Numerical “Particle-in-cell” methods: Theory and applications

Numerical "Particle-in-Cell" Methods: Theory and Applications

Volumn , Issue , 2012, Pages 1-252

  Grigoryev, Yu N ^a   Vshivkov, V A ^a   Fedoruk, M P ^a  

^a NONE

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EID: 85064835717     PISSN: None     EISSN: None     Source Type: Book    
DOI: 10.1515/9783110916706     Document Type: Book

References (126)

1. Kochin N.E. Kibel I.A. Roze N.V. Theoretical hydromechanics. P.I. Μ: FM, 1963, 583p (in Russian).
2. Leonard A. Vortex methods for flow simulation. J. Comp. Phys. 1980, V.37, p.289-335.
3. Vshivkov V.A., Kraeva, M.A., Malyshkin V.E. Parallel Implementation of the Particle-in-Cell Method//Programming and Computer Software, 1997, V.23, N.2, p.87-97.
4. Kraeva M.A. Malyshkin V.E. Implememntation of PIC Method on MIMD Multicomputers with Assembly Technology. In Proceed, of HPCN Europe 1997, LNCS, V.1255, Springer Verlag, p. 541-549.
5. Kraeva M.A. Malyshkin V.E. A Dynamical balance of assembly in the realization PIC - metod on MIMD multicomputers. Programmirovanie. 1999, N.l (in Russian).
6. Kraeva M.A. Malyshkin V.E. Assembly Technology for Parallel Realization of Numerical Models on MIMD-Multicomputers. In the special issue of the International Journal on Future Generation Computer Systems (Elsevier), devoted to Parallel Computing Technologies. 2001, V.17, p.755-765.
7. Hockney R.W. Eastwood J.W. Computer simulation using particles. McGraw-Hill Inc. 1981, p.523.
8. Christiansen J.P. Vortex. Two-dimensional hydrodynamics simulation code./Culham. Lab. Rep. CLM-106. HMSO. London: 1970.
9. Harlow F.H. Dickman D.O. Harris D.E. Martin R.E. Two dimensional hydrodynamic calculations./Los Alamos Scientific Lab. Rep. NLA-2301, 1959.
10. Harlow F.H. The particle-in-cell computing method in fluid dynamics. Meth. Comput. Phys. 1964, V.3, p.319-343.
11. Theoretical foundation and construction of the numerical algorithms for the problems of mathematical physics./Ed. by K.I. Babenko. Moscow: Nauka, 1979, p.295 (in Russian).
12. Yanenko N.N, Anuchina N.N., Petrenko V.E., Shokin Yu.I. On a calculation methods for gasdynamic problems with large deformations. Chislennye metody mexaniki sploshnoi sredyu. 1970, V.1, p.40-62 (in Russian).
13. Belotserkovsky O.M., Davyudov Yu.M. A coarse particle method in gas dynamic. Moscow: Nauka, 1982, p.392 (in Russian).
14. Batchelor J. Introduction to the fluid dynamics. Moscow: Mir, 1973, 757p (in Russian).
15. Lamb H. Hydrodynamics. Dover publications Inc. New York, 1954.
16. Rosenhead L. The formation of vortices from a surface of discontinuity. Proc. Roy. Soc. London A, 1931, v.134, p. 170-192.
17. Chorin A.J. Numerical study of slightly viscous flow. J. Fluid. Mech. 1973, v.57, p.785-796.
18. Baker G.R. The "Cloud in Cell" Technique Applied to the Roll Up of Vortex Sheets. J. Comp. Phys. 1979, V.31, N.1, p.76-95.
19. Christiansen J.P. Numerical simulation of Hydrodynamics by the Method of Point Vortices.//J. Comp. Phys. 1973, V.13, N.3, p.363-379.
20. Methods in computational physics. Plasma physics/Ed. by Alder B. Fernbach S. Rotenberg Μ. New York, Academic Press, 1970, V.9.
21. Buneman O. Dissipation of currents in ionized media. Phys. Rev. 1959, V. 115, N.3, p. 503-519.
22. Berezin Yu.A., Vshivkov V.A. Particle-in-cell method in rarefied plasma dynamics. Novosibirsk: Nauka, 1980, 94p (In Russian).
23. Kertis D. Monte Carlo method for iteration of linear operators. Uspexi matematicheskix nauk. 1957, V.12, p.149-174 (in Russian).
24. Haviland J.K. The solution of two molecular flow problems by the Monte-Karlo method. Methods in Comput.Phys. Adv. in Research and Appl. in Hydrodinamics. N-Y.: 1965, V.4, p.109-209.
25. Grigoryev Yu.N., Ivanov M.S., Kharitonova N.I. On a question about solving of nonlinear kinetic equations of the rarefied gas dynamic by the Monte Carlo method. Chislennye metody mexaniki sploshnoi sredyu. 1971, V.2. p.101-107 (in Russian).
26. Grigoryev Yu.N., Yanenko N.N., Ivanov M.S. Numerical simulation of the rarefied gas dynamics problems. Lect. Notes Phys.,141, Berlin- Heidelberg-N.-Y.-Tokyo, Spr.-Verlag, 1981, p.454-460.
27. Yanenko N.N., Grigoryev Yu.N. et al. Methods of statistical modeling and direct numerical integration of kinetic equations of gas theory. Development and application to the problems of rarefied gas dynamics. Proc. 13th Int. Symp. on RGD, V.l, N.-Y.-London: Plenum Press, 1985, p.371-382.
28. Belotserkovsky O.M. Yanitsky V.E. The statistical particle-in –cell method for solving rarefied gas dynamics problems. J. vychislitelnoi matematiki i matematicheskoi fiziki. 1975, V.15, N.5, p.1195-1208 (in Russian).
29. Belotserkovsky O.M. Numerical modeling in the mechanics of continuous media. Moscow: Nauka, 1984, 514p (in Russian).
30. Bird G.A. Molecular Gas Dynamics. Oxford: Clarendon Press, 1976.
31. Pullin D.I. Direct simulation Methods for Compressible Inviscid Ideal- Gas Flow. J. Comp. Phys. 1980, V.34, Ν 2. p.231-244.
32. Yanitsky V.E. Statistical particle method for solving of some problems of kinetic gas theory and turbulence. Dis. doct. fiz.-mat. nauk. Moscow: 1984, 271p. (in Russian).
33. Zenkevich O. Morgan K. Finite elements and approximations. Moscow: Mir, 1986 (in Russian).
34. G.A. Korn, T.M. Korn. Mathematical Handbook for scientists and engineers. N.-Y.: McGraw-Hill Book Co. 1968.
35. Hockney R.W. A computer experiment of anomalous diffusion. Phys. Fluids. 1966, V.9, N.9, p.1826-1835.
36. Burger P. Dunn D.A. Halstead A.S. Computer experiments on the randomization of electrons in a collisionless plasma. Phys. Fluids. 1965, V.8, N.12, p.2263-2272.
37. Birdsall C.K., Fuss D. Clouds-in-clouds, clouds-in-cells physics for many-body plasma simulation. J. Comp. Phys. 1969, V.3, N.4, p. 494-511.
38. Birdsall Ch.K., Langdon A.B. Plasma physics via computer simulation. McGraw-Hill, 1985.
39. Vshivkov V.A., Snytnikov V.N. On the particle method for solving of the Vlasov kinetic equations. J. vychislitelnoi matematiki i matematicheskoi fiziki. 1998, V.38, N.11, p.1877-1883 (in Russian).
40. Vshivkov V.A. Romanov D.V. Snytnikov V.N. A problem of selfheating of model plasma in the particle method. Vychislitelnye technologii. 1999, V.4, N.3, p.62-72 (in Russian).
41. Vshivkov V.A. The approximation properties of the particle-in-cell method. Comp. Maths. Math. Phys. 1996, V.36, N.4, p.509-515.
42. Potter D. An introduction to computational physics. John Wiley and Sons, Ltd. London, 1973.
43. Bluhm H. Hoppe P. Bachmann Η. et al. Progress in the development of a high power focusing B-applied extractortype ion diode for 1.5 TW pulse generator KALIF. Proc. 8th Internat. Conf. High-Power Particle Beams, Novosibirsk, 2-5 July 1990. World Scient. Singapore, p.451-456, 1990.
44. Matsumoto M. Kawata S. TRIPIC: triangular-mesh PIC code. J. Comput. Phys. 1990, V.87, p.488-493.
45. Thopmson J.F. Warsi Z.U.A. Mastin C.W. Boundary-fitted coordinate systems for numerical solution of partial differential equations – a review. J. Comput. Phys. 1982, V.47, p. 1-108.
46. Jones M.E. Electromagnetic PIC codes with body-fitted coordinates. Proc. 12th Internat. Conf. Numerical Simulation of Plasmas, San Francisco, 20-24 September 1984, IM3, 1987.
47. Westermann Τ. Numerical modelling of the stationary Maxwell - Lorentz system in technical devices. Intern. J. Num. Mod.: Electronic Network, Devices and Fields. 1994, V.7, p.43-67.
48. Halter E., Krauss M, C.-D. Münz et al. A concept for numerical solution of the Maxwell-Vlasov system. Forshungszentrum: Karlsruhe - Umwelt und Technik, FZKA 5564, 1995.
49. Assous F. Degond P. Segre J. A particle-tracking method for 3D electromagnetic PIC codes on unstructured meshes. Comput. Phys. Commun. 1992, V.72, p.105-114.
50. Sonnedrucker Ε. Ambrosiano J.J. Brandon S.T. A finite element formulation of the Darwin PIC model on unstructured grids. J. Comput. Phys, 1995, V.121, p.281-297.
51. Kruglyakova L.V. Neledov A.V. Tishkin V.F. Filatov A.Yu. Unstructured adaptive grids for problems of mathematical physics (review). Matematicheskoe modelirovanie. 1998, V.10, p.93-116 (in Russian).
52. Boldyrev A.S. Gasilov V.A. Olxovskaya V.G. On solving hyperbolic equations on unstructured grids. Matematicheskoe modelirovanie. 1996, V.8, p.51-78 (in Russian).
53. Lohner R. Ambrosiano J. A vectorized particle tracer for unstructured grids. J. Comput. Phys. 1990, V.91, p.22-31.
54. Westermann Τ. Localization shemes in 2D boundary-fitted grids. J. Comput. Phys. 1992, V.101, p.307-313.
55. Lohner R. Robust, vectorized search algorithms for interpolation on unstructured grids. J. Comput. Phys. 1995, V.118, p.380-387.
56. Fedoruk M. C.-D., Munz, Omnes P., Schneider R. A Maxwell-Lorentz solver for self-consistent particle-field simulations on unstructured grids. Karlsruhe, FZKA 6115, p. 1-81, 1998.
57. Dudnikova G.I. Romanov D.V. Fedoruk M.P. On the particle model on the unstructured grids. Vychislitelnye technologii. 1998, V.3, p.30-46 (in Russian).
58. Dudnikova G.I. Romanov D.V. Fedoruk M.P. On algorithms implementing the particle method on unstructured grids. Computational mathematics and mathematical physics. 2000, V.40, p.147-158.
59. Landau L.D. Lifshits E.M. Mechanics. Moscow: Nauka, 1988 (in Russian).
60. Assous F. Degond P. Heintze E. et al. On a finite element method for solving the three-dimensional Maxwell equations. J. Comput. Phys. 1993, V.109, p.222-237.
61. Ambrosiano J.J. Brandon S.T. Lohner R. DeVore C.R. Electromagnetics via the Taylor-Galerkin finite element method on unstructured grids. J. Comput. Phys. 1994, V.110, p.310-319.
62. Hermeline F. Two Coupled Particle-Finite Volume Methods Using Delaunay-Voronoi Meshes for the Approximation of Vlasov-Poisson and Vlasov-Maxwell Equations. J. Comput. Phys. 1993, 5. 106, p.1-18.
63. J.-P. Cioni, L. Fezoui, D. Issautier. High order upwind schemes for solving time-domain Maxwell equations. La Recherche Aerospatiale. 1994, N.5, p.319-328.
64. Rich M.A. A Method for Eulerian Fluid Dynamics. Los Alamos Sei. Lab. 1963, Rep. No LAMS-2826.
65. Gentry R.A., Martin R.E. and Daly B.J. An Eulerian Differensing Method for Unsteady Compressible Flow Problems. J. Comp.Phys. 1966, V.1, p.87-118.
66. Belotserkovsky O.M., Davydov Yu.M. A non-stationary "coarseparticle" method for gasdynamical computations. J. vychislitelnoi matematiki i matematicheskoi fiziki. 1971, V.11, N.1, p.182-207 (in Russian).
67. Sapojnikov G.A. A combine method of fluid fluxes and particle-in-cell for calculations of the gasdynamical flows/Sb. Voprosy razrabotki i ekspluatacii paketov prikladnyx programm. Novosibirsk: ITPM SO AN SSSR, 1981. p.89-97 (in Russian).
68. Godunov S.K. Ryabenkii V.S. Finite-difference schemes (Introduction to the theory). Moscow: Nauka, 1973, p.400 (in Russian).
69. Anuchina N.N. On the calculalation methods for flows of compressible fluid with large deformations. Chislennye metody mekhaniki sploshnoi sredy. 1970, V. 1, N. 4, p.3-84 (in Russian).
70. Petrils T. An approach to the inviscid rotational compressible flows./Prepr. B. Rol. Univ. Fac. Math. Res. Seminar. 1982, N.1. p.44-63.
71. Chorin A.J. Vortex sheet approximation of boundary layers. J. Comp. Phys. 1978, V.27, N.3 p.428-445.
72. Chorin A.J. Lectures on Turbulence Theory. Boston.: Publish/Perish, 1976. 159p.
73. Roshal A.S. The fast Fourier transform in computational physics. Izv. Vuzov. Radiofizika, 1976, V.XDC, N.10, p.1425-1454 (in Russian).
74. Yanenko N.N., Veretentsev A.N., Grigoryev Yu.N. A Hamiltonian formalism for a spatial system of small vortices in ideal fluid. Chislennye metody mexaniki sploshnoi sredyu. 1979. V.10, N.5, p.144-149 (in Russian).
75. Grigoryev Yu.N., Levinsky V.B., Yanenko N.N. Modeling of turbulence by ensembles of vorticies with inviscid interaction. Arch. Mech. 1984, V.36, N.2, p.279-292.
76. Williamson J.H. Initial particle distributions for simulated plasma. J. Comp. Phys. 1971, V.8, p.258-267.
77. Byers J.A., Grewal M. Perpendicularly propagating plasma cyclotron instabilities simulated with one-dimensional computer model. Phys. Fluids, 1970, V.13, p.1819-1830.
78. Matsuda Y. Crawford F.W. Computational study of nonlinear plasma waves. Phys. Fluids, 1975, V.18, p.1336-1353.
79. Boris J.P. Relativistic plasma simulation - optimization of a hybrid code coordinates. Procedings 4th International Conference on the Numerical Simulation of Plasmas, Washington, 20-24 September 1970, P.3-67.
80. Langdon A.B. Effects of the spatial grid in simulation plasma. J. Comp. Phys. 1970, V.6, p.247-267.
81. Langdon A.B. "Energy-conserving" plasma simulation algorithms. J. Comp. Phys. 1973, V.12, p.247-268.
82. White R.B., Fried B.D., Coroniti F.V. Structure of ion acoustic solitons and shock waves in a two-component plasma. Phys. Fluids. 1972, V.15, p.1484-1490.
83. Sagdeev R.Z. Cooperative processes and shock waves in a rarefied plasmas/Voprosyu teorii plasmy. Moscow: Atomizdat, 1964. Vyp.4, p.20-80 (in Russian).
84. Bernhardt P.A. Roussel-Dupre R.A. Haerendel G. Observations and theory of the AMPTE magnetotail barium released. J.Geophys.Res. 1987, V.92, p.5777-5794.
85. Berezin Yu.A., Vshivkov V.A. Shock waves of arbitrary amplitude in rarefied plasma with magnetic field. Fizika plasmy. 1977, V.3, p.365-370 (in Russian).
86. Vshivkov V.A. Fedoruk M.P. On algorithm for computation of shock waves in rarefied plasma propagating at arbitrary angle with respect to a magnetic field. Chislennye metody mexaniki sploshnoi sredyu. 1985, V.16, N.5, p.33-50 (in Russian).
87. Operation "Argus". Moscow: Atomizdat, 1960 (in Russian).
88. Operation "Star fish". Moscow: Atomizdat, 1964 (in Russian).
89. Dudnikova G.I., Orishich A.M., Ponomarenko A.G., Zakharov Yu.P., Vshivkov V.A. Laboratory and computer simulation of generation magnetosonic disturbances in magneto-spheric plasma In "Proc. XX ICPIG", Piza, 1991.
90. Borovsky J.E. Pongratz M.B. Roussel-Dupre R.A. Tan T. The laboratory simulation of unmagnetized supernova remnats: absence of a blast wave. Astrophys. J. 1984, V.280, p.802-808.
91. Golubev A.I., Solovyev A.A., Terekhin V.A. On a collisionless expansion of an ionized cloud in a homogeneous magnetized plasma. J. Prikl. mexan. and tex. fiz. 1978, N.5, p.33-42 (in Russian).
92. Bashurin V.P., Golubev A.I., Terekhin V.A. On a colisionless deceleration of an ionized cloud expanding into a homogeneous magnetized plasma. 1983. J. Prikl. mex. and tex. fiz. 1983, N.5, p.10-17 (in Russian).
93. Berezin Yu.A., Vshivkov V.A., Dudnikova G.I., Fedoruk M.P. Collisionless slowing down of a plasma cloud in nonuniform magnetized background. Sov. J. Plasma Phys. 1992, V.18, p.812-815.
94. Berezin Yu.A., Dudnikova G.I., Fedoruk M.P., Vshivkov V.A. Explosion phenomena in collisionless plasmas at super-alfvenic speed. Intern. J. Comput. Fluids. Dyn. 1998, V.10, p.117-126.
95. Bulanov S.V., Vshivkov V.A., Dudnikova G.I., Naumova N.M., Pegoraro F., Pogorelsky I.V. Laser acceleration of charged particles in nonuniform plasmas. Plasma Physics Reports. 1997, N.23, N.4, p. 259-269.
96. Askaryan G.A., Bulanov S.V., Dudnikova G.I., Esirkepov T.Zh., Lontano M., Meyer-ter-Vehn J., Pegoraro F., Pukhov A.M., Vshivkov V.A., Magnetic interaction of ultrashort high-intensity laser pulses in plasmas. Plasma Physics and Controlled Fusion, 1997, V.39, Ν. 5A, p. 137-144.
97. Pegoraro F. Bulanov S.V. Califano F. Esirkepov T.Zh. Lontano M. Meyer-ter-Vehn J. Naumova N.M. Pukhov A.M. Vshivkov V.A. Magnetic fields from high-intensity laser pulses in plasmas. Plasma Phys. Control. Fusion. 1997, V.39, p.B261-B272.
98. Vshivkov V.A. Bulanov S.V. Naumova N.M. Pegoraro F. Nonlinear electrodynamics of the interaction of ultra-intense laser pulses with a thin foil. Physics of Plasmas, 1998, V.5, N.7.
99. Bulanov S.V. Esirkepov T.Zh. Naumova N.M. Pegoraro F. Vshivkov V.A. Solitonlike electromagnetic waves behind a superintense laser pulse in a plasma. Physical Review Letters, 1999, V. 82, N.17, p.3440-3443.
100. Bulanov S.V. Califano F. Dudnikova G.I. Liseikina T.V. Naumova N.M. Pegoraro F. Vshivkov V.A. Sakai J.-I. Sakharov A.S. Laser acceleration of charged particles in inhomogeneous plasmas. II: Particle injection into acceleration phase due to nonlinear wake wave-breaking. Plasma Physics Reports, 1999, V.25, N.6, p.468-481.
101. Bulanov S.V. Vshivkov V.A. Dudnikova G.I. Esirkepov T.Zh. Califano F. Kamenets F.F. Liseikina T.V. Naumova N.M. Pegoraro F. Interaction of petawatt laser pulses with underdense plasmas. Plasma Physics Reports, 1999, V.25, N.9, p.701-714.
102. Esirkepov T.Zh. Sentoku Y. Mima K. Nishihara K. Califano F. Pegoraro F. Naumova N.M. Bulanov S.V. Ueshima Y. Liseikina T.V. Vshivkov V.A. Kato Y. Ion acceleration by superintense laser pulses in plasmas. JETF Letters, 1999, V.70, N.2, p.82-89.
103. Liseikina T.V. Califano F. Vshivkov V.A. Pegoraro F. Bulanov S.V. Small-scale electron density and magnetic-field structures in the wake of an ultraintense laser pulse. Physical Review E, 1999, V.60, N.5, p.5991-5997.
104. Pegoraro F. Bulanov S.V. Califano F. Esirkepov T.Zh. Lisejkina T.V. Lontano M. Naumova N.M. Ruhl H. Sakharov A.S. Vshivkov V.A. Coherent electromagnetic structures in relativistic plasmas. Physica Scripta, 2000, V.T84, p.89-93.
105. Pegoraro F. Bulanov S.V. Califano F. Esirkepov T.Zh. Liseikina T.V. Naumova N.M. Ruhl H. Vshivkov V.A. Ion acceleration regimes in underdense plasmas. IEEE Transaction on Plasma Science, 2000, V.28, N.4, p.1177-1183.
106. Sentoku Y. Liseikina T.V. Esirkepov T.Zh. Califano F. Naumova N.M. Ueshima Y. Vshivkov V.A. Kato Y. Mima K. Nishihara K. Pegoraro F. Bulanov S.V. High Density Collimated Beams of Relativistic Ions Produced by Petawatt Laser Pulses in Plasmas. Phys.Rev.E, 2000, V.62, N.5, p.7271-7281.
107. Vshivkov V.A. Dudnikova G.I. Algorithms for solution of the problem of interaction laser pulse with plasma. Vychislitelnye texnologii, 2001, V.6, N.2, p.47-63 (in Russian).
108. Bulanov S.V. Califano F. Dudnikova G.I. Esirkepov T.Zh. Inovenkov I.N. Kamenets F.F. Liseikina T.V. Lontano M. Mima K. Naumova N.M. Nishihara K. Pegoraro F. Ruhl H. Sakharov A.S. Sentoku Y. Vshivkov V.A. Zhakhovskii V.V. Relativistic Interaction of Laser Pulses with Plasmas/In: Reviews of Plasma Physics. New York: Kluwer Academic, 2001, V.21, p. 227-335.
109. Macchi Α. Cornolti F. Pegoraro F. Liseikina T. Ruhl H. Vshivkov V. Surface oscillations in overdense plasmas irradiated by ultrashort laser pulses. Phys. Rev. Lett. 2001, V.87, N.20.
110. Langdon A.B., Lasinski B.F. Electromagnetic and relativistic plasma simulation models. Meth. Comput. Phys. 1976, V.16, p.327-366.
111. Marder Β. A method for incorporating Gauss' law into electromagnetic PIC codes. J. Comput. Phys. 1987, V.68, p.48-55.
112. Villasenor J., Buneman O. Rigorous charge conservation for local electromagnetic field solver. Computer Phys. Comm. 1992, V.69, p. 306-316.
113. Cogan M.N. Introduction in the rarefied gas dynamics. Moscow: Nauka, 1967. p.440 (in Russian)
114. Mikhailov G.A. Some problems of Monte Carlo method theory. Novosibirsk: Nauka, 1974, p.144 (in Russian).
115. Kach M. Several probability problems of physics and mathematics. Moscow: Nauka, 1967. p.176 (in Russian).
116. Shakhov E.M. Kinetic model equations and numerical results/Rar. Gas Dyn.: Proc. 14th Symp. Rar. Gas Dyn. v.1. Tokyo, 1984, p.137-148.
117. Marchuk G.I. Yanenko N.N. A solution of multi-dimensional kinetic equations by the splitting method. Doklady AN SSSR. 1964, V.157, N.6 p.1241-1242 (in Russian).
118. McKean H.P. Speed of approach to equilibrium for Kac's caricature of a Maxwellian gas. Arc Rat. Mech and Anal. 1966, V.21, N.5, p.343-367.
119. Nanbu K. Direct simulation scheme derived from the Boltzmann equation I. Monocomponent gases. J. Phys. Soc. Jap. 1980, V.49, N.5. p.2042-2049.
120. Nanbu K. On the simulation method for BGK-equation. J. Phys. Soc. Jap. 1981, V.50, N.9. p.3154-3188.
121. Vladimirov V.G. Generalized functions in mathematical physics. Moscow: Nauka, 1979, 320p (in Russian).
122. Boyd I.D., Stark G.P.W. A comparison of the implementation and performance of the Nanbu and Bird direct simulation Monte-Carlo Methods. J. Phys. Fl. 1987, V.30, Ν. 12. p.3661-3668.
123. Bird G.A. Simulation of Multi-dimensional and chemically Reacting Flows. Rar. Gas Dyn. V.1. Paris: Comm. L'energie Atom, 1979, p.365-388.
124. Voloschuk V.M., Sedunov Yu.S. Coagulation processes in the dispersion systems. Leningrad: Gidrometeoizdat, 1975, 320p (in Russian).
125. Rogazinskii S.V. Monte Carlo method for solving of non-linear coagulation equation/Teoriya i prilojeniya statisticheskogo modelirovaniya, Novosibirsk: VC AN SSSR, 1985, p.137-147 (in Russian).
126. Lundgren T.S. Model equation for nonhomogeneous turbulence. J. Phys. Fl. 1979, V.12, N.3, p.485-497.