Numerical solution of Boltzmann's equation

Technical Report · Thu Apr 01 04:00:00 EST 1976

DOI:https://doi.org/10.2172/7184312· OSTI ID:7184312

Sod, G. A.

The numerical solution of Boltzmann's equation is considered for a gas model consisting of rigid spheres by means of Hilbert's expansion. If only the first two terms of the expansion are retained, Boltzmann's equation reduces to the Boltzmann-Hilbert integral equation. Successive terms in the Hilbert expansion are obtained by solving the same integral equation with a different source term. The Boltzmann-Hilbert integral equation is solved by a new very fast numerical method. The success of the method rests upon the simultaneous use of four judiciously chosen expansions; Hilbert's expansion for the distribution function, another expansion of the distribution function in terms of Hermite polynomials, the expansion of the kernel in terms of the eigenvalues and eigenfunctions of the Hilbert operator, and an expansion involved in solving a system of linear equations through a singular value decomposition. The numerical method is applied to the study of the shock structure in one space dimension. Numerical results are presented for Mach numbers of 1.1 and 1.6. 94 refs, 7 tables, 1 fig.

• Research Organization: California Univ., Berkeley (USA). Lawrence Berkeley Lab.
• DOE Contract Number: W-7405-ENG-48
• OSTI ID: 7184312
• Report Number(s): LBL-4662
• Country of Publication: United States
• Language: English

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71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
74 ATOMIC AND MOLECULAR PHYSICS
BOLTZMANN EQUATION
DIFFERENTIAL EQUATIONS
EQUATIONS
HILBERT TRANSFORMATION
INTEGRAL TRANSFORMATIONS
NUMERICAL SOLUTION
PERTURBATION THEORY
SHOCK WAVES
TRANSFORMATIONS
TRANSPORT THEORY