Geodesic
Conservative numerical method for solving the averaged Boltzmann equation
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 47 (2007) no. 11, pp. 1949-1957

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A method is proposed for averaging the Boltzmann kinetic equation with respect to transverse velocities. A system of two integro-differential equations for two desired functions depending only on the longitudinal velocity is derived. The gas particles are assumed to interact as absolutely hard spheres. The integrals in the equations are double. The reduction in the number of variables in the desired functions and the low multiplicity of the integrals ensure the computational efficiency of the averaged equations. A numerical method of discrete ordinates is developed that effectively solves the gas relaxation problem based on the averaged equations. The method is conservative, and the number of particles, momentum, and energy are conserved automatically at every time step. 
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     author = {V. A. Rykov and D. A. Shil'tsov},
     title = {Conservative numerical method for solving the averaged {Boltzmann} equation},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {1949--1957},
     publisher = {mathdoc},
     volume = {47},
     number = {11},
     year = {2007},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2007_47_11_a10/}
}
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V. A. Rykov; D. A. Shil'tsov. Conservative numerical method for solving the averaged Boltzmann equation. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 47 (2007) no. 11, pp. 1949-1957. http://geodesic.mathdoc.fr/item/ZVMMF_2007_47_11_a10/