Geodesic
Solution of the Boltzmann equation for unsteady flows with shock waves in narrow channels
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 50 (2010) no. 6, pp. 1148-1158

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Unsteady rarefied gas flows in narrow channels accompanied by shock wave formation and propagation were studied by solving the Boltzmann kinetic equation. The formation of a shock wave from an initial discontinuity of gas parameters, its propagation, damping, and reflection from the channel end face were analyzed. The Boltzmann equation was solved using finite differences. The collision integral was calculated on a fixed velocity grid by a conservative projection method. A detector of shock wave position was developed to keep track of the wave front. Parallel computations were implemented on a cluster of computers with the use of the MPI technology. Plots of shock wave damping and detailed flow fields are presented. 
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     author = {Yu. Yu. Kloss and F. G. Cheremisin and P. V. Shuvalov},
     title = {Solution of the {Boltzmann} equation for unsteady flows with shock waves in narrow channels},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {1148--1158},
     publisher = {mathdoc},
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     number = {6},
     year = {2010},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2010_50_6_a13/}
}
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Yu. Yu. Kloss; F. G. Cheremisin; P. V. Shuvalov. Solution of the Boltzmann equation for unsteady flows with shock waves in narrow channels. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 50 (2010) no. 6, pp. 1148-1158. http://geodesic.mathdoc.fr/item/ZVMMF_2010_50_6_a13/