Higher order approximation methods for the Boltzmann equation. (English) Zbl 0905.65134

The paper deals with a higher-order time difference method for the spatially nonhomogeneous Boltzmann equation \[ \begin{split} {\partial f\over \partial t}(X,\xi,t) +\xi \cdot {\partial f\over \partial X} (X,\xi,t) =Q\bigl(f(X, \xi,t), f(X,\xi,t) \bigr) [\xi],\\ 0<t\leq \Delta t, \quad f(X,\xi,0) =f_0(X,\xi). \end{split} \] The proposed method consists of two steps, as the first one is the same as the convection step in the splitting method while the second one takes into account not only the collision term but also its variation along the characteristic line.
Reviewer: E.Minchev (Sofia)


65R20 Numerical methods for integral equations
45K05 Integro-partial differential equations
Full Text: DOI


[1] Bird, G. A., Molecular Gas Dynamics, (1976)
[2] Bird, G. A., Molecular Gas Dynamics and the Direct Simulation of Gas Flows, (1994) · Zbl 0709.76511
[3] Cercignani, C.; Illner, R.; Pulvirenti, M., The Mathematical Theory of Dilute Gases, (1994) · Zbl 0813.76001
[4] Sone, Y.; Aoki, K., Molecular Gas Dynamics, (1994)
[5] Bogomolov, S. V., Convergence of the total-approximation method for the Boltzmann equation, U.S.S.R. Comput. Math. Math. Phys., 28, 79, (1988) · Zbl 0668.65118
[6] Illner, R., Approximation methods for the Boltzmann equation, Rarefied Gas Dynamics: Theory and Simulations, 551, (1994)
[7] Z.-Q. Tan, Y.-K. Chen, P. L. Varghese, J. R. Howell, New strategy to evaluate the collision integral of the Boltzmann equation, Rarefied Gas Dynamics: Theoretical and Computational Techniques, Muntz, E. P.Weaver, D. P.Campbell, D. H. AIAA, Washington, DC, 1989, 359
[8] F. G. Tcheremissine, Fast solutions of the Boltzmann equation, Rarefied Gas Dynamics, Beylich, A. E. VCH, Verlagsgesellshaft, Weinheim, 1991, 273
[9] Ohwada, T., Structure of normal shock waves: direct numerical analysis of the nonlinear Boltzmann equation for hard-sphere molecules, Phys. Fluids A, 5, 217, (1993) · Zbl 0768.76027
[10] Rogier, F.; Schneider, J., A direct method for solving the Boltzmann equation, Transp. Theor. Stat. Phys., 23, 313, (1994) · Zbl 0811.76050
[11] Ohwada, T.; Sone, Y.; Aoki, K., Numerical analysis of the shear and thermal creep flows of a rarefied gas over a plane wall on the basis of the linearized Boltzmann equation for hard-sphere molecules, Phys. Fluids A, 1, 1588, (1989) · Zbl 0695.76032
[12] Demeio, L., The inclusion of collisional effects in the splitting scheme, J. Comput. Phys, 99, 203, (1992) · Zbl 0744.76083
[13] Strang, G., On the construction and comparison of difference schemes, SIAM J. Numer. Anal, 5, 506, (1968) · Zbl 0184.38503
[14] Chu, C. K., Kinetic-theoretic description of the formation of a shock wave, Phys. Fluids, 8, 12, (1965)
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