Survey of Mathematical Approaches to Solving the Boltzmann Equation

It is not a simple task to fully review different methods in rarefied gas dynamics. Unfortunately no monographs are available which deal with mathematical approaches (including analytical and numerical methods) in this area. Nevertheless, there are some reviews and some chapters of books concerning this aspect. We can cite only a few works on these theme (see [1, 2, 3, 4, 5, 6, 7, 8]). Some of these survey papers were presented to conferences and symposia.

[1]  Anna Walsh.  STUDIES IN MOLECULAR DYNAMICS , 1965 .

[2]  J. Broadwell,et al.  Study of rarefied shear flow by the discrete velocity method , 1964, Journal of Fluid Mechanics.

[3]  Graeme A. Bird,et al.  Approach to Translational Equilibrium in a Rigid Sphere Gas , 1963 .

[4]  T. Carleman,et al.  Problèmes mathématiques dans la théorie cinétique des gaz , 1957 .

[5]  J. Ferziger,et al.  Mathematical theory of transport processes in gases , 1972 .

[6]  Rapoport Ia,et al.  [The stimulation of membrane disk formation on the outer segments of rat photoreceptor cells by using para-aminobenzoic acid]. , 1990 .

[7]  C. Cercignani.  The Boltzmann equation and its applications , 1988 .

[8]  S M Yen,et al.  NUMERICAL SOLUTION OF THE NONLINEAR BOLTZMANN EQUATION FOR NONEQUILIBRIUM GAS FLOW PROBLEMS , 1984 .

[9]  Y. Pomeau,et al.  Lattice-gas automata for the Navier-Stokes equation. , 1986, Physical review letters.

[10]  K. Suchy.  Calculation of Collision Integrals in the Moment Equations , 1963 .

[11]  B. Alder,et al.  Phase Transition for a Hard Sphere System , 1957 .

[12]  Alexandre J. Chorin.  Numerical solution of Boltzmann's equation , 1972 .

[13]  Bradford Sturtevant,et al.  Numerical study of discrete‐velocity gases , 1990 .

[14]  N Bellomo,et al.  Lecture Notes on Mathematical Theory of the Boltzmann Equation , 1995 .

[15]  S. N. Smirnov,et al.  An efficient stochastic algorithm for solving the Boltzmann equation , 1989 .

[16]  J. Broadwell,et al.  Shock Structure in a Simple Discrete Velocity Gas , 1964 .

[17]  Ein neuer Ansatz für die Behandlung gasdynamischer Probleme bei starken Abweichungen vom Thermodynamischen Gleichgewicht , 1961 .

[18]  S. Yen,et al.  Shock-wave structure and intermolecular collision laws , 1974, Journal of Fluid Mechanics.

[19]  M. L. Lavin,et al.  Application of the Monte Carlo Method to Heat Transfer in a Rarefied Gas , 1962 .

[20]  V. Aristov.  Solving the Boltzmann equation for discrete velocities , 1985 .

[21]  G. Bird.  Molecular Gas Dynamics and the Direct Simulation of Gas Flows , 1994 .

[22]  Yu.N. Grigor'ev,et al.  A spectral method of solving Boltzmann's kinetic equation numerically* , 1983 .

[23]  H. Grad.  On the kinetic theory of rarefied gases , 1949 .

[24]  W. Waluś.  COMPUTATIONAL METHODS FOR THE BOLTZMANN EQUATION , 1995 .

[25]  E. Atlee Jackson,et al.  Kinetic Models and the Linearized Boltzmann Equation , 1959 .

[26]  G. Sod.  A numerical solution of Boltzmann's equation , 1977 .

[27]  Katsuhisa Koura,et al.  Null‐collision technique in the direct‐simulation Monte Carlo method , 1986 .

[28]  Zanetti,et al.  Use of the Boltzmann equation to simulate lattice gas automata. , 1988, Physical review letters.

[29]  M. Krook.  Continuum equations in the dynamics of rarefied gases , 1959, Journal of Fluid Mechanics.

[30]  P. Bhatnagar,et al.  A Model for Collision Processes in Gases. I. Small Amplitude Processes in Charged and Neutral One-Component Systems , 1954 .

[31]  Theoretical Comparison of Stochastic Particle Methods in Rarefied Gas Dynamics , 1994 .

[32]  François Rogier,et al.  Une méthode déterministe pour la résolution de l'équation de Boltzmann homogène , 1992 .

[33]  H. Neunzert,et al.  On a simulation scheme for the Boltzmann equation , 1986 .

[34]  The numerical solution of Boltzmann's equation , 1973 .

[35]  Joel H. Ferziger,et al.  Shock‐Wave Structure using Nonlinear Model Boltzmann Equations , 1972 .

[36]  Bruce L. Hicks,et al.  MONTE CARLO EVALUATION OF THE BOLTZMANN COLLISION INTEGRAL , 1966 .

[37]  V. V. Aristov.  The method of variable meshes in the velocity space in the problem of a strong condensation shock , 1977 .

[38]  C. F. Curtiss,et al.  Molecular Theory Of Gases And Liquids , 1954 .

[39]  U. Frisch,et al.  Lattice gas models for 3D hydrodynamics , 1986 .

[40]  W. Wagner.  A convergence proof for Bird's direct simulation Monte Carlo method for the Boltzmann equation , 1992 .

[41]  The Boltzmann collision integrals for a combination of Maxwellians , 1969 .

[42]  Mikhail Naumovich Kogan,et al.  Rarefied Gas Dynamics , 1969 .

[43]  V. Aristov,et al.  The conservative splitting method for solving Boltzmann's equation , 1980 .

[44]  Taku Ohwada,et al.  Heat flow and temperature and density distributions in a rarefied gas between parallel plates with different temperatures. Finite‐difference analysis of the nonlinear Boltzmann equation for hard‐sphere molecules , 1996 .

[45]  John R. Howell,et al.  New numerical strategy to evaluate the collision integral of the Boltzmann equation , 1988 .

[46]  Laguerre polynomial solution of the nonlinear Boltzmann equation for the hard-sphere model , 1989 .

[47]  Nicola Bellomo,et al.  Mathematical Topics In Nonlinear Kinetic Theory , 1988 .

[48]  A. A. Arsen'yev.  On the approximation of the solution of the Boltzmann equation by solutions of the ito stochastic differential equations , 1987 .

[49]  Taku Ohwada,et al.  Structure of normal shock waves: Direct numerical analysis of the Boltzmann equation for hard-sphere molecules , 1993 .

[50]  K. Nanbu.  Theoretical basis of the direct simulation Monte Carlo method , 1986 .

[51]  R. Gatignol.  Théorie cinétique des gaz à répartition discrète de vitesses , 1975 .

[52]  Tai Tsun Wu,et al.  Exact solutions of the Boltzmann equation , 1977 .

[53]  A. V. Lukshin,et al.  A stochastic algorithm for solving the Boltzmann equation based on Euler's method , 1989 .

[54]  J. Banavar,et al.  Computer Simulation of Liquids , 1988 .

[55]  T. G. Cowling,et al.  The mathematical theory of non-uniform gases , 1939 .

[56]  Reinhard Illner,et al.  A convergence proof for Nanbu's simulation method for the full Boltzmann equation , 1989 .

[57]  M. Ivanov,et al.  Comparative analysis of direct simulation algorithms in rarefied gas dynamics , 1990 .

[58]  V. Aristov,et al.  Conservative splitting method for solving the Boltzmann equation , 1980 .

[59]  C. Sgarra,et al.  Numerical analysis of a shock-wave solution of the Enskog equation obtained via a Monte Carlo method , 1993 .

[60]  B. L. Hicks,et al.  The internal structure of shock waves , 1972, Journal of Fluid Mechanics.