×

Discretization and numerical schemes for steady kinetic model equations. (English) Zbl 0906.76062

Summary: There are still many open questions concerning the relationship between (steady) kinetic equations, random particle games designed for these equations, and transitions, e.g., to fluid dynamics and turbulence phenomena. The paper presents some first steps into the derivation of models which on one hand may be used for the design of efficient numerical schemes for steady gas kinetics, and on the other hand allow to study the interplay between particle schemes and physical phenomena.

MSC:

76M25 Other numerical methods (fluid mechanics) (MSC2010)
76P05 Rarefied gas flows, Boltzmann equation in fluid mechanics
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Cercignani, C., Mathematical models in rarefied gas dynamics, Surv. math. ind., 1, 119-143, (1991) · Zbl 0850.76620
[2] Mathé, P., Approximation theory of stochastic numerical methods, (1994), Habilitationsschrift, Fachbereich Mathematik FU Berlin
[3] Babovsky, H., A convergence proof for Nanbu’s Boltzmann simulation scheme, Eur. J. mech., B/fluids, 8, 41-55, (1989) · Zbl 0669.76096
[4] Babovsky, H.; Illner, R., A convergence proof for Nanbu’s simulation method for the full Boltzmann equation, SIAM journ. num. anal., 26, 45-64, (1989) · Zbl 0668.76086
[5] Babovsky, H., Systematic errors in stationary Boltzmann simulation schemes, (), 174-182, Washington
[6] S. Stefanov and C. Cercignani, Monte Carlo simulation of a channel flow of a rarefied gas, Eur. J. Mech. B/Fluids (to appear). · Zbl 0799.76065
[7] Breuer, H.P.; Petruccione, F., On a stochastic simulation method for fluctuating hydrodynamics, Transp. theory stat. phys., 23, 265-279, (1994) · Zbl 0808.60099
[8] Rannacher, R., Computation of viscous incompressible flows, () · Zbl 0798.76069
[9] Aoki, K.; Kanba, K.; Takata, S., Numerical analysis of a rarefied gas flow past a flat plate, Phys. fluids, 9, 1144-1161, (1997)
[10] Babovsky, H., Initial and boundary value problems in kinetic theory I, II, Transp. theory. stat. phys., 13, 455-497, (1984) · Zbl 0571.76071
[11] Kaniel, S.; Shinbrot, M., The Boltzmann equation, I: uniqueness and local existence, Comm. math. phys., 59, 65-84, (1978) · Zbl 0371.76061
[12] Bobylev, A.V.; Struckmeier, J., Numerical simulation of the stationary one-dimensional Boltzmann equation by particle methods, () · Zbl 0864.76082
[13] Illner, R.; Płatkowski, T., Discrete velocity models of the Boltzmann equation: A survey on the mathematical aspects of the theory, SIAM review, 30, 213-255, (1988) · Zbl 0668.76087
[14] Rogier, F.; Schneider, J., A direct method for solving the Boltzmann equation, Transp. theory stat. phys., 23, 313-338, (1994) · Zbl 0811.76050
[15] Alt, H.W., Lineare funktionalanalysis, (1985), Springer Berlin · Zbl 0577.46001
[16] Billingsley, P., Convergence of probability measures, (1968), John Wiley & Sons New York · Zbl 0172.21201
[17] Babovsky, H., On a simulation scheme for the Boltzmann equation, Math. meth. in the appl. sci., 8, 223-233, (1986) · Zbl 0609.76084
[18] Greengard, C.; Reyna, L., Conservation of expected momentum and energy in Monte Carlo particle simulation, Physics of fluids A, 4, 849-852, (1992) · Zbl 0753.76153
[19] H. Babovsky, A constructive approach to steady nonlinear kinetic equations, Journ. Comp. Appl. Math. (to appear). · Zbl 0906.76079
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.