, N.N., The method of fractional steps, (1971), Springer-Verlag New York · Zbl 0209.47103
[89] Zeldovich, Ya.B.; Frank-Kamenetskii, D.A., Zhur. fiz. khim., 12, 100, (1938)
[90] Zinsmeisrer, G.E., Bull. mech. eng., 7, 77, (1968)
[91] Zinsmeister, G.E.; Pau, S.S., Int. J. num. meth. eng., 10, 1057, (1976)
[92] Bell Telephone Education Comm, Diffusion along a rod, motion picture, Bell telephone laboratories for the semiconductor electronics, (1969), Newton, Mass.
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