Accurate numerical methods for the Boltzmann equation
We present accurate methods for the numerical solution of the Boltzmann equation of rarefied gas. The methods are based on a time splitting technique. On the one hand, the transport is solved by a third-order accurate (in space) Positive and Flux Conservative (PFC) method. On the other hand, the collision step is treated by a Fourier approximation of the collision integral, which guarantees spectral accuracy in velocity, coupled with high-order integrators in time preserving stationary states. Several space dependent numerical tests in 2D and 3D illustrate the accuracy and robustness of the methods.
KeywordsBoltzmann Equation Knudsen Number Riemann Problem Collision Operator Kernel Mode
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- 2.Bird, G.A.: Molecular gas dynamics. Clarendon Press, Oxford (1994)Google Scholar
- 11.Goldstein, D., Sturtevant, B. and Broadwell, J.E.: Investigation of the motion of discrete velocity gases. Rar. Gas. Dynam., Progress in Astronautics and Aeronautics 118 AIAA, Washington (1989)Google Scholar
- 14.Nanbu, K.: Direct simulation scheme derived from the Boltzmann equation. I. Monocomponent Gases. J. Phys. Soc. Japan 52, 2042–2049 (1983)Google Scholar
- 17.Pareschi, L: Computational methods and fast algorithms for Boltzmann equations, Chapter 7, Lecture Notes on the discretization of the Boltzmann equation, ed. N.Bellomo, World Scientific, 46 pp. (to appear)Google Scholar
- 19.Pareschi, L. and Russo, G.: On the stability of spectral methods for the homogeneous Boltzmann equation. Proceedings of the Fifth International Workshop on Mathematical Aspects of Fluid and Plasma Dynamics (Maui, HI, 1998). Transport Theory Statist. Phys. 29,431–447(2000)Google Scholar
- 24.Shu, C.-W.: Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws. Advanced Numerical Approximation of Nonlinear Hyperbolic Equations, B. Cockburn, C. Johnson, C.-W. Shu and E. Tadmor (Editor: A. Quarteroni), Lecture Notes in Mathematics, Springer 1697, 325–432 (1998)Google Scholar
- 27.Whitham, G. B.: Linear and nonlinear waves. Wiley Interscience (1974)Google Scholar