The 52 references with contexts in paper V. Titarev A., В. Титарев А. (2016) “Программный комплекс Несветай-3Д моделирования пространственных течений одноатомного разреженного газа // Software Package \Nesvetay-3D" for modeling three-dimensional flows of monatomic rarefied gas” / spz:neicon:technomag:y:2014:i:6:p:124-154

1
Shakhov E.M. [Generalization of the Krook kinetic relaxation equation].Izvestiya AN SSSR. Mekhanika Zhidkosti i Gaza, 1968, no. 5, pp. 142{145. (English translation:Fluid Dynamics, 1968, vol. 3, iss. 5, pp. 95{96. DOI:10.1007/BF01029546).
Total in-text references: 3
  1. In-text reference with the coordinate start=1626
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    ¬ ̇ÒÚÓˇ ̆ ‚ÂÏˇ ‰Îˇ ÏÓ‰ÂÎËÓ‚‡Ìˡ Ú ̃ÂÌËÈ Ó‰ÌÓ‡ÚÓÏÌÓ„Ó ‡ÁÂÊÂÌÌÓ„Ó „‡Á‡ ̄ËÓÍÓ ‡ÒÔÓÒÚ‡ÌÂÌË ÔÓÎÛ ̃ËÎÓ ÏÓ‰ÂÎ ̧ÌÓ Û‡‚ÌÂÌË ÌÂÔÓÎÌÓ„Ó ÚÂÚ ̧Â„Ó ÔË·ÎËÊÂÌˡ, ÔÓÎÛ ̃Ë‚ ̄Ë ̇Á‚‡ÌË ÏÓ‰ÂÎË ÿ‡ıÓ‚‡, ËÎËS-ÏÓ‰ÂÎË
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    . —‡‚ÌÂÌˡ Ò ‡Ò ̃ÂÚ‡ÏË ÔÓ ÚÓ ̃ÌÓÏÛ Û‡‚ÌÂÌË ̨ ¡ÓÎ ̧ˆÏ‡Ì‡, ÏÂÚÓ‰Û ÃÓÌÚÂ- ‡ÎÓ Ë Ò ̋ÍÒÔÂËÏÂÌڇΠ̧Ì ̊ÏË ‰‡ÌÌ ̊ÏË ÔÓ͇Á‡ÎË ıÓÓ ̄Û ̨ ÚÓ ̃ÌÓÒÚ ̧S-ÏÓ‰ÂÎË [3,4,5,6,7,8,9]. ”ÒÔ ̄ÌÓ ÔËÏÂÌÂÌËÂS-ÏÓ‰ÂÎË ÒÚËÏÛÎËÓ‚‡ÎÓ Â ӷӷ ̆ÂÌË ̇ Ú ̃ÂÌˡ ‰‚Ûı‡ÚÓÏÌÓ„Ó „‡Á‡ Ò Û ̃ÂÚÓÏ ‚‡ ̆‡ÚÂÎ ̧Ì ̊ı ÒÚÂÔÂÌÂÈ Ò‚Ó·Ó‰ ̊, Ë ‡Á‚ËÚË ÒÓÓÚ‚ÂÚÒÚ‚Û ̨ ̆Â„Ó ̃ËÒÎÂÌÌÓ„Ó ÏÂÚÓ‰‡ [10,11,12].

  2. In-text reference with the coordinate start=3650
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    ¬ ̇ÒÚÓˇ ̆ÂÈ ‡·ÓÚ ÔË‚Ó‰ËÚÒˇ Ó·ÁÓ ÔÓÒΉÌËı ÂÁÛÎ ̧Ú‡ÚÓ‚ ÔÓ ‡Á‡·ÓÚÍ [16,17,18] Ë ÔËÏÂÌÂÌË ̨ ÌÓ‚Ó„Ó ̃ËÒÎÂÌÌÓ„Ó ÏÂÚÓ‰‡  ̄ÂÌˡ ÍËÌÂÚË ̃ÂÒÍÓ„Ó Û‡‚ÌÂÌˡ ¡ÓÎ ̧ˆÏ‡Ì‡ Ò S-ÏÓ‰ÂÎ ̧Ì ̊Ï ËÌÚ„‡ÎÓÏ ÒÚÓÎÍÌÓ‚ÂÌËÈ
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    Ë ‡ÎËÁÛ ̨ ̆Â„Ó Â„Ó ÔÓ„‡ÏÏÌÓ„Ó ÍÓÏÔÎÂÍÒ‡ <ÕÂÒ‚ÂÚ‡È-3ƒ>. ŒÒÌÓ‚Ì ̊ÏË ·ÎÓ͇ÏË ÏÂÚÓ‰‡ ˇ‚Ρ ̨ÚÒˇ Í‚‡ÁËÏÓÌÓÚÓÌ̇ˇ ÒıÂχ ‚ÚÓÓ„Ó ÔÓˇ‰Í‡ ÚÓ ̃ÌÓÒÚË Ì‡ „Ë·Ë‰Ì ̊ı ÌÂÒÚÛÍÚÛËÓ‚‡ÌÌ ̊ı ÒÂÚ͇ı, ÍÓÌÒÂ‚‡Ú˂̇ˇ Ôӈ‰Û‡ ‡Ò ̃ÂÚ‡ χÍÓÒÍÓÔË ̃ÂÒÍËı Ô‡‡ÏÂÚÓ‚ „‡Á‡ Ë ÔÓÎÌÓÒÚ ̧ ̨ Ì¡‚Ì ̊È ÏÂÚÓ‰ ‰ËÒÍÂÚËÁ‡ˆËË ÔÓ ‚ÂÏÂÌË ·ÂÁ ËÚÂ‡ˆËÈ Ì‡ ‚ÂıÌÂÏ ÒÎÓÂ.

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    «‰ÂÒ ̧m| χÒÒ‡ ÏÓÎÂÍÛÎ ̊;Rg| „‡ÁÓ‚‡ˇ ÔÓÒÚÓˇÌ̇ˇ; Ô‰ÔÓ·„‡ÂÚÒˇ ÒÛÏÏËÓ‚‡ÌË ÔÓ ÔÓ‚ÚÓˇ ̨ ̆ËÏÒˇ „ ̃ÂÒÍËÏ Ë̉ÂÍÒ‡Ï ‚ Ô‰Â·ı ÓÚ1‰Ó3. ËÌÂÚË ̃ÂÒÍÓ Û‡‚ÌÂÌË ¡ÓÎ ̧ˆÏ‡Ì‡ ‰Îˇ ÙÛÌ͈ËË ‡ÒÔ‰ÂÎÂÌˡfÏÓÎÂÍÛÎ ÔÓ ÒÍÓÓÒÚˇÏ ÒS-ÏÓ‰ÂÎ ̧Ì ̊Ï ËÌÚ„‡ÎÓÏ ÒÚÓÎÍÌÓ‚ÂÌËÈ
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    ËÏÂÂÚ ‚ˉ ∂f ∂t +ξα ∂f ∂xα =J, J= p μ (f+−f), f+=fM [ 1 + 4 5 (1−Pr)Sαcα ( c2− 5 2 )] , fM= n (2πRgT)3/2 exp (−c2), Si= 1 n ∫ cic2f dξ,c= v √ 2RgT , c2=cβcβ, „‰Âμ=μ(T)| ‚ˇÁÍÓÒÚ ̧; ̃ËÒÎÓ œ‡Ì‰ÚΡ ‰Îˇ Ó‰ÌÓ‡ÚÓÏÌÓ„Ó „‡Á‡Pr = 2/3;Pr = 1ÒÓÓÚ‚ÂÚÒÚ‚ÛÂÚ ÏÓ‰ÂÎË ¡√ [19].

2
Shakhov E.M.Metod issledovaniya dvizheniy razrezhennogo gaza[Method of investigation of rarefied gas flows]. Moscow, Nauka Publ., 1974. 207 p. (in Russian).
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    ¬ ̇ÒÚÓˇ ̆ ‚ÂÏˇ ‰Îˇ ÏÓ‰ÂÎËÓ‚‡Ìˡ Ú ̃ÂÌËÈ Ó‰ÌÓ‡ÚÓÏÌÓ„Ó ‡ÁÂÊÂÌÌÓ„Ó „‡Á‡ ̄ËÓÍÓ ‡ÒÔÓÒÚ‡ÌÂÌË ÔÓÎÛ ̃ËÎÓ ÏÓ‰ÂÎ ̧ÌÓ Û‡‚ÌÂÌË ÌÂÔÓÎÌÓ„Ó ÚÂÚ ̧Â„Ó ÔË·ÎËÊÂÌˡ, ÔÓÎÛ ̃Ë‚ ̄Ë ̇Á‚‡ÌË ÏÓ‰ÂÎË ÿ‡ıÓ‚‡, ËÎËS-ÏÓ‰ÂÎË
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    . —‡‚ÌÂÌˡ Ò ‡Ò ̃ÂÚ‡ÏË ÔÓ ÚÓ ̃ÌÓÏÛ Û‡‚ÌÂÌË ̨ ¡ÓÎ ̧ˆÏ‡Ì‡, ÏÂÚÓ‰Û ÃÓÌÚÂ- ‡ÎÓ Ë Ò ̋ÍÒÔÂËÏÂÌڇΠ̧Ì ̊ÏË ‰‡ÌÌ ̊ÏË ÔÓ͇Á‡ÎË ıÓÓ ̄Û ̨ ÚÓ ̃ÌÓÒÚ ̧S-ÏÓ‰ÂÎË [3,4,5,6,7,8,9]. ”ÒÔ ̄ÌÓ ÔËÏÂÌÂÌËÂS-ÏÓ‰ÂÎË ÒÚËÏÛÎËÓ‚‡ÎÓ Â ӷӷ ̆ÂÌË ̇ Ú ̃ÂÌˡ ‰‚Ûı‡ÚÓÏÌÓ„Ó „‡Á‡ Ò Û ̃ÂÚÓÏ ‚‡ ̆‡ÚÂÎ ̧Ì ̊ı ÒÚÂÔÂÌÂÈ Ò‚Ó·Ó‰ ̊, Ë ‡Á‚ËÚË ÒÓÓÚ‚ÂÚÒÚ‚Û ̨ ̆Â„Ó ̃ËÒÎÂÌÌÓ„Ó ÏÂÚÓ‰‡ [10,11,12].

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    ¬ ̇ÒÚÓˇ ̆ÂÈ ‡·ÓÚ ÔË‚Ó‰ËÚÒˇ Ó·ÁÓ ÔÓÒΉÌËı ÂÁÛÎ ̧Ú‡ÚÓ‚ ÔÓ ‡Á‡·ÓÚÍ [16,17,18] Ë ÔËÏÂÌÂÌË ̨ ÌÓ‚Ó„Ó ̃ËÒÎÂÌÌÓ„Ó ÏÂÚÓ‰‡  ̄ÂÌˡ ÍËÌÂÚË ̃ÂÒÍÓ„Ó Û‡‚ÌÂÌˡ ¡ÓÎ ̧ˆÏ‡Ì‡ Ò S-ÏÓ‰ÂÎ ̧Ì ̊Ï ËÌÚ„‡ÎÓÏ ÒÚÓÎÍÌÓ‚ÂÌËÈ
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    Ë ‡ÎËÁÛ ̨ ̆Â„Ó Â„Ó ÔÓ„‡ÏÏÌÓ„Ó ÍÓÏÔÎÂÍÒ‡ <ÕÂÒ‚ÂÚ‡È-3ƒ>. ŒÒÌÓ‚Ì ̊ÏË ·ÎÓ͇ÏË ÏÂÚÓ‰‡ ˇ‚Ρ ̨ÚÒˇ Í‚‡ÁËÏÓÌÓÚÓÌ̇ˇ ÒıÂχ ‚ÚÓÓ„Ó ÔÓˇ‰Í‡ ÚÓ ̃ÌÓÒÚË Ì‡ „Ë·Ë‰Ì ̊ı ÌÂÒÚÛÍÚÛËÓ‚‡ÌÌ ̊ı ÒÂÚ͇ı, ÍÓÌÒÂ‚‡Ú˂̇ˇ Ôӈ‰Û‡ ‡Ò ̃ÂÚ‡ χÍÓÒÍÓÔË ̃ÂÒÍËı Ô‡‡ÏÂÚÓ‚ „‡Á‡ Ë ÔÓÎÌÓÒÚ ̧ ̨ Ì¡‚Ì ̊È ÏÂÚÓ‰ ‰ËÒÍÂÚËÁ‡ˆËË ÔÓ ‚ÂÏÂÌË ·ÂÁ ËÚÂ‡ˆËÈ Ì‡ ‚ÂıÌÂÏ ÒÎÓÂ.

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    «‰ÂÒ ̧m| χÒÒ‡ ÏÓÎÂÍÛÎ ̊;Rg| „‡ÁÓ‚‡ˇ ÔÓÒÚÓˇÌ̇ˇ; Ô‰ÔÓ·„‡ÂÚÒˇ ÒÛÏÏËÓ‚‡ÌË ÔÓ ÔÓ‚ÚÓˇ ̨ ̆ËÏÒˇ „ ̃ÂÒÍËÏ Ë̉ÂÍÒ‡Ï ‚ Ô‰Â·ı ÓÚ1‰Ó3. ËÌÂÚË ̃ÂÒÍÓ Û‡‚ÌÂÌË ¡ÓÎ ̧ˆÏ‡Ì‡ ‰Îˇ ÙÛÌ͈ËË ‡ÒÔ‰ÂÎÂÌˡfÏÓÎÂÍÛÎ ÔÓ ÒÍÓÓÒÚˇÏ ÒS-ÏÓ‰ÂÎ ̧Ì ̊Ï ËÌÚ„‡ÎÓÏ ÒÚÓÎÍÌÓ‚ÂÌËÈ
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    ËÏÂÂÚ ‚ˉ ∂f ∂t +ξα ∂f ∂xα =J, J= p μ (f+−f), f+=fM [ 1 + 4 5 (1−Pr)Sαcα ( c2− 5 2 )] , fM= n (2πRgT)3/2 exp (−c2), Si= 1 n ∫ cic2f dξ,c= v √ 2RgT , c2=cβcβ, „‰Âμ=μ(T)| ‚ˇÁÍÓÒÚ ̧; ̃ËÒÎÓ œ‡Ì‰ÚΡ ‰Îˇ Ó‰ÌÓ‡ÚÓÏÌÓ„Ó „‡Á‡Pr = 2/3;Pr = 1ÒÓÓÚ‚ÂÚÒÚ‚ÛÂÚ ÏÓ‰ÂÎË ¡√ [19].

3
Zhuk V.I., Rykov V.A., Shakhov E.M. [Kinetic models and the shock structure problem]. Izvestiya AN SSSR. Mekhanika Zhidkosti i Gaza, 1973, no. 4, pp. 135{141. (English translation: Fluid Dynamics, 1973, vol. 8, iss. 4, pp. 620{625. DOI:10.1007/BF01013101).
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    ¬ ̇ÒÚÓˇ ̆ ‚ÂÏˇ ‰Îˇ ÏÓ‰ÂÎËÓ‚‡Ìˡ Ú ̃ÂÌËÈ Ó‰ÌÓ‡ÚÓÏÌÓ„Ó ‡ÁÂÊÂÌÌÓ„Ó „‡Á‡ ̄ËÓÍÓ ‡ÒÔÓÒÚ‡ÌÂÌË ÔÓÎÛ ̃ËÎÓ ÏÓ‰ÂÎ ̧ÌÓ Û‡‚ÌÂÌË ÌÂÔÓÎÌÓ„Ó ÚÂÚ ̧Â„Ó ÔË·ÎËÊÂÌˡ, ÔÓÎÛ ̃Ë‚ ̄Ë ̇Á‚‡ÌË ÏÓ‰ÂÎË ÿ‡ıÓ‚‡, ËÎËS-ÏÓ‰ÂÎË [1,2]. —‡‚ÌÂÌˡ Ò ‡Ò ̃ÂÚ‡ÏË ÔÓ ÚÓ ̃ÌÓÏÛ Û‡‚ÌÂÌË ̨ ¡ÓÎ ̧ˆÏ‡Ì‡, ÏÂÚÓ‰Û ÃÓÌÚÂ- ‡ÎÓ Ë Ò ̋ÍÒÔÂËÏÂÌڇΠ̧Ì ̊ÏË ‰‡ÌÌ ̊ÏË ÔÓ͇Á‡ÎË ıÓÓ ̄Û ̨ ÚÓ ̃ÌÓÒÚ ̧S-ÏÓ‰ÂÎË
    Exact
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    . ”ÒÔ ̄ÌÓ ÔËÏÂÌÂÌËÂS-ÏÓ‰ÂÎË ÒÚËÏÛÎËÓ‚‡ÎÓ Â ӷӷ ̆ÂÌË ̇ Ú ̃ÂÌˡ ‰‚Ûı‡ÚÓÏÌÓ„Ó „‡Á‡ Ò Û ̃ÂÚÓÏ ‚‡ ̆‡ÚÂÎ ̧Ì ̊ı ÒÚÂÔÂÌÂÈ Ò‚Ó·Ó‰ ̊, Ë ‡Á‚ËÚË ÒÓÓÚ‚ÂÚÒÚ‚Û ̨ ̆Â„Ó ̃ËÒÎÂÌÌÓ„Ó ÏÂÚÓ‰‡ [10,11,12]. “ÂÏ Ì ÏÂÌÂÂ, ÌÂÒÏÓÚˇ ̇ ÓÚÌÓÒËÚÂÎ ̧ÌÛ ̨ ÔÓÒÚÓÚÛ,S-ÏÓ‰ÂÎ ̧ ˇ‚ΡÂÚÒˇ ÒÎÓÊÌ ̊Ï ËÌÚ„Ó-‰ËÙÙÂÂ̈ˇΠ̧Ì ̊Ï Û‡‚ÌÂÌËÂÏ ‚ ̊ÒÓÍÓÈ ‡ÁÏÂÌÓÒÚË, ̃ËÒÎÂÌÌÓ  ̄ÂÌË ÍÓÚÓÓ„Ó Ú·ÛÂÚ ‡Á‚ËÚˡ ‚ ̊ÒÓÍÓÚÓ ̃Ì ̊ı Ô‡‡

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Rykov V.A., Cheremisin F.G., Shakhov E.M. [Numerical investigations in rarefied gas dynamics].Zhurnal vychislitel'noy matematiki i matematicheskoy fiziki, 1980, vol. 20, no. 5, pp. 1266{1283. (English translation:USSR Computational Mathematics and Mathematical Physics, 1980, vol. 20, iss. 5, pp. 168{185. DOI:10.1016/0041-5553(80)90094-4).
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    ¬ ̇ÒÚÓˇ ̆ ‚ÂÏˇ ‰Îˇ ÏÓ‰ÂÎËÓ‚‡Ìˡ Ú ̃ÂÌËÈ Ó‰ÌÓ‡ÚÓÏÌÓ„Ó ‡ÁÂÊÂÌÌÓ„Ó „‡Á‡ ̄ËÓÍÓ ‡ÒÔÓÒÚ‡ÌÂÌË ÔÓÎÛ ̃ËÎÓ ÏÓ‰ÂÎ ̧ÌÓ Û‡‚ÌÂÌË ÌÂÔÓÎÌÓ„Ó ÚÂÚ ̧Â„Ó ÔË·ÎËÊÂÌˡ, ÔÓÎÛ ̃Ë‚ ̄Ë ̇Á‚‡ÌË ÏÓ‰ÂÎË ÿ‡ıÓ‚‡, ËÎËS-ÏÓ‰ÂÎË [1,2]. —‡‚ÌÂÌˡ Ò ‡Ò ̃ÂÚ‡ÏË ÔÓ ÚÓ ̃ÌÓÏÛ Û‡‚ÌÂÌË ̨ ¡ÓÎ ̧ˆÏ‡Ì‡, ÏÂÚÓ‰Û ÃÓÌÚÂ- ‡ÎÓ Ë Ò ̋ÍÒÔÂËÏÂÌڇΠ̧Ì ̊ÏË ‰‡ÌÌ ̊ÏË ÔÓ͇Á‡ÎË ıÓÓ ̄Û ̨ ÚÓ ̃ÌÓÒÚ ̧S-ÏÓ‰ÂÎË
    Exact
    [3,4,5,6,7,8,9]
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    . ”ÒÔ ̄ÌÓ ÔËÏÂÌÂÌËÂS-ÏÓ‰ÂÎË ÒÚËÏÛÎËÓ‚‡ÎÓ Â ӷӷ ̆ÂÌË ̇ Ú ̃ÂÌˡ ‰‚Ûı‡ÚÓÏÌÓ„Ó „‡Á‡ Ò Û ̃ÂÚÓÏ ‚‡ ̆‡ÚÂÎ ̧Ì ̊ı ÒÚÂÔÂÌÂÈ Ò‚Ó·Ó‰ ̊, Ë ‡Á‚ËÚË ÒÓÓÚ‚ÂÚÒÚ‚Û ̨ ̆Â„Ó ̃ËÒÎÂÌÌÓ„Ó ÏÂÚÓ‰‡ [10,11,12]. “ÂÏ Ì ÏÂÌÂÂ, ÌÂÒÏÓÚˇ ̇ ÓÚÌÓÒËÚÂÎ ̧ÌÛ ̨ ÔÓÒÚÓÚÛ,S-ÏÓ‰ÂÎ ̧ ˇ‚ΡÂÚÒˇ ÒÎÓÊÌ ̊Ï ËÌÚ„Ó-‰ËÙÙÂÂ̈ˇΠ̧Ì ̊Ï Û‡‚ÌÂÌËÂÏ ‚ ̊ÒÓÍÓÈ ‡ÁÏÂÌÓÒÚË, ̃ËÒÎÂÌÌÓ  ̄ÂÌË ÍÓÚÓÓ„Ó Ú·ÛÂÚ ‡Á‚ËÚˡ ‚ ̊ÒÓÍÓÚÓ ̃Ì ̊ı Ô‡‡

5
Shakhov E.M. [Transverse flow of a rarefield gas around a plate].Izvestiya AN SSSR. Mekhanika Zhidkosti i Gaza, 1972, no. 6, pp. 106{113. (English translation:Fluid Dynamics, 1972, vol. 7, iss. 6, pp. 961{966. DOI:10.1007/BF01176114).
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    ¬ ̇ÒÚÓˇ ̆ ‚ÂÏˇ ‰Îˇ ÏÓ‰ÂÎËÓ‚‡Ìˡ Ú ̃ÂÌËÈ Ó‰ÌÓ‡ÚÓÏÌÓ„Ó ‡ÁÂÊÂÌÌÓ„Ó „‡Á‡ ̄ËÓÍÓ ‡ÒÔÓÒÚ‡ÌÂÌË ÔÓÎÛ ̃ËÎÓ ÏÓ‰ÂÎ ̧ÌÓ Û‡‚ÌÂÌË ÌÂÔÓÎÌÓ„Ó ÚÂÚ ̧Â„Ó ÔË·ÎËÊÂÌˡ, ÔÓÎÛ ̃Ë‚ ̄Ë ̇Á‚‡ÌË ÏÓ‰ÂÎË ÿ‡ıÓ‚‡, ËÎËS-ÏÓ‰ÂÎË [1,2]. —‡‚ÌÂÌˡ Ò ‡Ò ̃ÂÚ‡ÏË ÔÓ ÚÓ ̃ÌÓÏÛ Û‡‚ÌÂÌË ̨ ¡ÓÎ ̧ˆÏ‡Ì‡, ÏÂÚÓ‰Û ÃÓÌÚÂ- ‡ÎÓ Ë Ò ̋ÍÒÔÂËÏÂÌڇΠ̧Ì ̊ÏË ‰‡ÌÌ ̊ÏË ÔÓ͇Á‡ÎË ıÓÓ ̄Û ̨ ÚÓ ̃ÌÓÒÚ ̧S-ÏÓ‰ÂÎË
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    [3,4,5,6,7,8,9]
    Suffix
    . ”ÒÔ ̄ÌÓ ÔËÏÂÌÂÌËÂS-ÏÓ‰ÂÎË ÒÚËÏÛÎËÓ‚‡ÎÓ Â ӷӷ ̆ÂÌË ̇ Ú ̃ÂÌˡ ‰‚Ûı‡ÚÓÏÌÓ„Ó „‡Á‡ Ò Û ̃ÂÚÓÏ ‚‡ ̆‡ÚÂÎ ̧Ì ̊ı ÒÚÂÔÂÌÂÈ Ò‚Ó·Ó‰ ̊, Ë ‡Á‚ËÚË ÒÓÓÚ‚ÂÚÒÚ‚Û ̨ ̆Â„Ó ̃ËÒÎÂÌÌÓ„Ó ÏÂÚÓ‰‡ [10,11,12]. “ÂÏ Ì ÏÂÌÂÂ, ÌÂÒÏÓÚˇ ̇ ÓÚÌÓÒËÚÂÎ ̧ÌÛ ̨ ÔÓÒÚÓÚÛ,S-ÏÓ‰ÂÎ ̧ ˇ‚ΡÂÚÒˇ ÒÎÓÊÌ ̊Ï ËÌÚ„Ó-‰ËÙÙÂÂ̈ˇΠ̧Ì ̊Ï Û‡‚ÌÂÌËÂÏ ‚ ̊ÒÓÍÓÈ ‡ÁÏÂÌÓÒÚË, ̃ËÒÎÂÌÌÓ  ̄ÂÌË ÍÓÚÓÓ„Ó Ú·ÛÂÚ ‡Á‚ËÚˡ ‚ ̊ÒÓÍÓÚÓ ̃Ì ̊ı Ô‡‡

6
Shakhov E.M. [Numerical methods of solution of approximating kinetic equations].Chislennye metody v dinamike razrezhennykh gazov: sb. Vyp. 2[Numerical methods in rarefied gas dynamics: coll. papers. Iss. 2]. Moscow, Computing Center of the USSR Academy of Sciences Publ., 1975, pp. 35{76. (in Russian).
Total in-text references: 1
  1. In-text reference with the coordinate start=1781
    Prefix
    ¬ ̇ÒÚÓˇ ̆ ‚ÂÏˇ ‰Îˇ ÏÓ‰ÂÎËÓ‚‡Ìˡ Ú ̃ÂÌËÈ Ó‰ÌÓ‡ÚÓÏÌÓ„Ó ‡ÁÂÊÂÌÌÓ„Ó „‡Á‡ ̄ËÓÍÓ ‡ÒÔÓÒÚ‡ÌÂÌË ÔÓÎÛ ̃ËÎÓ ÏÓ‰ÂÎ ̧ÌÓ Û‡‚ÌÂÌË ÌÂÔÓÎÌÓ„Ó ÚÂÚ ̧Â„Ó ÔË·ÎËÊÂÌˡ, ÔÓÎÛ ̃Ë‚ ̄Ë ̇Á‚‡ÌË ÏÓ‰ÂÎË ÿ‡ıÓ‚‡, ËÎËS-ÏÓ‰ÂÎË [1,2]. —‡‚ÌÂÌˡ Ò ‡Ò ̃ÂÚ‡ÏË ÔÓ ÚÓ ̃ÌÓÏÛ Û‡‚ÌÂÌË ̨ ¡ÓÎ ̧ˆÏ‡Ì‡, ÏÂÚÓ‰Û ÃÓÌÚÂ- ‡ÎÓ Ë Ò ̋ÍÒÔÂËÏÂÌڇΠ̧Ì ̊ÏË ‰‡ÌÌ ̊ÏË ÔÓ͇Á‡ÎË ıÓÓ ̄Û ̨ ÚÓ ̃ÌÓÒÚ ̧S-ÏÓ‰ÂÎË
    Exact
    [3,4,5,6,7,8,9]
    Suffix
    . ”ÒÔ ̄ÌÓ ÔËÏÂÌÂÌËÂS-ÏÓ‰ÂÎË ÒÚËÏÛÎËÓ‚‡ÎÓ Â ӷӷ ̆ÂÌË ̇ Ú ̃ÂÌˡ ‰‚Ûı‡ÚÓÏÌÓ„Ó „‡Á‡ Ò Û ̃ÂÚÓÏ ‚‡ ̆‡ÚÂÎ ̧Ì ̊ı ÒÚÂÔÂÌÂÈ Ò‚Ó·Ó‰ ̊, Ë ‡Á‚ËÚË ÒÓÓÚ‚ÂÚÒÚ‚Û ̨ ̆Â„Ó ̃ËÒÎÂÌÌÓ„Ó ÏÂÚÓ‰‡ [10,11,12]. “ÂÏ Ì ÏÂÌÂÂ, ÌÂÒÏÓÚˇ ̇ ÓÚÌÓÒËÚÂÎ ̧ÌÛ ̨ ÔÓÒÚÓÚÛ,S-ÏÓ‰ÂÎ ̧ ˇ‚ΡÂÚÒˇ ÒÎÓÊÌ ̊Ï ËÌÚ„Ó-‰ËÙÙÂÂ̈ˇΠ̧Ì ̊Ï Û‡‚ÌÂÌËÂÏ ‚ ̊ÒÓÍÓÈ ‡ÁÏÂÌÓÒÚË, ̃ËÒÎÂÌÌÓ  ̄ÂÌË ÍÓÚÓÓ„Ó Ú·ÛÂÚ ‡Á‚ËÚˡ ‚ ̊ÒÓÍÓÚÓ ̃Ì ̊ı Ô‡‡

7
Sharipov F., Seleznev V. Data on internal rarefied gas flows.J. Phys. Chem. Ref. Data, 1998, vol. 27, no. 3, pp. 657{706.
Total in-text references: 2
  1. In-text reference with the coordinate start=1781
    Prefix
    ¬ ̇ÒÚÓˇ ̆ ‚ÂÏˇ ‰Îˇ ÏÓ‰ÂÎËÓ‚‡Ìˡ Ú ̃ÂÌËÈ Ó‰ÌÓ‡ÚÓÏÌÓ„Ó ‡ÁÂÊÂÌÌÓ„Ó „‡Á‡ ̄ËÓÍÓ ‡ÒÔÓÒÚ‡ÌÂÌË ÔÓÎÛ ̃ËÎÓ ÏÓ‰ÂÎ ̧ÌÓ Û‡‚ÌÂÌË ÌÂÔÓÎÌÓ„Ó ÚÂÚ ̧Â„Ó ÔË·ÎËÊÂÌˡ, ÔÓÎÛ ̃Ë‚ ̄Ë ̇Á‚‡ÌË ÏÓ‰ÂÎË ÿ‡ıÓ‚‡, ËÎËS-ÏÓ‰ÂÎË [1,2]. —‡‚ÌÂÌˡ Ò ‡Ò ̃ÂÚ‡ÏË ÔÓ ÚÓ ̃ÌÓÏÛ Û‡‚ÌÂÌË ̨ ¡ÓÎ ̧ˆÏ‡Ì‡, ÏÂÚÓ‰Û ÃÓÌÚÂ- ‡ÎÓ Ë Ò ̋ÍÒÔÂËÏÂÌڇΠ̧Ì ̊ÏË ‰‡ÌÌ ̊ÏË ÔÓ͇Á‡ÎË ıÓÓ ̄Û ̨ ÚÓ ̃ÌÓÒÚ ̧S-ÏÓ‰ÂÎË
    Exact
    [3,4,5,6,7,8,9]
    Suffix
    . ”ÒÔ ̄ÌÓ ÔËÏÂÌÂÌËÂS-ÏÓ‰ÂÎË ÒÚËÏÛÎËÓ‚‡ÎÓ Â ӷӷ ̆ÂÌË ̇ Ú ̃ÂÌˡ ‰‚Ûı‡ÚÓÏÌÓ„Ó „‡Á‡ Ò Û ̃ÂÚÓÏ ‚‡ ̆‡ÚÂÎ ̧Ì ̊ı ÒÚÂÔÂÌÂÈ Ò‚Ó·Ó‰ ̊, Ë ‡Á‚ËÚË ÒÓÓÚ‚ÂÚÒÚ‚Û ̨ ̆Â„Ó ̃ËÒÎÂÌÌÓ„Ó ÏÂÚÓ‰‡ [10,11,12]. “ÂÏ Ì ÏÂÌÂÂ, ÌÂÒÏÓÚˇ ̇ ÓÚÌÓÒËÚÂÎ ̧ÌÛ ̨ ÔÓÒÚÓÚÛ,S-ÏÓ‰ÂÎ ̧ ˇ‚ΡÂÚÒˇ ÒÎÓÊÌ ̊Ï ËÌÚ„Ó-‰ËÙÙÂÂ̈ˇΠ̧Ì ̊Ï Û‡‚ÌÂÌËÂÏ ‚ ̊ÒÓÍÓÈ ‡ÁÏÂÌÓÒÚË, ̃ËÒÎÂÌÌÓ  ̄ÂÌË ÍÓÚÓÓ„Ó Ú·ÛÂÚ ‡Á‚ËÚˡ ‚ ̊ÒÓÍÓÚÓ ̃Ì ̊ı Ô‡‡

  2. In-text reference with the coordinate start=32901
    Prefix
    ¬ ̇ÒÚÓˇ ̆ÂÏ ‡Á‰ÂΠ‡ÒÒχÚË‚‡ ̨ÚÒˇ ÔËÏÂ ̊ ‡Ò ̃ÂÚÓ‚ Ú ̃ÂÌËÈ ‚ ÏËÍÓ͇̇·ı Ë  ̄ÂÌˡ Á‡‰‡ ̃ ‚Ì ̄ÌÂ„Ó Ó·ÚÂ͇Ìˡ ÚÂÎ ÒÎÓÊÌÓÈ ÙÓÏ ̊. “ ̃ÂÌˡ „‡Á‡ ‚ ÏËÍÓ͇̇·ı.»ÒÒΉӂ‡ÌË Ú ̃ÂÌËÈ ‡ÁÂÊÂÌÌ ̊ı „‡ÁÓ‚ ‚ ͇̇·ı Ë ÚÛ·‡ı ‚‡ÊÌÓ ‰Îˇ ÏÌÓ„Ëı ÔËÎÓÊÂÌËÈ. —Ó‚ÂÏÂÌÌÓ ÒÓÒÚÓˇÌË ‚ÓÔÓÒ‡ ÓÚ‡ÊÂÌÓ ‚ Ó·ÁÓ‡ı
    Exact
    [7,8]
    Suffix
    . ̇ÒÚÓˇ ̆ÂÏÛ ‚ÂÏÂÌË ‰ÓÒÚ‡ÚÓ ̃ÌÓ ÔÓÎÌÓ ‡Á‡·ÓÚ‡Ì ̊ ÏÂÚÓ‰ ̊ ‡Ò ̃ÂÚ‡ Ú ̃ÂÌËÈ ‚ ÚÛ·‡ı Ë Í‡Ì‡Î‡ı ·ÂÒÍÓÌ ̃ÌÓÈ ‰ÎËÌ ̊ Ë ÔÓËÁ‚ÓÎ ̧ÌÓ„Ó ÔÓÔÂ ̃ÌÓ„Ó Ò ̃ÂÌˡ ̇ ÓÒÌÓ‚Â ÎË̇ËÁÓ‚‡ÌÌ ̊ı ÍËÌÂÚË ̃ÂÒÍËı ÏÓ‰ÂÎÂÈ [39].

8
Sharipov F.M., Seleznev V.D.Dvizhenie razrezhennykh gazov v kanalakh i mikrokanalakh [Flows of rarefied gases in channels and microchannels]. Ekaterinburg, Russian Academy of Science, Ural Branch, Institute of Thermal Physics Publ., 2008. 232 p. (in Russian).
Total in-text references: 2
  1. In-text reference with the coordinate start=1781
    Prefix
    ¬ ̇ÒÚÓˇ ̆ ‚ÂÏˇ ‰Îˇ ÏÓ‰ÂÎËÓ‚‡Ìˡ Ú ̃ÂÌËÈ Ó‰ÌÓ‡ÚÓÏÌÓ„Ó ‡ÁÂÊÂÌÌÓ„Ó „‡Á‡ ̄ËÓÍÓ ‡ÒÔÓÒÚ‡ÌÂÌË ÔÓÎÛ ̃ËÎÓ ÏÓ‰ÂÎ ̧ÌÓ Û‡‚ÌÂÌË ÌÂÔÓÎÌÓ„Ó ÚÂÚ ̧Â„Ó ÔË·ÎËÊÂÌˡ, ÔÓÎÛ ̃Ë‚ ̄Ë ̇Á‚‡ÌË ÏÓ‰ÂÎË ÿ‡ıÓ‚‡, ËÎËS-ÏÓ‰ÂÎË [1,2]. —‡‚ÌÂÌˡ Ò ‡Ò ̃ÂÚ‡ÏË ÔÓ ÚÓ ̃ÌÓÏÛ Û‡‚ÌÂÌË ̨ ¡ÓÎ ̧ˆÏ‡Ì‡, ÏÂÚÓ‰Û ÃÓÌÚÂ- ‡ÎÓ Ë Ò ̋ÍÒÔÂËÏÂÌڇΠ̧Ì ̊ÏË ‰‡ÌÌ ̊ÏË ÔÓ͇Á‡ÎË ıÓÓ ̄Û ̨ ÚÓ ̃ÌÓÒÚ ̧S-ÏÓ‰ÂÎË
    Exact
    [3,4,5,6,7,8,9]
    Suffix
    . ”ÒÔ ̄ÌÓ ÔËÏÂÌÂÌËÂS-ÏÓ‰ÂÎË ÒÚËÏÛÎËÓ‚‡ÎÓ Â ӷӷ ̆ÂÌË ̇ Ú ̃ÂÌˡ ‰‚Ûı‡ÚÓÏÌÓ„Ó „‡Á‡ Ò Û ̃ÂÚÓÏ ‚‡ ̆‡ÚÂÎ ̧Ì ̊ı ÒÚÂÔÂÌÂÈ Ò‚Ó·Ó‰ ̊, Ë ‡Á‚ËÚË ÒÓÓÚ‚ÂÚÒÚ‚Û ̨ ̆Â„Ó ̃ËÒÎÂÌÌÓ„Ó ÏÂÚÓ‰‡ [10,11,12]. “ÂÏ Ì ÏÂÌÂÂ, ÌÂÒÏÓÚˇ ̇ ÓÚÌÓÒËÚÂÎ ̧ÌÛ ̨ ÔÓÒÚÓÚÛ,S-ÏÓ‰ÂÎ ̧ ˇ‚ΡÂÚÒˇ ÒÎÓÊÌ ̊Ï ËÌÚ„Ó-‰ËÙÙÂÂ̈ˇΠ̧Ì ̊Ï Û‡‚ÌÂÌËÂÏ ‚ ̊ÒÓÍÓÈ ‡ÁÏÂÌÓÒÚË, ̃ËÒÎÂÌÌÓ  ̄ÂÌË ÍÓÚÓÓ„Ó Ú·ÛÂÚ ‡Á‚ËÚˡ ‚ ̊ÒÓÍÓÚÓ ̃Ì ̊ı Ô‡‡

  2. In-text reference with the coordinate start=32901
    Prefix
    ¬ ̇ÒÚÓˇ ̆ÂÏ ‡Á‰ÂΠ‡ÒÒχÚË‚‡ ̨ÚÒˇ ÔËÏÂ ̊ ‡Ò ̃ÂÚÓ‚ Ú ̃ÂÌËÈ ‚ ÏËÍÓ͇̇·ı Ë  ̄ÂÌˡ Á‡‰‡ ̃ ‚Ì ̄ÌÂ„Ó Ó·ÚÂ͇Ìˡ ÚÂÎ ÒÎÓÊÌÓÈ ÙÓÏ ̊. “ ̃ÂÌˡ „‡Á‡ ‚ ÏËÍÓ͇̇·ı.»ÒÒΉӂ‡ÌË Ú ̃ÂÌËÈ ‡ÁÂÊÂÌÌ ̊ı „‡ÁÓ‚ ‚ ͇̇·ı Ë ÚÛ·‡ı ‚‡ÊÌÓ ‰Îˇ ÏÌÓ„Ëı ÔËÎÓÊÂÌËÈ. —Ó‚ÂÏÂÌÌÓ ÒÓÒÚÓˇÌË ‚ÓÔÓÒ‡ ÓÚ‡ÊÂÌÓ ‚ Ó·ÁÓ‡ı
    Exact
    [7,8]
    Suffix
    . ̇ÒÚÓˇ ̆ÂÏÛ ‚ÂÏÂÌË ‰ÓÒÚ‡ÚÓ ̃ÌÓ ÔÓÎÌÓ ‡Á‡·ÓÚ‡Ì ̊ ÏÂÚÓ‰ ̊ ‡Ò ̃ÂÚ‡ Ú ̃ÂÌËÈ ‚ ÚÛ·‡ı Ë Í‡Ì‡Î‡ı ·ÂÒÍÓÌ ̃ÌÓÈ ‰ÎËÌ ̊ Ë ÔÓËÁ‚ÓÎ ̧ÌÓ„Ó ÔÓÔÂ ̃ÌÓ„Ó Ò ̃ÂÌˡ ̇ ÓÒÌÓ‚Â ÎË̇ËÁÓ‚‡ÌÌ ̊ı ÍËÌÂÚË ̃ÂÒÍËı ÏÓ‰ÂÎÂÈ [39].

9
Graur I., Polikarpov A.Ph. Comparison of different kinetic models for the heat transfer problem. Heat and Mass Transfer, 2009, vol. 46, iss. 2, pp. 237{244. DOI:10.1007/s00231-009-0558-x
Total in-text references: 1
  1. In-text reference with the coordinate start=1781
    Prefix
    ¬ ̇ÒÚÓˇ ̆ ‚ÂÏˇ ‰Îˇ ÏÓ‰ÂÎËÓ‚‡Ìˡ Ú ̃ÂÌËÈ Ó‰ÌÓ‡ÚÓÏÌÓ„Ó ‡ÁÂÊÂÌÌÓ„Ó „‡Á‡ ̄ËÓÍÓ ‡ÒÔÓÒÚ‡ÌÂÌË ÔÓÎÛ ̃ËÎÓ ÏÓ‰ÂÎ ̧ÌÓ Û‡‚ÌÂÌË ÌÂÔÓÎÌÓ„Ó ÚÂÚ ̧Â„Ó ÔË·ÎËÊÂÌˡ, ÔÓÎÛ ̃Ë‚ ̄Ë ̇Á‚‡ÌË ÏÓ‰ÂÎË ÿ‡ıÓ‚‡, ËÎËS-ÏÓ‰ÂÎË [1,2]. —‡‚ÌÂÌˡ Ò ‡Ò ̃ÂÚ‡ÏË ÔÓ ÚÓ ̃ÌÓÏÛ Û‡‚ÌÂÌË ̨ ¡ÓÎ ̧ˆÏ‡Ì‡, ÏÂÚÓ‰Û ÃÓÌÚÂ- ‡ÎÓ Ë Ò ̋ÍÒÔÂËÏÂÌڇΠ̧Ì ̊ÏË ‰‡ÌÌ ̊ÏË ÔÓ͇Á‡ÎË ıÓÓ ̄Û ̨ ÚÓ ̃ÌÓÒÚ ̧S-ÏÓ‰ÂÎË
    Exact
    [3,4,5,6,7,8,9]
    Suffix
    . ”ÒÔ ̄ÌÓ ÔËÏÂÌÂÌËÂS-ÏÓ‰ÂÎË ÒÚËÏÛÎËÓ‚‡ÎÓ Â ӷӷ ̆ÂÌË ̇ Ú ̃ÂÌˡ ‰‚Ûı‡ÚÓÏÌÓ„Ó „‡Á‡ Ò Û ̃ÂÚÓÏ ‚‡ ̆‡ÚÂÎ ̧Ì ̊ı ÒÚÂÔÂÌÂÈ Ò‚Ó·Ó‰ ̊, Ë ‡Á‚ËÚË ÒÓÓÚ‚ÂÚÒÚ‚Û ̨ ̆Â„Ó ̃ËÒÎÂÌÌÓ„Ó ÏÂÚÓ‰‡ [10,11,12]. “ÂÏ Ì ÏÂÌÂÂ, ÌÂÒÏÓÚˇ ̇ ÓÚÌÓÒËÚÂÎ ̧ÌÛ ̨ ÔÓÒÚÓÚÛ,S-ÏÓ‰ÂÎ ̧ ˇ‚ΡÂÚÒˇ ÒÎÓÊÌ ̊Ï ËÌÚ„Ó-‰ËÙÙÂÂ̈ˇΠ̧Ì ̊Ï Û‡‚ÌÂÌËÂÏ ‚ ̊ÒÓÍÓÈ ‡ÁÏÂÌÓÒÚË, ̃ËÒÎÂÌÌÓ  ̄ÂÌË ÍÓÚÓÓ„Ó Ú·ÛÂÚ ‡Á‚ËÚˡ ‚ ̊ÒÓÍÓÚÓ ̃Ì ̊ı Ô‡‡

10
Rykov V.A. [A model kinetic equation for a gas with rotational degrees of freedom].Izvestiya AN SSSR. Mekhanika Zhidkosti i Gaza, 1975, no. 6, pp. 107{115. (English translation:Fluid Dynamics, 1975, vol. 10, iss. 6, pp. 959{966. DOI:10.1007/BF01023275).
Total in-text references: 3
  1. In-text reference with the coordinate start=1979
    Prefix
    —‡‚ÌÂÌˡ Ò ‡Ò ̃ÂÚ‡ÏË ÔÓ ÚÓ ̃ÌÓÏÛ Û‡‚ÌÂÌË ̨ ¡ÓÎ ̧ˆÏ‡Ì‡, ÏÂÚÓ‰Û ÃÓÌÚÂ- ‡ÎÓ Ë Ò ̋ÍÒÔÂËÏÂÌڇΠ̧Ì ̊ÏË ‰‡ÌÌ ̊ÏË ÔÓ͇Á‡ÎË ıÓÓ ̄Û ̨ ÚÓ ̃ÌÓÒÚ ̧S-ÏÓ‰ÂÎË [3,4,5,6,7,8,9]. ”ÒÔ ̄ÌÓ ÔËÏÂÌÂÌËÂS-ÏÓ‰ÂÎË ÒÚËÏÛÎËÓ‚‡ÎÓ Â ӷӷ ̆ÂÌË ̇ Ú ̃ÂÌˡ ‰‚Ûı‡ÚÓÏÌÓ„Ó „‡Á‡ Ò Û ̃ÂÚÓÏ ‚‡ ̆‡ÚÂÎ ̧Ì ̊ı ÒÚÂÔÂÌÂÈ Ò‚Ó·Ó‰ ̊, Ë ‡Á‚ËÚË ÒÓÓÚ‚ÂÚÒÚ‚Û ̨ ̆Â„Ó ̃ËÒÎÂÌÌÓ„Ó ÏÂÚÓ‰‡
    Exact
    [10,11,12]
    Suffix
    . “ÂÏ Ì ÏÂÌÂÂ, ÌÂÒÏÓÚˇ ̇ ÓÚÌÓÒËÚÂÎ ̧ÌÛ ̨ ÔÓÒÚÓÚÛ,S-ÏÓ‰ÂÎ ̧ ˇ‚ΡÂÚÒˇ ÒÎÓÊÌ ̊Ï ËÌÚ„Ó-‰ËÙÙÂÂ̈ˇΠ̧Ì ̊Ï Û‡‚ÌÂÌËÂÏ ‚ ̊ÒÓÍÓÈ ‡ÁÏÂÌÓÒÚË, ̃ËÒÎÂÌÌÓ  ̄ÂÌË ÍÓÚÓÓ„Ó Ú·ÛÂÚ ‡Á‚ËÚˡ ‚ ̊ÒÓÍÓÚÓ ̃Ì ̊ı Ô‡‡ÎÎÂÎ ̧Ì ̊ı ̃ËÒÎÂÌÌ ̊ı ÏÂÚÓ‰Ó‚  ̄ÂÌˡ.

  2. In-text reference with the coordinate start=19667
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    ÉÎÂÌÌÓÈ ̃‡ÒÚ ̧ ̨ ÏÂÚÓ‰‡ Õ ̧ ̨ÚÓ̇ ˇ‚ΡÂÚÒˇ ‚ ̊ ̃ËÒÎÂÌË χÚˈ ̊ flÍÓ·ËMÒËÒÚÂÏ ̊ Û‡‚ÌÂÌËÈ (16), Ô‰ÒÚ‡‚Ρ ̨ ̆ÂÈ ËÁ Ò·ˇ ‰ËÒÍÂÚÌ ̊ ÒÛÏÏ ̊ ÓÚ ÔÓËÁ‚Ó‰Ì ̊ıf(S)ÔÓ Ï‡ÍÓÒÍÓÔË ̃ÂÒÍËÏ ÔÂÂÏÂÌÌ ̊Ï. “Ó ̃ÌÓ ‚ ̊ ̃ËÒÎÂÌËÂMÚ·ÛÂÚ ÒÛ ̆ÂÒÚ‚ÂÌÌ ̊ı Á‡Ú‡Ú χ ̄ËÌÌÓ„Ó ‚ÂÏÂÌË, ÓÒÓ·ÂÌÌÓ ‰Îˇ ·ÓΠÒÎÓÊÌ ̊ı ÍËÌÂÚË ̃ÂÒÍËı ÏÓ‰ÂÎÂÈ, Ú‡ÍËı Í‡Í ÏÓ‰ÂÎ ̧ – ̊ÍÓ‚‡ ‰‚Ûı‡ÚÓÏÌÓ„Ó „‡Á‡
    Exact
    [10]
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    . —ÎÂ‰Ûˇ [31], ÔÂÂȉÂÏ ‚ ‚ ̊‡ÊÂÌËË ‰Îˇ χÚˈ ̊MÓÚ ̃ËÒÎÂÌÌÓ„Ó ËÌÚ„ËÓ‚‡Ìˡ Í ÚÓ ̃ÌÓÏÛ (‡Ì‡ÎËÚË ̃ÂÒÍÓÏÛ). ¬ ÂÁÛÎ ̧Ú‡ÚÂM‚ ̊‡Ê‡ÂÚÒˇ ‚ ˇ‚ÌÓÏ ‚ˉ ̃ÂÂÁ χÍÓÔ‡‡ÏÂÚ ̊ „‡Á‡. ‡Í Ô‡‚ËÎÓ, ËÒÔÓÎ ̧ÁÓ‚‡ÌË ÔË·ÎËÊÂÌÌÓÈ ÙÓÏÛÎ ̊ ‰Îˇ MÒÛ ̆ÂÒÚ‚ÂÌÌÓ ÛÏÂÌ ̧ ̄‡ÂÚ ‚ÂÏˇ Ò ̃ÂÚ‡ ‰Îˇδ≤1000, ÌÂÒÏÓÚˇ ̇ ÔÓÚÂ ̨ Í‚‡‰‡ÚË ̃ÌÓÈ ÒıÓ‰ËÏÓÒÚË ËÚÂ‡ˆËÈ.

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    ¬ ̇ÒÚÓˇ ̆ ‚ÂÏˇ ‚‰ÂÚÒˇ ‡·ÓÚ‡ ÔÓ ‡ÎËÁ‡ˆËË ‚ Ô‡ÍÂÚ Ì¡‚ÌÓÈ ‡ÁÌÓÒÚÌÓÈ ÒıÂÏ ̊ ‰Îˇ ÌÂÒÚ‡ˆËÓ̇Ì ̊ı Ú ̃ÂÌËÈ ‡ÁÂÊÂÌÌÓ„Ó „‡Á‡, ‡‰‡ÔÚ‡ˆËˇ ‰Îˇ ̋ÙÙÂÍÚË‚ÌÓ„Ó ÏÓ‰ÂÎËÓ‚‡Ìˡ Á‡‰‡ ̃ „ËÔÂÁ‚ÛÍÓ‚Ó„Ó Ó·ÚÂ͇Ìˡ ÚÂÎ ÒÎÓÊÌÓÈ ÔÓÒÚ‡ÌÒÚ‚ÂÌÌÓÈ ÙÓÏ ̊ Ë Ó·Ó· ̆ÂÌË ̇ ÒÎÛ ̃‡È ‰‚Ûı‡ÚÓÏÌÓ„Ó „‡Á‡ ̇ ÓÒÌÓ‚Â ÏÓ‰ÂÎË – ̊ÍÓ‚‡
    Exact
    [10,11,12]
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    . –‡·ÓÚ‡ ‚ ̊ÔÓÎÌÂ̇ ÔË ÔÓ‰‰ÂÊÍ –‘‘» (ÔÓÂÍÚ ̊ 13-01-00522-‡ Ë 14-08-00604-‡). ƒÎˇ ‡Ò ̃ÂÚÓ‚ ËÒÔÓÎ ̧ÁÓ‚‡ÎËÒ ̧ ÒÛÔÂÍÓÏÔ ̧ ̨ÚÂ ̊ ”ÌË‚ÂÒËÚÂÚ‡ ÂÌÙËΉ‡ (¬ÂÎËÍÓ·ËÚ‡Ìˡ), Õ»¬÷ Ã√” ËÏ. Ã.¬. ÀÓÏÓÌÓÒÓ‚‡ Ë Ã‘“».

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Rykov V.A., Titarev V.A., Shakhov E.M. [Shock wave structure in a diatomic gas based on a kinetic model].Izvestiya RAN. Mekhanika Zhidkosti i Gaza, 2008, no. 2, pp. 171{ 182. (English translation:Fluid Dynamics, 2008, vol. 43, iss. 2, pp. 316{326. DOI: 10.1134/S0015462808020178).
Total in-text references: 2
  1. In-text reference with the coordinate start=1979
    Prefix
    —‡‚ÌÂÌˡ Ò ‡Ò ̃ÂÚ‡ÏË ÔÓ ÚÓ ̃ÌÓÏÛ Û‡‚ÌÂÌË ̨ ¡ÓÎ ̧ˆÏ‡Ì‡, ÏÂÚÓ‰Û ÃÓÌÚÂ- ‡ÎÓ Ë Ò ̋ÍÒÔÂËÏÂÌڇΠ̧Ì ̊ÏË ‰‡ÌÌ ̊ÏË ÔÓ͇Á‡ÎË ıÓÓ ̄Û ̨ ÚÓ ̃ÌÓÒÚ ̧S-ÏÓ‰ÂÎË [3,4,5,6,7,8,9]. ”ÒÔ ̄ÌÓ ÔËÏÂÌÂÌËÂS-ÏÓ‰ÂÎË ÒÚËÏÛÎËÓ‚‡ÎÓ Â ӷӷ ̆ÂÌË ̇ Ú ̃ÂÌˡ ‰‚Ûı‡ÚÓÏÌÓ„Ó „‡Á‡ Ò Û ̃ÂÚÓÏ ‚‡ ̆‡ÚÂÎ ̧Ì ̊ı ÒÚÂÔÂÌÂÈ Ò‚Ó·Ó‰ ̊, Ë ‡Á‚ËÚË ÒÓÓÚ‚ÂÚÒÚ‚Û ̨ ̆Â„Ó ̃ËÒÎÂÌÌÓ„Ó ÏÂÚÓ‰‡
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    . “ÂÏ Ì ÏÂÌÂÂ, ÌÂÒÏÓÚˇ ̇ ÓÚÌÓÒËÚÂÎ ̧ÌÛ ̨ ÔÓÒÚÓÚÛ,S-ÏÓ‰ÂÎ ̧ ˇ‚ΡÂÚÒˇ ÒÎÓÊÌ ̊Ï ËÌÚ„Ó-‰ËÙÙÂÂ̈ˇΠ̧Ì ̊Ï Û‡‚ÌÂÌËÂÏ ‚ ̊ÒÓÍÓÈ ‡ÁÏÂÌÓÒÚË, ̃ËÒÎÂÌÌÓ  ̄ÂÌË ÍÓÚÓÓ„Ó Ú·ÛÂÚ ‡Á‚ËÚˡ ‚ ̊ÒÓÍÓÚÓ ̃Ì ̊ı Ô‡‡ÎÎÂÎ ̧Ì ̊ı ̃ËÒÎÂÌÌ ̊ı ÏÂÚÓ‰Ó‚  ̄ÂÌˡ.

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    ¬ ̇ÒÚÓˇ ̆ ‚ÂÏˇ ‚‰ÂÚÒˇ ‡·ÓÚ‡ ÔÓ ‡ÎËÁ‡ˆËË ‚ Ô‡ÍÂÚ Ì¡‚ÌÓÈ ‡ÁÌÓÒÚÌÓÈ ÒıÂÏ ̊ ‰Îˇ ÌÂÒÚ‡ˆËÓ̇Ì ̊ı Ú ̃ÂÌËÈ ‡ÁÂÊÂÌÌÓ„Ó „‡Á‡, ‡‰‡ÔÚ‡ˆËˇ ‰Îˇ ̋ÙÙÂÍÚË‚ÌÓ„Ó ÏÓ‰ÂÎËÓ‚‡Ìˡ Á‡‰‡ ̃ „ËÔÂÁ‚ÛÍÓ‚Ó„Ó Ó·ÚÂ͇Ìˡ ÚÂÎ ÒÎÓÊÌÓÈ ÔÓÒÚ‡ÌÒÚ‚ÂÌÌÓÈ ÙÓÏ ̊ Ë Ó·Ó· ̆ÂÌË ̇ ÒÎÛ ̃‡È ‰‚Ûı‡ÚÓÏÌÓ„Ó „‡Á‡ ̇ ÓÒÌÓ‚Â ÏÓ‰ÂÎË – ̊ÍÓ‚‡
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    . –‡·ÓÚ‡ ‚ ̊ÔÓÎÌÂ̇ ÔË ÔÓ‰‰ÂÊÍ –‘‘» (ÔÓÂÍÚ ̊ 13-01-00522-‡ Ë 14-08-00604-‡). ƒÎˇ ‡Ò ̃ÂÚÓ‚ ËÒÔÓÎ ̧ÁÓ‚‡ÎËÒ ̧ ÒÛÔÂÍÓÏÔ ̧ ̨ÚÂ ̊ ”ÌË‚ÂÒËÚÂÚ‡ ÂÌÙËΉ‡ (¬ÂÎËÍÓ·ËÚ‡Ìˡ), Õ»¬÷ Ã√” ËÏ. Ã.¬. ÀÓÏÓÌÓÒÓ‚‡ Ë Ã‘“».

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Larina I.N., Rykov V.A. [Computation of rarefied diatomic gas flows through a plane microchannel].Zhurnal vychislitel'noy matematiki i matematicheskoy fiziki, 2012, vol. 52, no. 4, pp. 720{732. (English translation:Computational Mathematics and Mathematical Physics, 2012, vol. 52, no. 4, pp. 637{648. DOI:10.1134/S0965542512040112).
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  1. In-text reference with the coordinate start=1979
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    —‡‚ÌÂÌˡ Ò ‡Ò ̃ÂÚ‡ÏË ÔÓ ÚÓ ̃ÌÓÏÛ Û‡‚ÌÂÌË ̨ ¡ÓÎ ̧ˆÏ‡Ì‡, ÏÂÚÓ‰Û ÃÓÌÚÂ- ‡ÎÓ Ë Ò ̋ÍÒÔÂËÏÂÌڇΠ̧Ì ̊ÏË ‰‡ÌÌ ̊ÏË ÔÓ͇Á‡ÎË ıÓÓ ̄Û ̨ ÚÓ ̃ÌÓÒÚ ̧S-ÏÓ‰ÂÎË [3,4,5,6,7,8,9]. ”ÒÔ ̄ÌÓ ÔËÏÂÌÂÌËÂS-ÏÓ‰ÂÎË ÒÚËÏÛÎËÓ‚‡ÎÓ Â ӷӷ ̆ÂÌË ̇ Ú ̃ÂÌˡ ‰‚Ûı‡ÚÓÏÌÓ„Ó „‡Á‡ Ò Û ̃ÂÚÓÏ ‚‡ ̆‡ÚÂÎ ̧Ì ̊ı ÒÚÂÔÂÌÂÈ Ò‚Ó·Ó‰ ̊, Ë ‡Á‚ËÚË ÒÓÓÚ‚ÂÚÒÚ‚Û ̨ ̆Â„Ó ̃ËÒÎÂÌÌÓ„Ó ÏÂÚÓ‰‡
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    . “ÂÏ Ì ÏÂÌÂÂ, ÌÂÒÏÓÚˇ ̇ ÓÚÌÓÒËÚÂÎ ̧ÌÛ ̨ ÔÓÒÚÓÚÛ,S-ÏÓ‰ÂÎ ̧ ˇ‚ΡÂÚÒˇ ÒÎÓÊÌ ̊Ï ËÌÚ„Ó-‰ËÙÙÂÂ̈ˇΠ̧Ì ̊Ï Û‡‚ÌÂÌËÂÏ ‚ ̊ÒÓÍÓÈ ‡ÁÏÂÌÓÒÚË, ̃ËÒÎÂÌÌÓ  ̄ÂÌË ÍÓÚÓÓ„Ó Ú·ÛÂÚ ‡Á‚ËÚˡ ‚ ̊ÒÓÍÓÚÓ ̃Ì ̊ı Ô‡‡ÎÎÂÎ ̧Ì ̊ı ̃ËÒÎÂÌÌ ̊ı ÏÂÚÓ‰Ó‚  ̄ÂÌˡ.

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    ¬ ̇ÒÚÓˇ ̆ ‚ÂÏˇ ‚‰ÂÚÒˇ ‡·ÓÚ‡ ÔÓ ‡ÎËÁ‡ˆËË ‚ Ô‡ÍÂÚ Ì¡‚ÌÓÈ ‡ÁÌÓÒÚÌÓÈ ÒıÂÏ ̊ ‰Îˇ ÌÂÒÚ‡ˆËÓ̇Ì ̊ı Ú ̃ÂÌËÈ ‡ÁÂÊÂÌÌÓ„Ó „‡Á‡, ‡‰‡ÔÚ‡ˆËˇ ‰Îˇ ̋ÙÙÂÍÚË‚ÌÓ„Ó ÏÓ‰ÂÎËÓ‚‡Ìˡ Á‡‰‡ ̃ „ËÔÂÁ‚ÛÍÓ‚Ó„Ó Ó·ÚÂ͇Ìˡ ÚÂÎ ÒÎÓÊÌÓÈ ÔÓÒÚ‡ÌÒÚ‚ÂÌÌÓÈ ÙÓÏ ̊ Ë Ó·Ó· ̆ÂÌË ̇ ÒÎÛ ̃‡È ‰‚Ûı‡ÚÓÏÌÓ„Ó „‡Á‡ ̇ ÓÒÌÓ‚Â ÏÓ‰ÂÎË – ̊ÍÓ‚‡
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    . –‡·ÓÚ‡ ‚ ̊ÔÓÎÌÂ̇ ÔË ÔÓ‰‰ÂÊÍ –‘‘» (ÔÓÂÍÚ ̊ 13-01-00522-‡ Ë 14-08-00604-‡). ƒÎˇ ‡Ò ̃ÂÚÓ‚ ËÒÔÓÎ ̧ÁÓ‚‡ÎËÒ ̧ ÒÛÔÂÍÓÏÔ ̧ ̨ÚÂ ̊ ”ÌË‚ÂÒËÚÂÚ‡ ÂÌÙËΉ‡ (¬ÂÎËÍÓ·ËÚ‡Ìˡ), Õ»¬÷ Ã√” ËÏ. Ã.¬. ÀÓÏÓÌÓÒÓ‚‡ Ë Ã‘“».

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Li Z.-H., Zhang H.-X. Numerical investigation from rarefied flow to continuum by solving the Boltzmann model equation.International Journal for Numerical Methods in Fluids, 2003, vol. 42, no. 4, pp. 361{382.
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  1. In-text reference with the coordinate start=2621
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    ¬ ·ÓÎ ̧ ̄ËÌÒÚ‚Â ËÏ ̨ ̆ËıÒˇ ‡·ÓÚ ‰Îˇ ̃ËÒÎÂÌÌÓ„Ó ËÌÚ„ËÓ‚‡Ìˡ ÍËÌÂÚË ̃ÂÒÍÓ„Ó Û‡‚ÌÂÌˡ Ò ÚÓ ̃Ì ̊Ï ÎË·Ó ÏÓ‰ÂÎ ̧Ì ̊Ï ËÌÚ„‡Î‡ÏË ÒÚÓÎÍÌÓ‚ÂÌËÈ ËÒÔÓÎ ̧ÁÛ ̨ÚÒˇ ˇ‚Ì ̊ ÏÂÚÓ‰ ̊ ‰ËÒÍÂÚËÁ‡ˆËË ÔÓ ‚ÂÏÂÌË Ë ‡Ò ̆ÂÔÎÂÌË Û‡‚ÌÂÌˡ ̇ ÔÓˆÂÒÒ ̊ ÔÂÂÌÓÒ‡ Ë ·ÍÒ‡ˆËË. »ÒÔÓÎ ̧ÁÛÂÏ ̊ ‡ÁÌÓÒÚÌ ̊ ÒıÂÏ ̊ ‡ÎËÁÓ‚‡Ì ̊ ̇ ÏÌÓ„Ó·ÎÓ ̃Ì ̊ı ÒÚÛÍÚÛËÓ‚‡ÌÌ ̊ı
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    , ÌÂÒÚÛÍÚÛËÓ‚‡ÌÌ ̊ı ‰Â͇ÚÓ‚ ̊ı [14] Ë ÚÂÚ‡ ̋‰‡Î ̧Ì ̊ı [15] ÒÂÚ͇ı. —ÓÁ‰‡ÌÌ ̊ ̇ ÓÒÌÓ‚Â ‰‡ÌÌ ̊ı ÏÂÚÓ‰Ó‚ ÍÓÏÔÎÂÍÒ ̊ ÔÓ„‡ÏÏ ÔÓÁ‚ÓΡ ̨Ú ÔÓ‚Ó‰ËÚ ̧ ̃ËÒÎÂÌÌÓ ÏÓ‰ÂÎËÓ‚‡ÌË Ú ̃ÂÌËÈ ‡ÁÂÊÂÌÌÓ„Ó „‡Á‡ ‰Îˇ Ó· ̇ÂÍÚÓ‚ ‰ÓÒÚ‡ÚÓ ̃ÌÓ ÒÎÓÊÌÓÈ ÔÓÒÚ‡ÌÒÚ‚ÂÌÌÓÈ ÙÓÏ ̊.

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Kolobov V.I., Arslanbekov R.R., Aristov V.V., Frolova A.A., Zabelok S.A. Unified solver for rarefied and continuum flows with adaptive mesh and algorithm refinement.Journal of Computational Physics, 2007, vol. 223, no. 2, pp. 589{608. DOI:10.1016/j.jcp.2006.09.021
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  1. In-text reference with the coordinate start=2659
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    ¬ ·ÓÎ ̧ ̄ËÌÒÚ‚Â ËÏ ̨ ̆ËıÒˇ ‡·ÓÚ ‰Îˇ ̃ËÒÎÂÌÌÓ„Ó ËÌÚ„ËÓ‚‡Ìˡ ÍËÌÂÚË ̃ÂÒÍÓ„Ó Û‡‚ÌÂÌˡ Ò ÚÓ ̃Ì ̊Ï ÎË·Ó ÏÓ‰ÂÎ ̧Ì ̊Ï ËÌÚ„‡Î‡ÏË ÒÚÓÎÍÌÓ‚ÂÌËÈ ËÒÔÓÎ ̧ÁÛ ̨ÚÒˇ ˇ‚Ì ̊ ÏÂÚÓ‰ ̊ ‰ËÒÍÂÚËÁ‡ˆËË ÔÓ ‚ÂÏÂÌË Ë ‡Ò ̆ÂÔÎÂÌË Û‡‚ÌÂÌˡ ̇ ÔÓˆÂÒÒ ̊ ÔÂÂÌÓÒ‡ Ë ·ÍÒ‡ˆËË. »ÒÔÓÎ ̧ÁÛÂÏ ̊ ‡ÁÌÓÒÚÌ ̊ ÒıÂÏ ̊ ‡ÎËÁÓ‚‡Ì ̊ ̇ ÏÌÓ„Ó·ÎÓ ̃Ì ̊ı ÒÚÛÍÚÛËÓ‚‡ÌÌ ̊ı [13], ÌÂÒÚÛÍÚÛËÓ‚‡ÌÌ ̊ı ‰Â͇ÚÓ‚ ̊ı
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    Ë ÚÂÚ‡ ̋‰‡Î ̧Ì ̊ı [15] ÒÂÚ͇ı. —ÓÁ‰‡ÌÌ ̊ ̇ ÓÒÌÓ‚Â ‰‡ÌÌ ̊ı ÏÂÚÓ‰Ó‚ ÍÓÏÔÎÂÍÒ ̊ ÔÓ„‡ÏÏ ÔÓÁ‚ÓΡ ̨Ú ÔÓ‚Ó‰ËÚ ̧ ̃ËÒÎÂÌÌÓ ÏÓ‰ÂÎËÓ‚‡ÌË Ú ̃ÂÌËÈ ‡ÁÂÊÂÌÌÓ„Ó „‡Á‡ ‰Îˇ Ó· ̇ÂÍÚÓ‚ ‰ÓÒÚ‡ÚÓ ̃ÌÓ ÒÎÓÊÌÓÈ ÔÓÒÚ‡ÌÒÚ‚ÂÌÌÓÈ ÙÓÏ ̊.

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Anikin Yu.A., Kloss Yu.Yu., Martynov D.V., Cheremisin F.G. [Computer Simulation and Analysis of the Knudsen experiment of the 1910 year].Nano- i mikrosistemnaya tekhnika { Journal of NANO and MICROSYSTEM TECHNIQUE, 2010, no. 8, pp. 6{14.
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  1. In-text reference with the coordinate start=2684
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    ̄ËÌÒÚ‚Â ËÏ ̨ ̆ËıÒˇ ‡·ÓÚ ‰Îˇ ̃ËÒÎÂÌÌÓ„Ó ËÌÚ„ËÓ‚‡Ìˡ ÍËÌÂÚË ̃ÂÒÍÓ„Ó Û‡‚ÌÂÌˡ Ò ÚÓ ̃Ì ̊Ï ÎË·Ó ÏÓ‰ÂÎ ̧Ì ̊Ï ËÌÚ„‡Î‡ÏË ÒÚÓÎÍÌÓ‚ÂÌËÈ ËÒÔÓÎ ̧ÁÛ ̨ÚÒˇ ˇ‚Ì ̊ ÏÂÚÓ‰ ̊ ‰ËÒÍÂÚËÁ‡ˆËË ÔÓ ‚ÂÏÂÌË Ë ‡Ò ̆ÂÔÎÂÌË Û‡‚ÌÂÌˡ ̇ ÔÓˆÂÒÒ ̊ ÔÂÂÌÓÒ‡ Ë ·ÍÒ‡ˆËË. »ÒÔÓÎ ̧ÁÛÂÏ ̊ ‡ÁÌÓÒÚÌ ̊ ÒıÂÏ ̊ ‡ÎËÁÓ‚‡Ì ̊ ̇ ÏÌÓ„Ó·ÎÓ ̃Ì ̊ı ÒÚÛÍÚÛËÓ‚‡ÌÌ ̊ı [13], ÌÂÒÚÛÍÚÛËÓ‚‡ÌÌ ̊ı ‰Â͇ÚÓ‚ ̊ı [14] Ë ÚÂÚ‡ ̋‰‡Î ̧Ì ̊ı
    Exact
    [15]
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    ÒÂÚ͇ı. —ÓÁ‰‡ÌÌ ̊ ̇ ÓÒÌÓ‚Â ‰‡ÌÌ ̊ı ÏÂÚÓ‰Ó‚ ÍÓÏÔÎÂÍÒ ̊ ÔÓ„‡ÏÏ ÔÓÁ‚ÓΡ ̨Ú ÔÓ‚Ó‰ËÚ ̧ ̃ËÒÎÂÌÌÓ ÏÓ‰ÂÎËÓ‚‡ÌË Ú ̃ÂÌËÈ ‡ÁÂÊÂÌÌÓ„Ó „‡Á‡ ‰Îˇ Ó· ̇ÂÍÚÓ‚ ‰ÓÒÚ‡ÚÓ ̃ÌÓ ÒÎÓÊÌÓÈ ÔÓÒÚ‡ÌÒÚ‚ÂÌÌÓÈ ÙÓÏ ̊. Œ‰Ì‡ÍÓ ËÒÔÓÎ ̧ÁÓ‚‡ÌË ˇ‚Ì ̊ı ÏÂÚÓ‰Ó‚ ‰ËÒÍÂÚËÁ‡ˆËË ÔÓ ‚ÂÏÂÌË Ë ÔÓÒÚ‡ÌÒÚ‚ÂÌÌ ̊ı ÒÂÚÓÍ ·ÂÁ ‡‰‡ÔÚ‡ˆËË Í ÔÓ‚ÂıÌÓÒÚË Ú· ‚‰ÂÚ Í ·ÓÎ ̧ ̄ËÏ Á‡Ú‡Ú‡Ï χ ̄ËÌÌÓ„Ó ‚ÂÏÂÌË. œӂ‰ÂÌË ‚ ̊ ̃ËÒÎÂÌËÈ Ò „‡‡ÌÚ

16
Titarev V.A. [Implicit numerical method for computing three-dimensional rarefied gas flows on unstructured meshes].Zhurnal vychislitel'noy matematiki i matematicheskoy fiziki, 2010, vol. 50, no. 10, pp. 1811{1826. (English translation:Computational Mathematics and Mathematical Physics, 2010, vol. 50, no. 10, pp. 1719{1733. DOI:10.1134/S0965542510100088).
Total in-text references: 3
  1. In-text reference with the coordinate start=3516
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    “‡ÍËÏ Ó·‡ÁÓÏ, ‡ÍÚۇΠ̧ÌÓÈ ˇ‚ΡÂÚÒˇ Á‡‰‡ ̃‡ ‡Á‡·ÓÚÍË ·ÓΠ̋ÙÙÂÍÚË‚Ì ̊ı Ë ÛÌË‚Â҇Π̧Ì ̊ı ÏÂÚÓ‰Ó‚  ̄ÂÌˡ ÔÓÒÚ‡ÌÒÚ‚ÂÌÌ ̊ı Á‡‰‡ ̃ ÚÂÓËË ‡ÁÂÊÂÌÌ ̊ı „‡ÁÓ‚ ̇ ÓÒÌÓ‚Â ̃ËÒÎÂÌÌÓ„Ó  ̄ÂÌˡ ÍËÌÂÚË ̃ÂÒÍËı Û‡‚ÌÂÌËÈ. ¬ ̇ÒÚÓˇ ̆ÂÈ ‡·ÓÚ ÔË‚Ó‰ËÚÒˇ Ó·ÁÓ ÔÓÒΉÌËı ÂÁÛÎ ̧Ú‡ÚÓ‚ ÔÓ ‡Á‡·ÓÚÍÂ
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    [16,17,18]
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    Ë ÔËÏÂÌÂÌË ̨ ÌÓ‚Ó„Ó ̃ËÒÎÂÌÌÓ„Ó ÏÂÚÓ‰‡  ̄ÂÌˡ ÍËÌÂÚË ̃ÂÒÍÓ„Ó Û‡‚ÌÂÌˡ ¡ÓÎ ̧ˆÏ‡Ì‡ Ò S-ÏÓ‰ÂÎ ̧Ì ̊Ï ËÌÚ„‡ÎÓÏ ÒÚÓÎÍÌÓ‚ÂÌËÈ [1,2] Ë ‡ÎËÁÛ ̨ ̆Â„Ó Â„Ó ÔÓ„‡ÏÏÌÓ„Ó ÍÓÏÔÎÂÍÒ‡ <ÕÂÒ‚ÂÚ‡È-3ƒ>. ŒÒÌÓ‚Ì ̊ÏË ·ÎÓ͇ÏË ÏÂÚÓ‰‡ ˇ‚Ρ ̨ÚÒˇ Í‚‡ÁËÏÓÌÓÚÓÌ̇ˇ ÒıÂχ ‚ÚÓÓ„Ó ÔÓˇ‰Í‡ ÚÓ ̃ÌÓÒÚË Ì‡ „Ë·Ë‰Ì ̊ı ÌÂÒÚÛÍÚÛËÓ‚‡ÌÌ ̊ı ÒÂÚ͇ı, ÍÓÌÒÂ‚‡Ú˂̇ˇ Ôӈ‰Û‡ ‡Ò ̃ÂÚ‡ χÍÓÒÍÓÔË ̃ÂÒÍËı Ô‡‡ÏÂÚÓ‚ „‡Á‡ Ë ÔÓÎÌÓÒÚ ̧ ̨

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    Œ‰Ì‡ÍÓ ‰Îˇ ‡ÎËÁ‡ˆËË Ì‡ ÒÓ‚ÂÏÂÌÌ ̊ı ÏÌÓ„Óˇ‰ÂÌ ̊ı ÔÓˆÂÒÒÓ‡ı Ò ‚ÂÍÚÓÌ ̊ÏË „ËÒÚ‡ÏË Û‰Ó·Ì ËÒÔÓÎ ̧ÁÓ‚‡Ú ̧ Á‡ÔËÒ ̧ ÍËÌÂÚË ̃ÂÒÍÓ„Ó Û‡‚ÌÂÌˡ Í‡Í ‚ÂÍÚÓÌÓ„Ó Á‡ÍÓ̇ ÒÓı‡ÌÂÌˡ (6). œÓÏËÏÓ Û‰Ó·ÒÚ‚‡ ÔÓ„‡ÏÏÌÓÈ ‡ÎËÁ‡ˆËË, ‚ÂÍÚÓ̇ˇ Á‡ÔËÒ ̧ (6) ‰‡ÂÚ ‚ÓÁÏÓÊÌÓÒÚ ̧ ÌÓ‚ÓÈ ÔÓ Ò‡‚ÌÂÌË ̨ ‡ÌÌËÏË ‡·ÓÚ‡ÏË
    Exact
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    ËÌÚÂÔÂÚ‡ˆËË Ì¡‚ÌÓÈ ‡ÁÌÓÒÚÌÓÈ ÒıÂÏ ̊ ‰Îˇ ÍËÌÂÚË ̃ÂÒÍÓ„Ó Û‡‚ÌÂÌˡ. –‡Ò ̃ÂÚ̇ˇ ӷ·ÒÚ ̧ ‚ ÙËÁË ̃ÂÒÍËı ÔÂÂÏÂÌÌ ̊ı ‡Á·Ë‚‡ÂÚÒˇ ̇ ÍÓÌÚÓÎ ̧Ì ̊ ӷ ̇ÂÏ ̊ (ˇ ̃ÂÈÍË)ViÚÂÚ‡ ̋‰‡Î ̧ÌÓÈ, ÔË‡Ïˉ‡Î ̧ÌÓÈ, „ÂÍÒ‡ ̋‰‡Î ̧ÌÓÈ ËÎË ÔËÁχÚË ̃ÂÒÍÓÈ ÙÓÏ ̊.

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    –ËÒ. 1.ÿ‡·ÎÓÌ ÏÂÚÓ‰‡ ̇ËÏÂÌ ̧ ̄Ëı Í‚‡‰‡ÚÓ‚ ‚ ÙËÁË ̃ÂÒÍÓÈ (Ò΂‡) Ë ÎÓ͇Π̧ÌÓÈ (ÒÔ‡‚‡) ÒËÒÚÂχı ÍÓÓ‰ËÌ‡Ú ‰Îˇ „ÂÍÒ‡ ̋‰‡Î ̧ÌÓÈ ÒÂÚÍË œÓ‰Ó·ÌÓ ÓÔËÒ‡ÌË Ôӈ‰Û ̊ ÔÓÒÚÓÂÌˡ ÍÛÒÓ ̃ÌÓ-ÔÓÎËÌÓÏˇΠ̧ÌÓ„Ó Ô‰ÒÚ‡‚ÎÂÌˡ  ̄ÂÌˡ ÓÔËÒ‡ÌÓ ‚ [23] ‰Îˇ ÚÂÚ‡ ̋‰‡Î ̧Ì ̊ı ˇ ̃ÂÂÍ Ë ‚
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    [16,24]
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    ‰Îˇ ÚÂıÏÂÌ ̊ı ˇ ̃ÂÂÍ ÔÓËÁ‚ÓÎ ̧ÌÓÈ ÙÓÏ ̊. ƒÎˇ ÎËÌÂÈÌÓÈ ÒıÂÏ ̊ ÏÓÊÌÓ ÔÓÎÓÊËÚ ̧ Ò‰ÌË Á̇ ̃ÂÌˡ ÙÛÌ͈ËË ‡ÒÔ‰ÂÎÂÌˡfnilÔÓ Í‡Ê‰ÓÈ „‡ÌËlˇ ̃ÂÈÍË ‡‚Ì ̊ÏË Ò‰ÌËÏ Á̇ ̃ÂÌˡÏpnilÔÓÎÛ ̃ÂÌÌÓ„Ó ÂÍÓÌÒÚÛ͈ËÓÌÌÓ„Ó ÏÌÓ„Ó ̃ÎÂ̇pni(x). œË ̋ÚÓÏpnilÔ‰ÒÚ‡‚Ρ ̨ÚÒˇ ‚ ‚ˉ ÎËÌÂÈÌÓÈ ÍÓÏ·Ë̇ˆËË Á̇ ̃ÂÌËÈ ÙÛÌ͈ËË ‡ÒÔ‰ÂÎÂÌˡ ‚ ˇ ̃ÂÈ͇ı ̄‡·ÎÓ̇, ÍÓ ̋ÙÙˈËÂÌÚ ̊ ÍÓÚÓÓÈ ÓÔ‰ÂΡ ̨ÚÒˇ ÚÓÎ ̧ÍÓ „ÂÓÏÂÚËÂÈ ÔÓ

17
Titarev V.A. Efficient deterministic modelling of three-dimensional rarefied gas flows.Commun. Comput. Phys., 2012, vol. 12, no. 1, pp. 161{192.
Total in-text references: 4
  1. In-text reference with the coordinate start=3516
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    “‡ÍËÏ Ó·‡ÁÓÏ, ‡ÍÚۇΠ̧ÌÓÈ ˇ‚ΡÂÚÒˇ Á‡‰‡ ̃‡ ‡Á‡·ÓÚÍË ·ÓΠ̋ÙÙÂÍÚË‚Ì ̊ı Ë ÛÌË‚Â҇Π̧Ì ̊ı ÏÂÚÓ‰Ó‚  ̄ÂÌˡ ÔÓÒÚ‡ÌÒÚ‚ÂÌÌ ̊ı Á‡‰‡ ̃ ÚÂÓËË ‡ÁÂÊÂÌÌ ̊ı „‡ÁÓ‚ ̇ ÓÒÌÓ‚Â ̃ËÒÎÂÌÌÓ„Ó  ̄ÂÌˡ ÍËÌÂÚË ̃ÂÒÍËı Û‡‚ÌÂÌËÈ. ¬ ̇ÒÚÓˇ ̆ÂÈ ‡·ÓÚ ÔË‚Ó‰ËÚÒˇ Ó·ÁÓ ÔÓÒΉÌËı ÂÁÛÎ ̧Ú‡ÚÓ‚ ÔÓ ‡Á‡·ÓÚÍÂ
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    Ë ÔËÏÂÌÂÌË ̨ ÌÓ‚Ó„Ó ̃ËÒÎÂÌÌÓ„Ó ÏÂÚÓ‰‡  ̄ÂÌˡ ÍËÌÂÚË ̃ÂÒÍÓ„Ó Û‡‚ÌÂÌˡ ¡ÓÎ ̧ˆÏ‡Ì‡ Ò S-ÏÓ‰ÂÎ ̧Ì ̊Ï ËÌÚ„‡ÎÓÏ ÒÚÓÎÍÌÓ‚ÂÌËÈ [1,2] Ë ‡ÎËÁÛ ̨ ̆Â„Ó Â„Ó ÔÓ„‡ÏÏÌÓ„Ó ÍÓÏÔÎÂÍÒ‡ <ÕÂÒ‚ÂÚ‡È-3ƒ>. ŒÒÌÓ‚Ì ̊ÏË ·ÎÓ͇ÏË ÏÂÚÓ‰‡ ˇ‚Ρ ̨ÚÒˇ Í‚‡ÁËÏÓÌÓÚÓÌ̇ˇ ÒıÂχ ‚ÚÓÓ„Ó ÔÓˇ‰Í‡ ÚÓ ̃ÌÓÒÚË Ì‡ „Ë·Ë‰Ì ̊ı ÌÂÒÚÛÍÚÛËÓ‚‡ÌÌ ̊ı ÒÂÚ͇ı, ÍÓÌÒÂ‚‡Ú˂̇ˇ Ôӈ‰Û‡ ‡Ò ̃ÂÚ‡ χÍÓÒÍÓÔË ̃ÂÒÍËı Ô‡‡ÏÂÚÓ‚ „‡Á‡ Ë ÔÓÎÌÓÒÚ ̧ ̨

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    ¬ ÂÁÛÎ ̧Ú‡Ú ÏÓ‰ÂÎËÓ‚‡ÌË ÔÂÂıÓ‰Ì ̊ı Ë ÒÔÎÓ ̄ÌÓÒÂ‰Ì ̊ı Ú ̃ÂÌËÈ „‡Á‡ ÏÓÊÂÚ Ú·ӂ‡Ú ̧ ÌÂÔËÂÏÎÂÏÓ ·ÓÎ ̧ ̄Ó„Ó ̃ËÒ· ÛÁÎÓ‚Nξ‚ ÒÍÓÓÒÚÌÓÈ ÒÂÚÍÂ. ÓÓÚÍËÈ Ó·ÁÓ ÒÛ ̆ÂÒÚ‚Û ̨ ̆Ëı ÒÔÓÒÓ·Ó‚ ÔÓÒÚÓÂÌˡ ÍÓÌÒÂ‚‡ÚË‚Ì ̊ı ÒıÂÏ ‰Îˇ ÏÓ‰ÂÎ ̧ÌÓ„Ó ÍËÌÂÚË ̃ÂÒÍÓ„Ó Û‡‚ÌÂÌˡ Ô˂‰ÂÌ ‚
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    [17]
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    . ¬ ̇ÒÚÓˇ ̆ÂÈ ‡·ÓÚ ËÒÔÓÎ ̧ÁÛÂÚÒˇ ÓÚÌÓÒËÚÂÎ ̧ÌÓ ÌÓ‚ ̊È ÏÂÚÓ‰ ÔÓÒÚÓÂÌˡ ÍÓÌÒÂ‚‡ÚË‚ÌÓÈ ‡ÔÔÓÍÒËχˆËË ËÌÚ„‡Î‡ ÒÚÓÎÍÌÓ‚ÂÌËÈ [29,30,31]. ÇÍÓÔ‡‡ÏÂÚ ̊ „‡Á‡ ̇ıÓ‰ˇÚÒˇ Í‡Í  ̄ÂÌË ÒËÒÚÂÏ ̊ Û‡‚ÌÂÌËÈ, ËÒÔÓÎ ̧ÁÛÂÏÓÈ ‰Îˇ ÔÓÒÚÓÂÌˡS-ÏÓ‰ÂÎ ̧ÌÓ„Ó Û‡‚ÌÂÌˡ: H(W) = ∑Nξ j=1        1 ξj ξ2j vjv2j        (f (S) j−fj)ωj+        0 0 0 2 Prq        =0,W=        n u T q 

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    ¬ ÚÂÍÛ ̆ÂÈ ‚ÂÒËË <ÕÂÒ‚ÂÚ‡È-3ƒ> ‡ÎËÁÓ‚‡Ì ̊ Ó·‡ ÔÓ‰ıÓ‰‡ Í Ó„‡ÌËÁ‡ˆËË ‡Ò ̃ÂÚÓ‚ Ò MPI, Ò ÌÂÍÓÚÓ ̊ÏË ÏÓ‰ËÙË͇ˆËˇÏË ÔÓ Ò‡‚ÌÂÌË ̨ Ò ‡ÌÌËÏË ‚ÂÒˡÏË Ô‡ÍÂÚ‡. ƒÎˇ ÔÂ‚Ó„Ó ÔÓ‰ıÓ‰‡ ËÒÔÓÎ ̧ÁÛÂÚÒˇ ·ÓΠ̄ËÓÍÓ ÔÂÂÍ ̊ÚË ·ÎÓÍÓ‚ ÒÂÚÍË, Ò ÚÂÏ, ̃ÚÓ· ̊ ÔÓÒÚ‡ÌÒÚ‚ÂÌ̇ˇ ÂÍÓÌÒÚÛÍˆËˇ ‰‡ÌÌ ̊ı Ïӄ· ÔÓ‚Ó‰ËÚÒˇ Ë ‚ ÌÂÔÓÒ‰ÒÚ‚ÂÌÌ ̊ı ÒÓÒ‰ˇı Í „‡Ìˈ ·ÎÓ͇. ¬ ·ÓΠ‡ÌÌËı ‚ÂÒˡı Ô‡ÍÂÚ‡
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    ËÒÔÓÎ ̧ÁÓ‚‡ÎÓÒ ̧ ÏËÌËχΠ̧ÌÓ ‚ÓÁÏÓÊÌÓ ÔÂÂÍ ̊ÚË ·ÎÓÍÓ‚, ÌÓ ÔË ̋ÚÓÏ ‰ÓÔÓÎÌËÚÂÎ ̧ÌÓ Ú·ӂ‡ÎÒˇ Ó·ÏÂÌ ‰‡ÌÌ ̊ÏË Ì‡ „‡Ìˇı „‡ÌË ̃Ì ̊ı ˇ ̃ÂÂÍ, ̃ÚÓ ÒÛ ̆ÂÒÚ‚ÂÌÌÓ ÛÒÎÓÊÌˇÎÓ ÔÓ„‡ÏÏÌ ̊È ÍÓ‰. ƒÎˇ ‚ÚÓÓ„Ó ÔÓ‰ıÓ‰‡ ‡ÎËÁÓ‚‡Ì‡ „˷͇ˇ Ôӈ‰Û‡ Ó·ÏÂ̇ ‰‡ÌÌ ̊ÏË Ì‡ ̋Ú‡Ô ‡Ò ̃ÂÚ ̊ χÍÓÒÍÓÔË ̃ÂÒÍËı ‚ÂÎË ̃ËÌ: MPIAllReduce ‚ ̊Á ̊‚‡ÂÚÒˇ ÚÓÎ ̧ÍÓ ‰Îˇ ÚÂı ˇ ̃ÂÂÍ, ‚ ÍÓÚÓ ̊ı ËÚÂ‡ˆËÓÌÌ ̊È ÔÓˆÂÒÒ (16) Ì ÒÓ ̄ÂÎ

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    . œË ‡Ì‡ÎËÁ ÂÁÛÎ ̧Ú‡ÚÓ‚ Û‰Ó·Ì ÔÓÎ ̧ÁÓ‚‡Ú ̧Òˇ ‚ÂÎË ̃ËÌÓÈQ, Ô‰ÒÚ‡‚Ρ ̨ ̆ÂÈ ÒÓ·ÓÈ ‡ÒıÓ‰ ̇M„‡Á‡ ̃ÂÂÁ Ò ̃ÂÌË ÚÛ· ̊, ÌÓÏËÓ‚‡ÌÌ ̊È Ì‡ ‚ÂÎË ̃ËÌÛ ‡ÒıÓ‰‡ ̇M отв 0ÔË Ò‚Ó·Ó‰ÌÓ-ÏÓÎÂÍÛΡÌÓÏ ËÒÚ ̃ÂÌËË ‚ ‚‡ÍÛÛÏ ̃ÂÂÁ ÓÚ‚ÂÒÚË [44,45]: Q= ̇M / ̇M отв 0, M ̇отв 0= √ π/2.(22) ŒÒÌÓ‚Ì ̊Ï ˇ‚ΡÂÚÒˇ ÒÎÛ ̃‡È ÍÛ„ÎÓÈ ÚÛ· ̊ ÔÓÒÚÓˇÌÌÓ„Ó ‡‰ËÛÒ‡, ‡‚ÌÓ„Ó ‡‰ËÛÒÛ ‚ıÓ‰ÌÓ„Ó Ò ̃ÂÌˡa. ¬ ˆËÍΠ‡·ÓÚ
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     ̄ÂÌË Á‡‰‡ ̃Ë · ̊ÎÓ ÔÓÒÚÓÂÌÓ Ô‡ÍÂÚÓÏ <ÕÂÒ‚ÂÚ‡È-3ƒ> ‰Îˇ χÎÓ„Ó ÓÚÌÓ ̄ÂÌˡ ‰‡‚ÎÂÌËÈ|p10/p20|−1-1(ÎË̇ËÁÓ‚‡Ì̇ˇ Á‡‰‡ ̃‡), ‡ Ú‡ÍÊ ‰‚Ûı ÌÂÎËÌÂÈÌ ̊ı ÂÊËÏÓ‚p10/p20= 2(ÛÏÂÂÌÌ ̊È ÔÂÂÔ‡‰ ‰‡‚ÎÂÌËÈ) Ëp10/p20=∞ (Ú ̃ÂÌË ‚ ‚‡ÍÛÛÏ).

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Titarev V.A., Dumbser M., Utyuzhnikov S.V. Construction and comparison of parallel implicit kinetic solvers in three spatial dimensions.Journal of Computational Physics, 2014, vol. 256, no. 1, pp. 17{33. DOI:10.1016/j.jcp.2013.08.051
Total in-text references: 4
  1. In-text reference with the coordinate start=3516
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    “‡ÍËÏ Ó·‡ÁÓÏ, ‡ÍÚۇΠ̧ÌÓÈ ˇ‚ΡÂÚÒˇ Á‡‰‡ ̃‡ ‡Á‡·ÓÚÍË ·ÓΠ̋ÙÙÂÍÚË‚Ì ̊ı Ë ÛÌË‚Â҇Π̧Ì ̊ı ÏÂÚÓ‰Ó‚  ̄ÂÌˡ ÔÓÒÚ‡ÌÒÚ‚ÂÌÌ ̊ı Á‡‰‡ ̃ ÚÂÓËË ‡ÁÂÊÂÌÌ ̊ı „‡ÁÓ‚ ̇ ÓÒÌÓ‚Â ̃ËÒÎÂÌÌÓ„Ó  ̄ÂÌˡ ÍËÌÂÚË ̃ÂÒÍËı Û‡‚ÌÂÌËÈ. ¬ ̇ÒÚÓˇ ̆ÂÈ ‡·ÓÚ ÔË‚Ó‰ËÚÒˇ Ó·ÁÓ ÔÓÒΉÌËı ÂÁÛÎ ̧Ú‡ÚÓ‚ ÔÓ ‡Á‡·ÓÚÍÂ
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    Ë ÔËÏÂÌÂÌË ̨ ÌÓ‚Ó„Ó ̃ËÒÎÂÌÌÓ„Ó ÏÂÚÓ‰‡  ̄ÂÌˡ ÍËÌÂÚË ̃ÂÒÍÓ„Ó Û‡‚ÌÂÌˡ ¡ÓÎ ̧ˆÏ‡Ì‡ Ò S-ÏÓ‰ÂÎ ̧Ì ̊Ï ËÌÚ„‡ÎÓÏ ÒÚÓÎÍÌÓ‚ÂÌËÈ [1,2] Ë ‡ÎËÁÛ ̨ ̆Â„Ó Â„Ó ÔÓ„‡ÏÏÌÓ„Ó ÍÓÏÔÎÂÍÒ‡ <ÕÂÒ‚ÂÚ‡È-3ƒ>. ŒÒÌÓ‚Ì ̊ÏË ·ÎÓ͇ÏË ÏÂÚÓ‰‡ ˇ‚Ρ ̨ÚÒˇ Í‚‡ÁËÏÓÌÓÚÓÌ̇ˇ ÒıÂχ ‚ÚÓÓ„Ó ÔÓˇ‰Í‡ ÚÓ ̃ÌÓÒÚË Ì‡ „Ë·Ë‰Ì ̊ı ÌÂÒÚÛÍÚÛËÓ‚‡ÌÌ ̊ı ÒÂÚ͇ı, ÍÓÌÒÂ‚‡Ú˂̇ˇ Ôӈ‰Û‡ ‡Ò ̃ÂÚ‡ χÍÓÒÍÓÔË ̃ÂÒÍËı Ô‡‡ÏÂÚÓ‚ „‡Á‡ Ë ÔÓÎÌÓÒÚ ̧ ̨

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    ̧ÌÓ Ú·ӂ‡ÎÒˇ Ó·ÏÂÌ ‰‡ÌÌ ̊ÏË Ì‡ „‡Ìˇı „‡ÌË ̃Ì ̊ı ˇ ̃ÂÂÍ, ̃ÚÓ ÒÛ ̆ÂÒÚ‚ÂÌÌÓ ÛÒÎÓÊÌˇÎÓ ÔÓ„‡ÏÏÌ ̊È ÍÓ‰. ƒÎˇ ‚ÚÓÓ„Ó ÔÓ‰ıÓ‰‡ ‡ÎËÁÓ‚‡Ì‡ „˷͇ˇ Ôӈ‰Û‡ Ó·ÏÂ̇ ‰‡ÌÌ ̊ÏË Ì‡ ̋Ú‡Ô ‡Ò ̃ÂÚ ̊ χÍÓÒÍÓÔË ̃ÂÒÍËı ‚ÂÎË ̃ËÌ: MPIAllReduce ‚ ̊Á ̊‚‡ÂÚÒˇ ÚÓÎ ̧ÍÓ ‰Îˇ ÚÂı ˇ ̃ÂÂÍ, ‚ ÍÓÚÓ ̊ı ËÚÂ‡ˆËÓÌÌ ̊È ÔÓˆÂÒÒ (16) Ì ÒÓ ̄ÂÎÒˇ. —‡‚ÌÂÌË ‰‚Ûı ÔÓ‰ıÓ‰Ó‚ Í Ó„‡ÌËÁ‡ˆËË Ô‡‡ÎÎÂÎ ̧Ì ̊ı ‚ ̊ ̃ËÒÎÂÌËÈ Ôӂ‰ÂÌÓ ‚
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    . œÓ͇Á‡Ì‡ ıÓÓ ̄‡ˇ Ï‡Ò ̄Ú‡·ËÛÂÏÓÒÚ ̧ ÍÓ‰‡ ̇ 4-ı ˇ‰ÂÌ ̊ı ÔÓˆÂÒÒÓ‡ı Intel Xeon ÔË ËÒÔÓÎ ̧ÁÓ‚‡ÌËË ÔË ËÒÔÓÎ ̧ÁÓ‚‡ÌËË ‰Ó 512 ‚ ̊ ̃ËÒÎËÚÂÎ ̧Ì ̊ı ˇ‰Â. 4. œËÎÓÊÂÌˡ — ÔÓÏÓ ̆ ̧ ̨ ‚ ̊ ̃ËÒÎËÚÂÎ ̧ÌÓ„Ó Ô‡ÍÂÚ‡ <ÕÂÒ‚ÂÚ‡È-3ƒ> ÏÓ„ÛÚ ÏÓ‰ÂÎËÓ‚‡Ú ̧Òˇ Í‡Í ÔÓÒÚ ̊ Ú ̃ÂÌˡ ‡ÁÂÊÂÌÌÓ„Ó „‡Á‡, Ú‡Í Ë Ú ̃ÂÌˡ ‚ ÒÎÓÊÌ ̊ı „ÂÓÏÂÚË ̃ÂÒÍËı ӷ·ÒÚˇı.

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    Ò ̃ÂÌˡa. ¬ ˆËÍΠ‡·ÓÚ [17,27,43,46]  ̄ÂÌË Á‡‰‡ ̃Ë · ̊ÎÓ ÔÓÒÚÓÂÌÓ Ô‡ÍÂÚÓÏ <ÕÂÒ‚ÂÚ‡È-3ƒ> ‰Îˇ χÎÓ„Ó ÓÚÌÓ ̄ÂÌˡ ‰‡‚ÎÂÌËÈ|p10/p20|−1-1(ÎË̇ËÁÓ‚‡Ì̇ˇ Á‡‰‡ ̃‡), ‡ Ú‡ÍÊ ‰‚Ûı ÌÂÎËÌÂÈÌ ̊ı ÂÊËÏÓ‚p10/p20= 2(ÛÏÂÂÌÌ ̊È ÔÂÂÔ‡‰ ‰‡‚ÎÂÌËÈ) Ëp10/p20=∞ (Ú ̃ÂÌË ‚ ‚‡ÍÛÛÏ). –‡ÒÒχÚË‚‡ÎËÒ ̧ ÚÛ· ̊ ‡ÁÌÓÈ ÓÚÌÓÒËÚÂÎ ̧ÌÓÈ ‰ÎËÌ ̊1≤L/a≤50. Շ˷ÓΠËÌÚÂÂÒÌ ̊Ï ˇ‚ΡÂÚÒˇ ÒÎÛ ̃‡È ËÒÚ ̃ÂÌˡ ‚ ‚‡ÍÛÛÏ. ¬ ‡·ÓÚÂ
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    · ̊· Ôӂ‰Â̇ ÒÂˡ ‚ÂËÙË͇ˆËÓÌÌ ̊ı ‡Ò ̃ÂÚÓ‚ Ò ˆÂÎ ̧ ̨ ÔÓ‰ÂÏÓÌÒÚËÓ‚‡Ú ̧ ÚÓ ̃ÌÓÒÚ ̧ Ë ÒıÓ‰ËÏÓÒÚ ̧ ÏÂÚÓ‰‡  ̄ÂÌˡ ‰Îˇ ÍÓÓÚÍÓÈ ÚÛ· ̊L/a= 1. ¬ ‡Ò ̃ÂÚ‡ı ËÒÔÓÎ ̧ÁÓ‚‡Î‡Ò ̧ ÔÓÒΉӂ‡ÚÂÎ ̧ÌÓÒÚ ̧ ËÁ ÚÂı ÔÓÒÚ‡ÌÒÚ‚ÂÌÌ ̊ı ÒÂÚÓÍ. œÂ‚‡ˇ ÒÂÚ͇ ÒÓÒÚӡ· ËÁ5600„ÂÍÒ‡ ̋‰Ó‚; ÔË ̋ÚÓÏ ‚‰ÓÎ ̧ ÚÛ· ̊ ËÒÔÓÎ ̧ÁÓ‚‡ÎÓÒ ̧10ˇ ̃ÂÂÍ, ‚ ÔÓÔÂ ̃ÌÓÏ Ò ̃ÂÌËË |116ˇ ̃ÂÂÍ. ÿ‡„ ÒÂÚÍË ÔÓ ÌÓχÎË Í ÔÓ‚ÂıÌÓÒÚË ÚÛ· ̊ 

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    — ÓÒÚÓÏ Á̇ ̃ÂÌˡ Ô‡‡ÏÂÚ‡ ‡ÁÂÊÂÌÌÓÒÚË (ÛÏÂÌ ̧ ̄ÂÌˡ ̃ËÒ· ÌÛ‰ÒÂ̇) ÒÚÛÍÚÛ‡ Û‰‡ÌÓÈ ‚ÓÎÌ ̊ ÒÚ‡ÌÓ‚ËÚÒˇ ·ÓΠ‚ ̊‡ÊÂÌÌÓÈ Ë ÔÓÁËˆËˇ χıÓ‚ÒÍÓ„Ó ‰ËÒ͇ ÛÒڇ̇‚ÎË‚‡ÂÚÒˇ ÓÍÓÎÓz≈9. «‡‰‡ ̃Ë ‚Ì ̄ÌÂ„Ó Ó·ÚÂ͇Ìˡ ÚÂÎ ÒÎÓÊÌÓÈ ÙÓÏ ̊.¬ ͇ ̃ÂÒÚ‚Â ËÎÎ ̨ÒÚ‡ˆËË ÔËÏÂÌÂÌˡ Ô‡ÍÂÚ‡ ÔÓ„‡ÏÏ Í Á‡‰‡ ̃‡Ï ‚Ì ̄ÌÂÈ ‡ ̋Ó‰Ë̇ÏËÍË ÔÓÒÚÓËÏ ÔÓΠڠ̃ÂÌˡ ‚ÓÍÛ„ ‰‚Ûı „ÂÓÏÂÚË ̃ÂÒÍËı ÏÓ‰ÂÎÂÈ
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    . œÂ‚‡ˇ ÏÓ‰ÂÎ ̧ Ô‰ÒÚ‡‚ΡÂÚ ÒÓ·ÓÈ ÏÓ‰ÂÎ ̧ ‡ÍÂÚ ̊ Ò ‰‚ÛÏˇ Í ̊Î ̧ˇÏË Ë ̃ÂÚ ̊ ̧Ïˇ ÒÚ‡·ËÎËÁ‡ÚÓ‡ÏË. √ÂÓÏÂÚˡ ÏÓ‰ÂÎË ÔÓ‰Ó·ÌÓ ÓÔË҇̇ ‚ [50]. ƒË‡ÏÂÚ ‡ÍÂÚ ̊ Ë Á̇ ̃ÂÌˡ ‰‡‚ÎÂÌˡ Ë ÚÂÏÔÂ‡ÚÛ ̊ ‚ ̇·Â„‡ ̨ ̆ÂÏ ÔÓÚÓÍ ‚ ̊·Ë‡ ̨ÚÒˇ ‚ ͇ ̃ÂÒÚ‚Â Ï‡Ò ̄Ú‡·Ó‚ ‰ÎËÌ ̊l∗,p∗,T∗.

19
Bhatnagar P.L., Gross E.P., Krook M. A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems.Phys. Rev., 1954, vol. 94, no. 3, pp. 511{525.
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    ËÌÂÚË ̃ÂÒÍÓ Û‡‚ÌÂÌË ¡ÓÎ ̧ˆÏ‡Ì‡ ‰Îˇ ÙÛÌ͈ËË ‡ÒÔ‰ÂÎÂÌˡfÏÓÎÂÍÛÎ ÔÓ ÒÍÓÓÒÚˇÏ ÒS-ÏÓ‰ÂÎ ̧Ì ̊Ï ËÌÚ„‡ÎÓÏ ÒÚÓÎÍÌÓ‚ÂÌËÈ [1,2] ËÏÂÂÚ ‚ˉ ∂f ∂t +ξα ∂f ∂xα =J, J= p μ (f+−f), f+=fM [ 1 + 4 5 (1−Pr)Sαcα ( c2− 5 2 )] , fM= n (2πRgT)3/2 exp (−c2), Si= 1 n ∫ cic2f dξ,c= v √ 2RgT , c2=cβcβ, „‰Âμ=μ(T)| ‚ˇÁÍÓÒÚ ̧; ̃ËÒÎÓ œ‡Ì‰ÚΡ ‰Îˇ Ó‰ÌÓ‡ÚÓÏÌÓ„Ó „‡Á‡Pr = 2/3;Pr = 1ÒÓÓÚ‚ÂÚÒÚ‚ÛÂÚ ÏÓ‰ÂÎË ¡√
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    . ¬ ‡·ÓÚ ‡ÒÒχÚË‚‡ ̨ÚÒˇ ÌÂÒÍÓÎ ̧ÍÓ ‚‡ˇÌÚÓ‚ Á‡‰‡Ìˡ „‡ÌË ̃Ì ̊ı ÛÒÎÓ‚ËÈ Ì‡ ÔÓ‚ÂıÌÓÒÚË Ú·. œÂ‚Ó ÛÒÎÓ‚Ë ÒÓÓÚ‚ÂÚÒÚ‚ÛÂÚ ‰ËÙÙÛÁÌÓÏÛ ÓÚ‡ÊÂÌË ̨ Ò ÔÓÎÌÓÈ ÚÂÔÎÓ‚ÓÈ ‡ÍÍÓÏÓ‰‡ˆËÂÈ Í ÚÂÏÔÂ‡ÚÛ ÔÓ‚ÂıÌÓÒÚËTsur. œË ̋ÚÓÏ ÙÛÌÍˆËˇ ‡ÒÔ‰ÂÎÂÌˡ ÓÚ‡ÊÂÌÌ ̊ı ÏÓÎÂÍÛÎ Á‡ÔËÒ ̊‚‡ÂÚÒˇ ‚ ‚ˉÂ: fw= nw (2πRgTsur)3/2 exp ( − ξ2 2RgTsur ) , nw= √ 2π RgTsur Ni, Ni=− ∫ ξn<0 ξnf dξ,(1) „‰Âξn| ÔÓÂÍˆËˇ ‚ÂÍÚÓ‡

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    f (S) j−fj)ωj+        0 0 0 2 Prq        =0,W=        n u T q        .(16) ƒÎˇ ͇ʉÓÈ ÔÓÒÚ‡ÌÒÚ‚ÂÌÌÓÈ ˇ ̃ÂÈÍËiÒËÒÚÂχ ËÁ ‚ÓÒ ̧ÏË Û‡‚ÌÂÌËÈ (16)  ̄‡ÂÚÒˇ ÏÂÚÓ‰ÓÏ Õ ̧ ̨ÚÓ̇ M(Ws−1)(Ws−Ws−1) =−H(Ws−1), s= 1,2, . . . , M= ∂H ∂W . ¬ ͇ ̃ÂÒڂ ̇ ̃‡Î ̧ÌÓ„Ó ÔË·ÎËÊÂÌˡ ËÒÔÓÎ ̧ÁÛ ̨ÚÒˇ Á̇ ̃ÂÌˡ, ÔÓÎÛ ̃ÂÌÌ ̊ ËÁ ÔË·ÎËÊÂÌˡ (15). ¬ ̃‡ÒÚÌÓÏ ÒÎÛ ̃‡ÂPr = 1(ÏÓ‰ÂÎ ̧ ¡√
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    ) ÙÛÌÍˆËˇf(S)Ì ÒÓ‰ÂÊËÚ Á̇ ̃ÂÌËÈ ‚ÂÍÚÓ‡ ÔÓÚÓÍ ÚÂÔ·, Ú‡Í ̃ÚÓ ÔÓÒΉÌË ÚË Û‡‚ÌÂÌˡ ÒËÒÚÂÏ ̊ (16) ÒÚ‡ÌÓ‚ˇÚÒˇ ËÁ· ̊ÚÓ ̃Ì ̊ÏË. ¬ ‰‡ÌÌÓÏ ÒÎÛ ̃‡Â ÒËÒÚÂχ (16) ÒÓ‚Ô‡‰‡ÂÚ Ò ÏÂÚÓ‰ÓÏ, Ô‰ÎÓÊÂÌÌ ̊Ï ‚ [32,33] ‰Îˇ ¡√ -ÏÓ‰ÂÎË.

20
Godunov S.K. [A difference method for numerical calculation of discontinuous solutions of the equations of hydrodynamics].Matematicheskiy sbornik, 1959, vol. 47, no. 3, pp. 271{306. (English translation:Sbornik: Mathematics, 1959, vol. 47, pp. 357{393).
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    ̄Ì ̨ ̨ ‰ËÌË ̃ÌÛ ̨ ÌÓχΠ̧ nil„‡ÌËlˇ ̃ÂÈÍËVi. œӂ‰ÂÏ ÔË·ÎËÊÂÌÌÛ ̨ ÎË̇ËÁ‡ˆË ̨ (7) ÔÓ ‚ÂÏÂÌË. ƒÎˇ ÏÓ‰ÂÎ ̧ÌÓ„Ó ËÌÚ„‡Î‡ ÒÚÓÎÍÌÓ‚ÂÌËÈ ·Û‰ÂÏ ÓÔÛÒÍ‡Ú ̧ ‰ÓÒÚ‡ÚÓ ̃ÌÓ ÒÎÓÊÌÛ ̨ Á‡‚ËÒËÏÓÒÚ ̧ χÍÓÒÍÓÔË ̃ÂÒÍËı ‚ÂÎË ̃ËÌ ÓÚfiÚ‡Í ̃ÚÓ Jn+1i≈Jni−νni∆fi. ‘ÓÏÛ· ‰Îˇ ̃ËÒÎÂÌÌÓ„Ó ÔÓÚÓ͇ (8) ÒÓ‰ÂÊËÚ ËÌÚ„‡Î ÓÚ ÙÛÌ͈ËË ‡ÒÔ‰ÂÎÂÌˡ ÔÓ „‡ÌË ˇ ̃ÂÈÍË. ‡Í Ó· ̊ ̃ÌÓ ‚ ÔÓÚË‚ÓÔÓÚÓ ̃Ì ̊ı ÒıÂχı ÚËÔ‡ √Ó‰ÛÌÓ‚‡
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    , ̇ ͇ʉÓÈ „‡ÌË ÂÒÚ ̧ ‰‚‡ Á̇ ̃ÂÌˡ  ̄ÂÌˡ. œÂ‚ÓÈ Á̇ ̃ÂÌËÂf−ÔÓÎÛ ̃‡ÂÚÒˇ ̋ÍÒÚ‡ÔÓΡˆËÂÈ Ì‡ „‡Ì ̧ ‚ ‡Ï͇ı ˇ ̃ÂÈÍËVi. ¬ÚÓÓ Á̇ ̃ÂÌËÂf+ÒÓÓÚ‚ÂÚÒÚ‚ÛÂÚ ̋ÍÒÚ‡ÔÓΡˆËË Ì‡ „‡ÌËAilÒ̇ÛÊË ˇ ̃ÂÈÍË. ◊ËÒÎÂÌÌ ̊È ÔÓÚÓÍ ̃ÂÂÁ „‡Ì ̧lˇ ̃ÂÈÍËVi‚ ̊ ̃ËÒΡÂÚÒˇ ÔÓ ÙÓÏÛΠΦnil=G(fnil,fnil,l1)|Ail|,(9) „‰Âil| ÌÓÏÂ ˇ ̃ÂÈÍË, ÒÓÒ‰ÌÂÈ Í „‡ÌËl(ÂÒÎË Ú‡Í‡ˇ ÒÛ ̆ÂÒÚ‚ÛÂÚ);l1| ÌÓÏÂ ÒÓÓÚ‚ÂÚÒÚ‚Û ̨ ̆ÂÈ „‡ÌË ˇ ̃ÂÈÍËi

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    „‡ÌËlˇ ̃ÂÈÍË ‡‚Ì ̊ÏË Ò‰ÌËÏ Á̇ ̃ÂÌˡÏpnilÔÓÎÛ ̃ÂÌÌÓ„Ó ÂÍÓÌÒÚÛ͈ËÓÌÌÓ„Ó ÏÌÓ„Ó ̃ÎÂ̇pni(x). œË ̋ÚÓÏpnilÔ‰ÒÚ‡‚Ρ ̨ÚÒˇ ‚ ‚ˉ ÎËÌÂÈÌÓÈ ÍÓÏ·Ë̇ˆËË Á̇ ̃ÂÌËÈ ÙÛÌ͈ËË ‡ÒÔ‰ÂÎÂÌˡ ‚ ˇ ̃ÂÈ͇ı ̄‡·ÎÓ̇, ÍÓ ̋ÙÙˈËÂÌÚ ̊ ÍÓÚÓÓÈ ÓÔ‰ÂΡ ̨ÚÒˇ ÚÓÎ ̧ÍÓ „ÂÓÏÂÚËÂÈ ÔÓÒÚ‡ÌÒÚ‚ÂÌÌÓÈ ÒÂÚÍË. ‡Í ËÁ‚ÂÒÚÌÓ, ÎËÌÂÈÌ ̊ ÒıÂÏ ̊ ‚ ̊ÒÓÍÓ„Ó ÔÓˇ‰Í‡ ÚÓ ̃ÌÓÒÚË ÌÂÏÓÌÓÚÓÌÌ ̊ ‚ ӷ·ÒÚˇı · ̊ÒÚÓ„Ó ËÁÏÂÌÂÌˡ  ̄ÂÌˡ
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    . ƒÎˇ ÔÓ‰‡‚ÎÂÌˡ Ô‡‡ÁËÚÌ ̊ı ÓÒˆËÎΡˆËÈ ‚ ·ÎÓÍ ÂÍÓÌÒÚÛ͈ËË ‚‚Ó‰ËÚÒˇ Ú‡Í Ì‡Á ̊‚‡ÂÏ ̊È Ó„‡ÌË ̃ËÚÂÎ ̧ ̇ÍÎÓÌÓ‚ [21]. ƒÎˇ ‡Ò ̃ÂÚ‡ ÒÚ‡ˆËÓ̇Ì ̊ı Ú ̃ÂÌËÈ Û‰Ó·ÂÌ Ó„‡ÌË ̃ËÚÂÎ ̧ ̇ÍÎÓÌÓ‚, Ô‰ÎÓÊÂÌÌ ̊È ‚ [25].

21
Kolgan V.P. [Application of the minimum-derivative principle in the construction of finitedifference schemes for numerical analysis of discontinuous solutions in gas dynamics]. Uchenye zapiski TsAGI, 1972, vol. 3, no. 6, pp. 68{77. (in Russian).
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    ̄Ì ̨ ̨ ‰ËÌË ̃ÌÛ ̨ ÌÓχΠ̧ nil„‡ÌËlˇ ̃ÂÈÍËVi. œӂ‰ÂÏ ÔË·ÎËÊÂÌÌÛ ̨ ÎË̇ËÁ‡ˆË ̨ (7) ÔÓ ‚ÂÏÂÌË. ƒÎˇ ÏÓ‰ÂÎ ̧ÌÓ„Ó ËÌÚ„‡Î‡ ÒÚÓÎÍÌÓ‚ÂÌËÈ ·Û‰ÂÏ ÓÔÛÒÍ‡Ú ̧ ‰ÓÒÚ‡ÚÓ ̃ÌÓ ÒÎÓÊÌÛ ̨ Á‡‚ËÒËÏÓÒÚ ̧ χÍÓÒÍÓÔË ̃ÂÒÍËı ‚ÂÎË ̃ËÌ ÓÚfiÚ‡Í ̃ÚÓ Jn+1i≈Jni−νni∆fi. ‘ÓÏÛ· ‰Îˇ ̃ËÒÎÂÌÌÓ„Ó ÔÓÚÓ͇ (8) ÒÓ‰ÂÊËÚ ËÌÚ„‡Î ÓÚ ÙÛÌ͈ËË ‡ÒÔ‰ÂÎÂÌˡ ÔÓ „‡ÌË ˇ ̃ÂÈÍË. ‡Í Ó· ̊ ̃ÌÓ ‚ ÔÓÚË‚ÓÔÓÚÓ ̃Ì ̊ı ÒıÂχı ÚËÔ‡ √Ó‰ÛÌÓ‚‡
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    , ̇ ͇ʉÓÈ „‡ÌË ÂÒÚ ̧ ‰‚‡ Á̇ ̃ÂÌˡ  ̄ÂÌˡ. œÂ‚ÓÈ Á̇ ̃ÂÌËÂf−ÔÓÎÛ ̃‡ÂÚÒˇ ̋ÍÒÚ‡ÔÓΡˆËÂÈ Ì‡ „‡Ì ̧ ‚ ‡Ï͇ı ˇ ̃ÂÈÍËVi. ¬ÚÓÓ Á̇ ̃ÂÌËÂf+ÒÓÓÚ‚ÂÚÒÚ‚ÛÂÚ ̋ÍÒÚ‡ÔÓΡˆËË Ì‡ „‡ÌËAilÒ̇ÛÊË ˇ ̃ÂÈÍË. ◊ËÒÎÂÌÌ ̊È ÔÓÚÓÍ ̃ÂÂÁ „‡Ì ̧lˇ ̃ÂÈÍËVi‚ ̊ ̃ËÒΡÂÚÒˇ ÔÓ ÙÓÏÛΠΦnil=G(fnil,fnil,l1)|Ail|,(9) „‰Âil| ÌÓÏÂ ˇ ̃ÂÈÍË, ÒÓÒ‰ÌÂÈ Í „‡ÌËl(ÂÒÎË Ú‡Í‡ˇ ÒÛ ̆ÂÒÚ‚ÛÂÚ);l1| ÌÓÏÂ ÒÓÓÚ‚ÂÚÒÚ‚Û ̨ ̆ÂÈ „‡ÌË ˇ ̃ÂÈÍËi

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    ‡Í ËÁ‚ÂÒÚÌÓ, ÎËÌÂÈÌ ̊ ÒıÂÏ ̊ ‚ ̊ÒÓÍÓ„Ó ÔÓˇ‰Í‡ ÚÓ ̃ÌÓÒÚË ÌÂÏÓÌÓÚÓÌÌ ̊ ‚ ӷ·ÒÚˇı · ̊ÒÚÓ„Ó ËÁÏÂÌÂÌˡ  ̄ÂÌˡ [20]. ƒÎˇ ÔÓ‰‡‚ÎÂÌˡ Ô‡‡ÁËÚÌ ̊ı ÓÒˆËÎΡˆËÈ ‚ ·ÎÓÍ ÂÍÓÌÒÚÛ͈ËË ‚‚Ó‰ËÚÒˇ Ú‡Í Ì‡Á ̊‚‡ÂÏ ̊È Ó„‡ÌË ̃ËÚÂÎ ̧ ̇ÍÎÓÌÓ‚
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    . ƒÎˇ ‡Ò ̃ÂÚ‡ ÒÚ‡ˆËÓ̇Ì ̊ı Ú ̃ÂÌËÈ Û‰Ó·ÂÌ Ó„‡ÌË ̃ËÚÂÎ ̧ ̇ÍÎÓÌÓ‚, Ô‰ÎÓÊÂÌÌ ̊È ‚ [25]. »Á‚ÂÒÚÌÓ, ̃ÚÓ ÚÓ ̃ÌÓÒÚ ̧ ‡ÔÔÓÍÒËχˆËË ÏÂÚÓ‰Ó‚ ̇ËÏÂÌ ̧ ̄Ëı Í‚‡‰‡ÚÓ‚ Ô‡‰‡ÂÚ ‰Îˇ ÒËÎ ̧ÌÓ ‚ ̊ÚˇÌÛÚ ̊ı ˇ ̃ÂÂÍ [26].

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    ËÓ‚‡ÌÌ ̊È) ÒÔÓÒÓ· ‚ ̊ ̃ËÒÎÂÌˡ Á̇ ̃ÂÌËÈ Ì‡ „‡Ìˇı ˇ ̃ÂÂÍ [27]: fnil=fni+ψ1d(SnL,SnR)∆l,(14) „‰Â∆l| ‡ÒÒÚÓˇÌË ÓÚ ˆÂÌÚ‡ ˇ ̃ÂÈÍËi‰Ó ˆÂÌÚ‡ „‡ÌËl;SnLËSnR| ΂‡ˇ Ë Ô‡‚‡ˇ ÓˆÂÌÍË ÔÓËÁ‚Ó‰ÌÓÈ (̇ÍÎÓ̇)  ̄ÂÌˡ;ψ1d(a, b)| Ó‰ÌÓÏÂÌ ̊È Ó„‡ÌË ̃ËÚÂÎ ̧ ̇ÍÎÓÌÓ‚, ÔËÏÂÌˇÂÏ ̊È ÔÓÍÓÏÔÓÌÂÌÚÌÓ. ¬ ̇ÒÚÓˇ ̆ÂÏ ÏÂÚӉ ÏÓ„ÛÚ ËÒÔÓÎ ̧ÁÓ‚‡Ú ̧ ‡ÁÌ ̊ ӄ‡ÌË ̃ËÚÂÎË, ‚ÍÎ ̨ ̃‡ˇ Í·ÒÒË ̃ÂÒÍËÈ Ó„‡ÌË ̃ËÚÂÎ ̧ Ó΄‡Ì‡
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    (ÚËÔ‡ minmod), monotonized central (MC) Ó„‡ÌË ̃ËÚÂÎ ̧ [28], ‡ Ú‡ÍÊ ӄ‡ÌË ̃ËÚÂÎ ̧ ‚‡Ì ¿Î·‡‰‡ [22]. ƒ‡Î ‰Îˇ ‚ÒÂı „‡ÌÂÈ ˇ ̃ÂÂÍ ÒÂÚÍË, ÒÓÓÚ‚ÂÚÒÚ‚Û ̨ ̆Ëı „‡ÌË ̃ÌÓÏÛ ÛÒÎÓ‚Ë ̨, ÙÓÏÛ· ‰Îˇ ̃ËÒÎÂÌÌÓ„Ó ÔÓÚÓ͇ (9) ‚ˉÓËÁÏÂÌˇÂÚÒˇ Ú‡Í, ̃ÚÓ· ̊ ‡ÔÔÓÍÒËÏËÓ‚‡Ú ̧ „‡ÌË ̃ÌÓ ÛÒÎÓ‚ËÂ. œË ̋ÚÓÏ Ï‡ÍÓÒÍÓÔË ̃ÂÒÍË ԇ‡ÏÂÚ ̊, ‚ıÓ‰ˇ ̆Ë ‚ ÛÒÎÓ‚Ë ‰ËÙÙÛÁÌÓ„Ó ÓÚ‡ÊÂÌˡ (5), ‚ ̊ ̃ËÒΡ ̨ÚÒˇ Ú‡ÍËÏ Ó·‡ÁÓÏ, ̃ÚÓ

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Toro E.F.Riemann solvers and numerical methods for fluid dynamics. 3rded. Springer Berlin Heidelberg, 2009. 724 p. DOI:10.1007/b79761
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    ̄Ì ̨ ̨ ‰ËÌË ̃ÌÛ ̨ ÌÓχΠ̧ nil„‡ÌËlˇ ̃ÂÈÍËVi. œӂ‰ÂÏ ÔË·ÎËÊÂÌÌÛ ̨ ÎË̇ËÁ‡ˆË ̨ (7) ÔÓ ‚ÂÏÂÌË. ƒÎˇ ÏÓ‰ÂÎ ̧ÌÓ„Ó ËÌÚ„‡Î‡ ÒÚÓÎÍÌÓ‚ÂÌËÈ ·Û‰ÂÏ ÓÔÛÒÍ‡Ú ̧ ‰ÓÒÚ‡ÚÓ ̃ÌÓ ÒÎÓÊÌÛ ̨ Á‡‚ËÒËÏÓÒÚ ̧ χÍÓÒÍÓÔË ̃ÂÒÍËı ‚ÂÎË ̃ËÌ ÓÚfiÚ‡Í ̃ÚÓ Jn+1i≈Jni−νni∆fi. ‘ÓÏÛ· ‰Îˇ ̃ËÒÎÂÌÌÓ„Ó ÔÓÚÓ͇ (8) ÒÓ‰ÂÊËÚ ËÌÚ„‡Î ÓÚ ÙÛÌ͈ËË ‡ÒÔ‰ÂÎÂÌˡ ÔÓ „‡ÌË ˇ ̃ÂÈÍË. ‡Í Ó· ̊ ̃ÌÓ ‚ ÔÓÚË‚ÓÔÓÚÓ ̃Ì ̊ı ÒıÂχı ÚËÔ‡ √Ó‰ÛÌÓ‚‡
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    , ̇ ͇ʉÓÈ „‡ÌË ÂÒÚ ̧ ‰‚‡ Á̇ ̃ÂÌˡ  ̄ÂÌˡ. œÂ‚ÓÈ Á̇ ̃ÂÌËÂf−ÔÓÎÛ ̃‡ÂÚÒˇ ̋ÍÒÚ‡ÔÓΡˆËÂÈ Ì‡ „‡Ì ̧ ‚ ‡Ï͇ı ˇ ̃ÂÈÍËVi. ¬ÚÓÓ Á̇ ̃ÂÌËÂf+ÒÓÓÚ‚ÂÚÒÚ‚ÛÂÚ ̋ÍÒÚ‡ÔÓΡˆËË Ì‡ „‡ÌËAilÒ̇ÛÊË ˇ ̃ÂÈÍË. ◊ËÒÎÂÌÌ ̊È ÔÓÚÓÍ ̃ÂÂÁ „‡Ì ̧lˇ ̃ÂÈÍËVi‚ ̊ ̃ËÒΡÂÚÒˇ ÔÓ ÙÓÏÛΠΦnil=G(fnil,fnil,l1)|Ail|,(9) „‰Âil| ÌÓÏÂ ˇ ̃ÂÈÍË, ÒÓÒ‰ÌÂÈ Í „‡ÌËl(ÂÒÎË Ú‡Í‡ˇ ÒÛ ̆ÂÒÚ‚ÛÂÚ);l1| ÌÓÏÂ ÒÓÓÚ‚ÂÚÒÚ‚Û ̨ ̆ÂÈ „‡ÌË ˇ ̃ÂÈÍËi

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    ¬ ̇ÒÚÓˇ ̆ÂÏ ÏÂÚӉ ÏÓ„ÛÚ ËÒÔÓÎ ̧ÁÓ‚‡Ú ̧ ‡ÁÌ ̊ ӄ‡ÌË ̃ËÚÂÎË, ‚ÍÎ ̨ ̃‡ˇ Í·ÒÒË ̃ÂÒÍËÈ Ó„‡ÌË ̃ËÚÂÎ ̧ Ó΄‡Ì‡ [21] (ÚËÔ‡ minmod), monotonized central (MC) Ó„‡ÌË ̃ËÚÂÎ ̧ [28], ‡ Ú‡ÍÊ ӄ‡ÌË ̃ËÚÂÎ ̧ ‚‡Ì ¿Î·‡‰‡
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    . ƒ‡Î ‰Îˇ ‚ÒÂı „‡ÌÂÈ ˇ ̃ÂÂÍ ÒÂÚÍË, ÒÓÓÚ‚ÂÚÒÚ‚Û ̨ ̆Ëı „‡ÌË ̃ÌÓÏÛ ÛÒÎÓ‚Ë ̨, ÙÓÏÛ· ‰Îˇ ̃ËÒÎÂÌÌÓ„Ó ÔÓÚÓ͇ (9) ‚ˉÓËÁÏÂÌˇÂÚÒˇ Ú‡Í, ̃ÚÓ· ̊ ‡ÔÔÓÍÒËÏËÓ‚‡Ú ̧ „‡ÌË ̃ÌÓ ÛÒÎÓ‚ËÂ. œË ̋ÚÓÏ Ï‡ÍÓÒÍÓÔË ̃ÂÒÍË ԇ‡ÏÂÚ ̊, ‚ıÓ‰ˇ ̆Ë ‚ ÛÒÎÓ‚Ë ‰ËÙÙÛÁÌÓ„Ó ÓÚ‡ÊÂÌˡ (5), ‚ ̊ ̃ËÒΡ ̨ÚÒˇ Ú‡ÍËÏ Ó·‡ÁÓÏ, ̃ÚÓ ÛÒÎÓ‚Ë ÌÂÔÓÚÂ͇Ìˡ ‚ ̊ÔÓÎÌˇÂÚÒˇ ÚÓʉÂÒÚ‚ÂÌÌÓ.

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Dumbser M., Kaser M. Arbitrary high order non-oscillatory finite volume schemes on unstructured meshes for linear hyperbolic systems.Journal of Computational Physics, 2007, vol. 221, no. 2, pp. 693{723. DOI:10.1016/j.jcp.2006.06.043
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    –ËÒ. 1.ÿ‡·ÎÓÌ ÏÂÚÓ‰‡ ̇ËÏÂÌ ̧ ̄Ëı Í‚‡‰‡ÚÓ‚ ‚ ÙËÁË ̃ÂÒÍÓÈ (Ò΂‡) Ë ÎÓ͇Π̧ÌÓÈ (ÒÔ‡‚‡) ÒËÒÚÂχı ÍÓÓ‰ËÌ‡Ú ‰Îˇ „ÂÍÒ‡ ̋‰‡Î ̧ÌÓÈ ÒÂÚÍË œÓ‰Ó·ÌÓ ÓÔËÒ‡ÌË Ôӈ‰Û ̊ ÔÓÒÚÓÂÌˡ ÍÛÒÓ ̃ÌÓ-ÔÓÎËÌÓÏˇΠ̧ÌÓ„Ó Ô‰ÒÚ‡‚ÎÂÌˡ  ̄ÂÌˡ ÓÔËÒ‡ÌÓ ‚
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    ‰Îˇ ÚÂÚ‡ ̋‰‡Î ̧Ì ̊ı ˇ ̃ÂÂÍ Ë ‚ [16,24] ‰Îˇ ÚÂıÏÂÌ ̊ı ˇ ̃ÂÂÍ ÔÓËÁ‚ÓÎ ̧ÌÓÈ ÙÓÏ ̊. ƒÎˇ ÎËÌÂÈÌÓÈ ÒıÂÏ ̊ ÏÓÊÌÓ ÔÓÎÓÊËÚ ̧ Ò‰ÌË Á̇ ̃ÂÌˡ ÙÛÌ͈ËË ‡ÒÔ‰ÂÎÂÌˡfnilÔÓ Í‡Ê‰ÓÈ „‡ÌËlˇ ̃ÂÈÍË ‡‚Ì ̊ÏË Ò‰ÌËÏ Á̇ ̃ÂÌˡÏpnilÔÓÎÛ ̃ÂÌÌÓ„Ó ÂÍÓÌÒÚÛ͈ËÓÌÌÓ„Ó ÏÌÓ„Ó ̃ÎÂ̇pni(x). œË ̋ÚÓÏpnilÔ‰ÒÚ‡‚Ρ ̨ÚÒˇ ‚ ‚ˉ ÎËÌÂÈÌÓÈ ÍÓÏ·Ë̇ˆËË Á̇ ̃ÂÌËÈ ÙÛÌ͈ËË ‡ÒÔ‰ÂÎÂÌˡ ‚ ˇ ̃ÂÈ͇ı ̄‡·ÎÓ̇, ÍÓ ̋ÙÙˈËÂÌÚ ̊ ÍÓÚ

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Tsoutsanis P., Titarev V.A., Drikakis D. WENO schemes on arbitrary mixed-element unstructured meshes in three space dimensions.Journal of Computational Physics, 2010, vol. 230, no. 4, pp. 1585{1601. DOI:10.1016/j.jcp.2010.11.023
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    –ËÒ. 1.ÿ‡·ÎÓÌ ÏÂÚÓ‰‡ ̇ËÏÂÌ ̧ ̄Ëı Í‚‡‰‡ÚÓ‚ ‚ ÙËÁË ̃ÂÒÍÓÈ (Ò΂‡) Ë ÎÓ͇Π̧ÌÓÈ (ÒÔ‡‚‡) ÒËÒÚÂχı ÍÓÓ‰ËÌ‡Ú ‰Îˇ „ÂÍÒ‡ ̋‰‡Î ̧ÌÓÈ ÒÂÚÍË œÓ‰Ó·ÌÓ ÓÔËÒ‡ÌË Ôӈ‰Û ̊ ÔÓÒÚÓÂÌˡ ÍÛÒÓ ̃ÌÓ-ÔÓÎËÌÓÏˇΠ̧ÌÓ„Ó Ô‰ÒÚ‡‚ÎÂÌˡ  ̄ÂÌˡ ÓÔËÒ‡ÌÓ ‚ [23] ‰Îˇ ÚÂÚ‡ ̋‰‡Î ̧Ì ̊ı ˇ ̃ÂÂÍ Ë ‚
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    ‰Îˇ ÚÂıÏÂÌ ̊ı ˇ ̃ÂÂÍ ÔÓËÁ‚ÓÎ ̧ÌÓÈ ÙÓÏ ̊. ƒÎˇ ÎËÌÂÈÌÓÈ ÒıÂÏ ̊ ÏÓÊÌÓ ÔÓÎÓÊËÚ ̧ Ò‰ÌË Á̇ ̃ÂÌˡ ÙÛÌ͈ËË ‡ÒÔ‰ÂÎÂÌˡfnilÔÓ Í‡Ê‰ÓÈ „‡ÌËlˇ ̃ÂÈÍË ‡‚Ì ̊ÏË Ò‰ÌËÏ Á̇ ̃ÂÌˡÏpnilÔÓÎÛ ̃ÂÌÌÓ„Ó ÂÍÓÌÒÚÛ͈ËÓÌÌÓ„Ó ÏÌÓ„Ó ̃ÎÂ̇pni(x). œË ̋ÚÓÏpnilÔ‰ÒÚ‡‚Ρ ̨ÚÒˇ ‚ ‚ˉ ÎËÌÂÈÌÓÈ ÍÓÏ·Ë̇ˆËË Á̇ ̃ÂÌËÈ ÙÛÌ͈ËË ‡ÒÔ‰ÂÎÂÌˡ ‚ ˇ ̃ÂÈ͇ı ̄‡·ÎÓ̇, ÍÓ ̋ÙÙˈËÂÌÚ ̊ ÍÓÚÓÓÈ ÓÔ‰ÂΡ ̨ÚÒˇ ÚÓÎ ̧ÍÓ „ÂÓÏÂÚËÂÈ ÔÓ

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Venkatakrishnan V. On the accuracy of limiters and convergence to steady-state solutions. 31stAerospace Science Meeting and Exhibit, Reno, NV, January 11{14, 1993. AIAA paper 93-0880. DOI:10.2514/6.1993-880
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    ‡Í ËÁ‚ÂÒÚÌÓ, ÎËÌÂÈÌ ̊ ÒıÂÏ ̊ ‚ ̊ÒÓÍÓ„Ó ÔÓˇ‰Í‡ ÚÓ ̃ÌÓÒÚË ÌÂÏÓÌÓÚÓÌÌ ̊ ‚ ӷ·ÒÚˇı · ̊ÒÚÓ„Ó ËÁÏÂÌÂÌˡ  ̄ÂÌˡ [20]. ƒÎˇ ÔÓ‰‡‚ÎÂÌˡ Ô‡‡ÁËÚÌ ̊ı ÓÒˆËÎΡˆËÈ ‚ ·ÎÓÍ ÂÍÓÌÒÚÛ͈ËË ‚‚Ó‰ËÚÒˇ Ú‡Í Ì‡Á ̊‚‡ÂÏ ̊È Ó„‡ÌË ̃ËÚÂÎ ̧ ̇ÍÎÓÌÓ‚ [21]. ƒÎˇ ‡Ò ̃ÂÚ‡ ÒÚ‡ˆËÓ̇Ì ̊ı Ú ̃ÂÌËÈ Û‰Ó·ÂÌ Ó„‡ÌË ̃ËÚÂÎ ̧ ̇ÍÎÓÌÓ‚, Ô‰ÎÓÊÂÌÌ ̊È ‚
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    . »Á‚ÂÒÚÌÓ, ̃ÚÓ ÚÓ ̃ÌÓÒÚ ̧ ‡ÔÔÓÍÒËχˆËË ÏÂÚÓ‰Ó‚ ̇ËÏÂÌ ̧ ̄Ëı Í‚‡‰‡ÚÓ‚ Ô‡‰‡ÂÚ ‰Îˇ ÒËÎ ̧ÌÓ ‚ ̊ÚˇÌÛÚ ̊ı ˇ ̃ÂÂÍ [26]. “‡ÍÊÂ, ‰Îˇ ÌÂÍÓÚÓ ̊ı Í·ÒÒÓ‚ Á‡‰‡ ̃, ̇ÔËÏÂ, ÔË ‡Ò ̃ÂÚ Ú ̃ÂÌËÈ ‚ ‰ÎËÌÌ ̊ı ͇̇·ı, Ú ̃ÂÌË ÏÓÊÂÚ ËÏÂÚ ̧ ‚ ̊‰ÂÎÂÌÌÓ ̇Ô‡‚ÎÂÌËÂ, „‡‰ËÂÌÚ ̊ ‚‰ÓÎ ̧ ÍÓÚÓÓ„Ó Á̇ ̃ËÚÂÎ ̧ÌÓ ÏÂÌ ̧ ̄ ÔÓÔÂ ̃Ì ̊ı ÔÓËÁ‚Ó‰Ì ̊ı, ̃ÚÓ Ì ËÒÔÓÎ ̧ÁÛÂÚÒˇ ‚ ÔÓÎÌÓÒÚ ̧ ̨ ÚÂıÏÂÌÓÏ ÏÂÚÓ‰Â. œÓ ̋ÚÓÏÛ ‰Îˇ „

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Petrovskaya N.B., Volkov A.V. [The impact of grid geometry on the accuracy of higher order finite-volume and finite-element schemes].Matematicheskoe modelirovanie, 2010, vol. 22, no. 3, pp. 145{160. (in Russian).
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    ̊ÒÓÍÓ„Ó ÔÓˇ‰Í‡ ÚÓ ̃ÌÓÒÚË ÌÂÏÓÌÓÚÓÌÌ ̊ ‚ ӷ·ÒÚˇı · ̊ÒÚÓ„Ó ËÁÏÂÌÂÌˡ  ̄ÂÌˡ [20]. ƒÎˇ ÔÓ‰‡‚ÎÂÌˡ Ô‡‡ÁËÚÌ ̊ı ÓÒˆËÎΡˆËÈ ‚ ·ÎÓÍ ÂÍÓÌÒÚÛ͈ËË ‚‚Ó‰ËÚÒˇ Ú‡Í Ì‡Á ̊‚‡ÂÏ ̊È Ó„‡ÌË ̃ËÚÂÎ ̧ ̇ÍÎÓÌÓ‚ [21]. ƒÎˇ ‡Ò ̃ÂÚ‡ ÒÚ‡ˆËÓ̇Ì ̊ı Ú ̃ÂÌËÈ Û‰Ó·ÂÌ Ó„‡ÌË ̃ËÚÂÎ ̧ ̇ÍÎÓÌÓ‚, Ô‰ÎÓÊÂÌÌ ̊È ‚ [25]. »Á‚ÂÒÚÌÓ, ̃ÚÓ ÚÓ ̃ÌÓÒÚ ̧ ‡ÔÔÓÍÒËχˆËË ÏÂÚÓ‰Ó‚ ̇ËÏÂÌ ̧ ̄Ëı Í‚‡‰‡ÚÓ‚ Ô‡‰‡ÂÚ ‰Îˇ ÒËÎ ̧ÌÓ ‚ ̊ÚˇÌÛÚ ̊ı ˇ ̃ÂÂÍ
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    [26]
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    . “‡ÍÊÂ, ‰Îˇ ÌÂÍÓÚÓ ̊ı Í·ÒÒÓ‚ Á‡‰‡ ̃, ̇ÔËÏÂ, ÔË ‡Ò ̃ÂÚ Ú ̃ÂÌËÈ ‚ ‰ÎËÌÌ ̊ı ͇̇·ı, Ú ̃ÂÌË ÏÓÊÂÚ ËÏÂÚ ̧ ‚ ̊‰ÂÎÂÌÌÓ ̇Ô‡‚ÎÂÌËÂ, „‡‰ËÂÌÚ ̊ ‚‰ÓÎ ̧ ÍÓÚÓÓ„Ó Á̇ ̃ËÚÂÎ ̧ÌÓ ÏÂÌ ̧ ̄ ÔÓÔÂ ̃Ì ̊ı ÔÓËÁ‚Ó‰Ì ̊ı, ̃ÚÓ Ì ËÒÔÓÎ ̧ÁÛÂÚÒˇ ‚ ÔÓÎÌÓÒÚ ̧ ̨ ÚÂıÏÂÌÓÏ ÏÂÚÓ‰Â. œÓ ̋ÚÓÏÛ ‰Îˇ „ÂÍÒ‡ ̋‰‡Î ̧Ì ̊ı ˇ ̃ÂÂÍ Ô‰ÛÒÏÓÚÂ̇ ‚ÓÁÏÓÊÌÓÒÚ ̧ ÔÂÂÍÎ ̨ ̃ÂÌˡ ̇ ÎÓ͇Π̧ÌÓ-Ó‰ÌÓÏÂÌ ̊È (ÒÚÛÍÚÛËÓ‚‡ÌÌ ̊È) ÒÔÓ

27
Titarev V.A., Shakhov E.M. Computational study of a rarefied gas flow through a long circular pipe into vacuum.Vacuum. Spec. iss. \Vacuum Gas Dynamics. Theory, experiments and practical applications", 2012, vol. 86, no. 11, pp. 1709{1716. DOI:10.1016/j.vacuum.2012.02.026
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  1. In-text reference with the coordinate start=16618
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    , ÔË ‡Ò ̃ÂÚ Ú ̃ÂÌËÈ ‚ ‰ÎËÌÌ ̊ı ͇̇·ı, Ú ̃ÂÌË ÏÓÊÂÚ ËÏÂÚ ̧ ‚ ̊‰ÂÎÂÌÌÓ ̇Ô‡‚ÎÂÌËÂ, „‡‰ËÂÌÚ ̊ ‚‰ÓÎ ̧ ÍÓÚÓÓ„Ó Á̇ ̃ËÚÂÎ ̧ÌÓ ÏÂÌ ̧ ̄ ÔÓÔÂ ̃Ì ̊ı ÔÓËÁ‚Ó‰Ì ̊ı, ̃ÚÓ Ì ËÒÔÓÎ ̧ÁÛÂÚÒˇ ‚ ÔÓÎÌÓÒÚ ̧ ̨ ÚÂıÏÂÌÓÏ ÏÂÚÓ‰Â. œÓ ̋ÚÓÏÛ ‰Îˇ „ÂÍÒ‡ ̋‰‡Î ̧Ì ̊ı ˇ ̃ÂÂÍ Ô‰ÛÒÏÓÚÂ̇ ‚ÓÁÏÓÊÌÓÒÚ ̧ ÔÂÂÍÎ ̨ ̃ÂÌˡ ̇ ÎÓ͇Π̧ÌÓ-Ó‰ÌÓÏÂÌ ̊È (ÒÚÛÍÚÛËÓ‚‡ÌÌ ̊È) ÒÔÓÒÓ· ‚ ̊ ̃ËÒÎÂÌˡ Á̇ ̃ÂÌËÈ Ì‡ „‡Ìˇı ˇ ̃ÂÂÍ
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    [27]
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    : fnil=fni+ψ1d(SnL,SnR)∆l,(14) „‰Â∆l| ‡ÒÒÚÓˇÌË ÓÚ ˆÂÌÚ‡ ˇ ̃ÂÈÍËi‰Ó ˆÂÌÚ‡ „‡ÌËl;SnLËSnR| ΂‡ˇ Ë Ô‡‚‡ˇ ÓˆÂÌÍË ÔÓËÁ‚Ó‰ÌÓÈ (̇ÍÎÓ̇)  ̄ÂÌˡ;ψ1d(a, b)| Ó‰ÌÓÏÂÌ ̊È Ó„‡ÌË ̃ËÚÂÎ ̧ ̇ÍÎÓÌÓ‚, ÔËÏÂÌˇÂÏ ̊È ÔÓÍÓÏÔÓÌÂÌÚÌÓ.

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    ¬ ̉‡‚ÌÂÈ ‡·ÓÚ [41] ËÒÚ ̃ÂÌË „‡Á‡ ̃ÂÂÁ ÍÓÓÚÍÛ ̨ ÍÛ„ÎÛ ̨ ÚÛ·Û Ô‰ÎÓÊÂÌÓ ‚ ‚ˉ ӉÌÓÈ ËÁ Òڇ̉‡ÚÌ ̊ı ‚ÂËÙË͇ˆËÓÌÌ ̊ı Á‡‰‡ ̃, ̇ ÍÓÚÓÓÈ ÏÓÊÌÓ Ò‡‚ÌË‚‡Ú ̧ ÂÁÛÎ ̧Ú‡Ú ̊ Ò ̃ÂÚ‡ ÔÓ ‡ÁÌ ̊Ï ÏÂÚÓ‰‡Ï Ë ÍÓÏÔÎÂÍÒ‡Ï ÔÓ„‡ÏÏ. ƒÂڇΠ̧̇ˇ ÔÓÒÚ‡Ìӂ͇ Á‡‰‡ ̃Ë Ó Ú ̃ÂÌËË „‡Á‡ ̃ÂÂÁ ÚÛ·Û ÍÓÌ ̃ÌÓÈ ‰ÎËÌ ̊ ËÁÎÓÊÂ̇ ‚ ‡·ÓÚ‡ı
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    [27,40,43]
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    . –‡ÒÒχÚË‚‡ÂÚÒˇ ÒÚ‡ˆËÓ̇ÌÓ Ú ̃ÂÌË ‡ÁÂÊÂÌÌÓ„Ó „‡Á‡ ËÁ ÂÁÂ‚Û‡‡ 1‚ ÂÁÂ‚Û‡2 ̃ÂÂÁ ÚÛ·Û ‰ÎËÌ ̊LË ÍÛ„Ó‚Ó„Ó ÔÓÔÂ ̃ÌÓ„Ó Ò ̃ÂÌˡ Ò ‡‰ËÛÒÓÏ ‚ıÓ‰ÌÓ„Ó Ú ̃ÂÌˡa. ¬ Ó· ̆ÂÏ ÒÎÛ ̃‡Â ‡‰ËÛÒ ÚÛ· ̊ ÏÓÊÂÚ ÏÂÌˇÚ ̧Òˇ ÔÓ ‰ÎËÌÂ.

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    . œË ‡Ì‡ÎËÁ ÂÁÛÎ ̧Ú‡ÚÓ‚ Û‰Ó·Ì ÔÓÎ ̧ÁÓ‚‡Ú ̧Òˇ ‚ÂÎË ̃ËÌÓÈQ, Ô‰ÒÚ‡‚Ρ ̨ ̆ÂÈ ÒÓ·ÓÈ ‡ÒıÓ‰ ̇M„‡Á‡ ̃ÂÂÁ Ò ̃ÂÌË ÚÛ· ̊, ÌÓÏËÓ‚‡ÌÌ ̊È Ì‡ ‚ÂÎË ̃ËÌÛ ‡ÒıÓ‰‡ ̇M отв 0ÔË Ò‚Ó·Ó‰ÌÓ-ÏÓÎÂÍÛΡÌÓÏ ËÒÚ ̃ÂÌËË ‚ ‚‡ÍÛÛÏ ̃ÂÂÁ ÓÚ‚ÂÒÚË [44,45]: Q= ̇M / ̇M отв 0, M ̇отв 0= √ π/2.(22) ŒÒÌÓ‚Ì ̊Ï ˇ‚ΡÂÚÒˇ ÒÎÛ ̃‡È ÍÛ„ÎÓÈ ÚÛ· ̊ ÔÓÒÚÓˇÌÌÓ„Ó ‡‰ËÛÒ‡, ‡‚ÌÓ„Ó ‡‰ËÛÒÛ ‚ıÓ‰ÌÓ„Ó Ò ̃ÂÌˡa. ¬ ˆËÍΠ‡·ÓÚ
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    [17,27,43,46]
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     ̄ÂÌË Á‡‰‡ ̃Ë · ̊ÎÓ ÔÓÒÚÓÂÌÓ Ô‡ÍÂÚÓÏ <ÕÂÒ‚ÂÚ‡È-3ƒ> ‰Îˇ χÎÓ„Ó ÓÚÌÓ ̄ÂÌˡ ‰‡‚ÎÂÌËÈ|p10/p20|−1-1(ÎË̇ËÁÓ‚‡Ì̇ˇ Á‡‰‡ ̃‡), ‡ Ú‡ÍÊ ‰‚Ûı ÌÂÎËÌÂÈÌ ̊ı ÂÊËÏÓ‚p10/p20= 2(ÛÏÂÂÌÌ ̊È ÔÂÂÔ‡‰ ‰‡‚ÎÂÌËÈ) Ëp10/p20=∞ (Ú ̃ÂÌË ‚ ‚‡ÍÛÛÏ).

28
Leer B. van. Towards the ultimate conservative difference scheme V. A second order sequel to Godunov's method.Journal of Computational Physics, 1979, vol. 32, iss. 1, pp. 101{136. DOI:10.1016/0021-9991(79)90145-1
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  1. In-text reference with the coordinate start=17011
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    [27]: fnil=fni+ψ1d(SnL,SnR)∆l,(14) „‰Â∆l| ‡ÒÒÚÓˇÌË ÓÚ ˆÂÌÚ‡ ˇ ̃ÂÈÍËi‰Ó ˆÂÌÚ‡ „‡ÌËl;SnLËSnR| ΂‡ˇ Ë Ô‡‚‡ˇ ÓˆÂÌÍË ÔÓËÁ‚Ó‰ÌÓÈ (̇ÍÎÓ̇)  ̄ÂÌˡ;ψ1d(a, b)| Ó‰ÌÓÏÂÌ ̊È Ó„‡ÌË ̃ËÚÂÎ ̧ ̇ÍÎÓÌÓ‚, ÔËÏÂÌˇÂÏ ̊È ÔÓÍÓÏÔÓÌÂÌÚÌÓ. ¬ ̇ÒÚÓˇ ̆ÂÏ ÏÂÚӉ ÏÓ„ÛÚ ËÒÔÓÎ ̧ÁÓ‚‡Ú ̧ ‡ÁÌ ̊ ӄ‡ÌË ̃ËÚÂÎË, ‚ÍÎ ̨ ̃‡ˇ Í·ÒÒË ̃ÂÒÍËÈ Ó„‡ÌË ̃ËÚÂÎ ̧ Ó΄‡Ì‡ [21] (ÚËÔ‡ minmod), monotonized central (MC) Ó„‡ÌË ̃ËÚÂÎ ̧
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    [28]
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    , ‡ Ú‡ÍÊ ӄ‡ÌË ̃ËÚÂÎ ̧ ‚‡Ì ¿Î·‡‰‡ [22]. ƒ‡Î ‰Îˇ ‚ÒÂı „‡ÌÂÈ ˇ ̃ÂÂÍ ÒÂÚÍË, ÒÓÓÚ‚ÂÚÒÚ‚Û ̨ ̆Ëı „‡ÌË ̃ÌÓÏÛ ÛÒÎÓ‚Ë ̨, ÙÓÏÛ· ‰Îˇ ̃ËÒÎÂÌÌÓ„Ó ÔÓÚÓ͇ (9) ‚ˉÓËÁÏÂÌˇÂÚÒˇ Ú‡Í, ̃ÚÓ· ̊ ‡ÔÔÓÍÒËÏËÓ‚‡Ú ̧ „‡ÌË ̃ÌÓ ÛÒÎÓ‚ËÂ. œË ̋ÚÓÏ Ï‡ÍÓÒÍÓÔË ̃ÂÒÍË ԇ‡ÏÂÚ ̊, ‚ıÓ‰ˇ ̆Ë ‚ ÛÒÎÓ‚Ë ‰ËÙÙÛÁÌÓ„Ó ÓÚ‡ÊÂÌˡ (5), ‚ ̊ ̃ËÒΡ ̨ÚÒˇ Ú‡ÍËÏ Ó·‡ÁÓÏ, ̃ÚÓ ÛÒÎÓ‚Ë ÌÂÔÓÚÂ͇Ìˡ ‚ ̊ÔÓÎÌˇÂÚÒˇ ÚÓʉÂÒÚ‚ÂÌÌÓ.

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Titarev V.A. Conservative numerical methods for model kinetic equations.Computers and Fluids, 2007, vol. 36, no. 9, pp. 1446{1459.
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  1. In-text reference with the coordinate start=18049
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    ̨ÚÒˇ Í‡Í ËÌÚ„‡Î ̊ ÓÚ ÙÛÌ͈ËË ‡ÒÔ‰ÂÎÂÌˡ ÔÓ ÏÓÎÂÍÛΡÌÓÈ ÒÍÓÓÒÚË (4). œˇÏ‡ˇ ‡ÔÔÓÍÒËχˆËˇ ‚ ̊‡ÊÂÌËÈ (4) ÔÓÒ‰ÒÚ‚ÓÏ Í‚‡‰‡ÚÛÌÓÈ ÙÓÏÛÎ ̊ Ò ‚ÂÒ‡ÏËωj(Ë̉ÂÍÒ ̊iËn ÓÔÛ ̆ÂÌ ̊ ‰Îˇ ÔÓÒÚÓÚ ̊) eve        n nu 3 2 nT+nu2 q        = ∑Nξ j=1        1 ξj ξ2j 1 2 vjv2j        fjωj(15) ÔË‚Ó‰ËÚ Í Ì‡Û ̄ÂÌË ̨ ‚ ̃ËÒÎÂÌÌÓÈ ÒıÂÏ Á‡ÍÓÌÓ‚ ÒÓı‡ÌÂÌˡ χÒÒ ̊, ËÏÔÛÎ ̧Ò‡ Ë ̋ÌÂ„ËË
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    [29]
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    . œÓˇ‚Ρ ̨ ̆ËÈÒˇ ÔË ̃ËÒÎÂÌÌÓÏ Ò ̃ÂÚ ԇ‡ÁËÚÌ ̊È ËÒÚÓ ̃ÌËÍ ÔˇÏÓ ÔÓÔÓˆËÓ̇ÎÂÌ Ô‡‡ÏÂÚÛ ‡ÁÂÊÂÌÌÓÒÚËδ. ¬ ÂÁÛÎ ̧Ú‡Ú ÏÓ‰ÂÎËÓ‚‡ÌË ÔÂÂıÓ‰Ì ̊ı Ë ÒÔÎÓ ̄ÌÓÒÂ‰Ì ̊ı Ú ̃ÂÌËÈ „‡Á‡ ÏÓÊÂÚ Ú·ӂ‡Ú ̧ ÌÂÔËÂÏÎÂÏÓ ·ÓÎ ̧ ̄Ó„Ó ̃ËÒ· ÛÁÎÓ‚Nξ‚ ÒÍÓÓÒÚÌÓÈ ÒÂÚÍÂ.

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    ÓÓÚÍËÈ Ó·ÁÓ ÒÛ ̆ÂÒÚ‚Û ̨ ̆Ëı ÒÔÓÒÓ·Ó‚ ÔÓÒÚÓÂÌˡ ÍÓÌÒÂ‚‡ÚË‚Ì ̊ı ÒıÂÏ ‰Îˇ ÏÓ‰ÂÎ ̧ÌÓ„Ó ÍËÌÂÚË ̃ÂÒÍÓ„Ó Û‡‚ÌÂÌˡ Ô˂‰ÂÌ ‚ [17]. ¬ ̇ÒÚÓˇ ̆ÂÈ ‡·ÓÚ ËÒÔÓÎ ̧ÁÛÂÚÒˇ ÓÚÌÓÒËÚÂÎ ̧ÌÓ ÌÓ‚ ̊È ÏÂÚÓ‰ ÔÓÒÚÓÂÌˡ ÍÓÌÒÂ‚‡ÚË‚ÌÓÈ ‡ÔÔÓÍÒËχˆËË ËÌÚ„‡Î‡ ÒÚÓÎÍÌÓ‚ÂÌËÈ
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    [29,30,31]
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    . ÇÍÓÔ‡‡ÏÂÚ ̊ „‡Á‡ ̇ıÓ‰ˇÚÒˇ Í‡Í  ̄ÂÌË ÒËÒÚÂÏ ̊ Û‡‚ÌÂÌËÈ, ËÒÔÓÎ ̧ÁÛÂÏÓÈ ‰Îˇ ÔÓÒÚÓÂÌˡS-ÏÓ‰ÂÎ ̧ÌÓ„Ó Û‡‚ÌÂÌˡ: H(W) = ∑Nξ j=1        1 ξj ξ2j vjv2j        (f (S) j−fj)ωj+        0 0 0 2 Prq        =0,W=        n u T q        .(16) ƒÎˇ ͇ʉÓÈ ÔÓÒÚ‡ÌÒÚ‚ÂÌÌÓÈ ˇ ̃ÂÈÍËiÒËÒÚÂχ ËÁ ‚ÓÒ ̧ÏË Û‡‚ÌÂÌËÈ (16)  ̄‡ÂÚÒˇ ÏÂÚÓ‰ÓÏ Õ ̧ ̨ÚÓ̇ M(Ws−1)(Ws−Ws−1) =−H(

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Titarev V.A., Shakhov E.M. [Numerical Calculation of the Hypersonic Rarefied Transverse Flow Past a Cold Flat Plate].Izvestiya RAN. Mekhanika Zhidkosti i Gaza, 2005, no. 5, pp. 139{154. (English translation:Fluid Dynamics, 2005, vol. 40, iss. 5, pp. 790{804. DOI: 10.1007/s10697-005-0117-1).
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    ÓÓÚÍËÈ Ó·ÁÓ ÒÛ ̆ÂÒÚ‚Û ̨ ̆Ëı ÒÔÓÒÓ·Ó‚ ÔÓÒÚÓÂÌˡ ÍÓÌÒÂ‚‡ÚË‚Ì ̊ı ÒıÂÏ ‰Îˇ ÏÓ‰ÂÎ ̧ÌÓ„Ó ÍËÌÂÚË ̃ÂÒÍÓ„Ó Û‡‚ÌÂÌˡ Ô˂‰ÂÌ ‚ [17]. ¬ ̇ÒÚÓˇ ̆ÂÈ ‡·ÓÚ ËÒÔÓÎ ̧ÁÛÂÚÒˇ ÓÚÌÓÒËÚÂÎ ̧ÌÓ ÌÓ‚ ̊È ÏÂÚÓ‰ ÔÓÒÚÓÂÌˡ ÍÓÌÒÂ‚‡ÚË‚ÌÓÈ ‡ÔÔÓÍÒËχˆËË ËÌÚ„‡Î‡ ÒÚÓÎÍÌÓ‚ÂÌËÈ
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    [29,30,31]
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    . ÇÍÓÔ‡‡ÏÂÚ ̊ „‡Á‡ ̇ıÓ‰ˇÚÒˇ Í‡Í  ̄ÂÌË ÒËÒÚÂÏ ̊ Û‡‚ÌÂÌËÈ, ËÒÔÓÎ ̧ÁÛÂÏÓÈ ‰Îˇ ÔÓÒÚÓÂÌˡS-ÏÓ‰ÂÎ ̧ÌÓ„Ó Û‡‚ÌÂÌˡ: H(W) = ∑Nξ j=1        1 ξj ξ2j vjv2j        (f (S) j−fj)ωj+        0 0 0 2 Prq        =0,W=        n u T q        .(16) ƒÎˇ ͇ʉÓÈ ÔÓÒÚ‡ÌÒÚ‚ÂÌÌÓÈ ˇ ̃ÂÈÍËiÒËÒÚÂχ ËÁ ‚ÓÒ ̧ÏË Û‡‚ÌÂÌËÈ (16)  ̄‡ÂÚÒˇ ÏÂÚÓ‰ÓÏ Õ ̧ ̨ÚÓ̇ M(Ws−1)(Ws−Ws−1) =−H(

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Titarev V.A. [Numerical method for computing two-dimensional unsteady rarefied gas flows in arbitrarily shaped domains].Zhurnal vychislitel'noy matematiki i matematicheskoy fiziki, 2009, vol. 49, no. 7, pp. 1255{1270. (English translation:Computational Mathematics and Mathematical Physics, 2009, vol. 49, iss. 7, pp. 1197{1211. DOI:10.1134/S0965542509070112).
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    ÓÓÚÍËÈ Ó·ÁÓ ÒÛ ̆ÂÒÚ‚Û ̨ ̆Ëı ÒÔÓÒÓ·Ó‚ ÔÓÒÚÓÂÌˡ ÍÓÌÒÂ‚‡ÚË‚Ì ̊ı ÒıÂÏ ‰Îˇ ÏÓ‰ÂÎ ̧ÌÓ„Ó ÍËÌÂÚË ̃ÂÒÍÓ„Ó Û‡‚ÌÂÌˡ Ô˂‰ÂÌ ‚ [17]. ¬ ̇ÒÚÓˇ ̆ÂÈ ‡·ÓÚ ËÒÔÓÎ ̧ÁÛÂÚÒˇ ÓÚÌÓÒËÚÂÎ ̧ÌÓ ÌÓ‚ ̊È ÏÂÚÓ‰ ÔÓÒÚÓÂÌˡ ÍÓÌÒÂ‚‡ÚË‚ÌÓÈ ‡ÔÔÓÍÒËχˆËË ËÌÚ„‡Î‡ ÒÚÓÎÍÌÓ‚ÂÌËÈ
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    . ÇÍÓÔ‡‡ÏÂÚ ̊ „‡Á‡ ̇ıÓ‰ˇÚÒˇ Í‡Í  ̄ÂÌË ÒËÒÚÂÏ ̊ Û‡‚ÌÂÌËÈ, ËÒÔÓÎ ̧ÁÛÂÏÓÈ ‰Îˇ ÔÓÒÚÓÂÌˡS-ÏÓ‰ÂÎ ̧ÌÓ„Ó Û‡‚ÌÂÌˡ: H(W) = ∑Nξ j=1        1 ξj ξ2j vjv2j        (f (S) j−fj)ωj+        0 0 0 2 Prq        =0,W=        n u T q        .(16) ƒÎˇ ͇ʉÓÈ ÔÓÒÚ‡ÌÒÚ‚ÂÌÌÓÈ ˇ ̃ÂÈÍËiÒËÒÚÂχ ËÁ ‚ÓÒ ̧ÏË Û‡‚ÌÂÌËÈ (16)  ̄‡ÂÚÒˇ ÏÂÚÓ‰ÓÏ Õ ̧ ̨ÚÓ̇ M(Ws−1)(Ws−Ws−1) =−H(

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    ÉÎÂÌÌÓÈ ̃‡ÒÚ ̧ ̨ ÏÂÚÓ‰‡ Õ ̧ ̨ÚÓ̇ ˇ‚ΡÂÚÒˇ ‚ ̊ ̃ËÒÎÂÌË χÚˈ ̊ flÍÓ·ËMÒËÒÚÂÏ ̊ Û‡‚ÌÂÌËÈ (16), Ô‰ÒÚ‡‚Ρ ̨ ̆ÂÈ ËÁ Ò·ˇ ‰ËÒÍÂÚÌ ̊ ÒÛÏÏ ̊ ÓÚ ÔÓËÁ‚Ó‰Ì ̊ıf(S)ÔÓ Ï‡ÍÓÒÍÓÔË ̃ÂÒÍËÏ ÔÂÂÏÂÌÌ ̊Ï. “Ó ̃ÌÓ ‚ ̊ ̃ËÒÎÂÌËÂMÚ·ÛÂÚ ÒÛ ̆ÂÒÚ‚ÂÌÌ ̊ı Á‡Ú‡Ú χ ̄ËÌÌÓ„Ó ‚ÂÏÂÌË, ÓÒÓ·ÂÌÌÓ ‰Îˇ ·ÓΠÒÎÓÊÌ ̊ı ÍËÌÂÚË ̃ÂÒÍËı ÏÓ‰ÂÎÂÈ, Ú‡ÍËı Í‡Í ÏÓ‰ÂÎ ̧ – ̊ÍÓ‚‡ ‰‚Ûı‡ÚÓÏÌÓ„Ó „‡Á‡ [10]. —ÎÂ‰Ûˇ
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    , ÔÂÂȉÂÏ ‚ ‚ ̊‡ÊÂÌËË ‰Îˇ χÚˈ ̊MÓÚ ̃ËÒÎÂÌÌÓ„Ó ËÌÚ„ËÓ‚‡Ìˡ Í ÚÓ ̃ÌÓÏÛ (‡Ì‡ÎËÚË ̃ÂÒÍÓÏÛ). ¬ ÂÁÛÎ ̧Ú‡ÚÂM‚ ̊‡Ê‡ÂÚÒˇ ‚ ˇ‚ÌÓÏ ‚ˉ ̃ÂÂÁ χÍÓÔ‡‡ÏÂÚ ̊ „‡Á‡. ‡Í Ô‡‚ËÎÓ, ËÒÔÓÎ ̧ÁÓ‚‡ÌË ÔË·ÎËÊÂÌÌÓÈ ÙÓÏÛÎ ̊ ‰Îˇ MÒÛ ̆ÂÒÚ‚ÂÌÌÓ ÛÏÂÌ ̧ ̄‡ÂÚ ‚ÂÏˇ Ò ̃ÂÚ‡ ‰Îˇδ≤1000, ÌÂÒÏÓÚˇ ̇ ÔÓÚÂ ̨ Í‚‡‰‡ÚË ̃ÌÓÈ ÒıÓ‰ËÏÓÒÚË ËÚÂ‡ˆËÈ.

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Mieussens L. Discrete velocity model and implicit scheme for the BGK equation of rarefied gas dynamics.Math. Models and Meth. Appl. Sci., 2000, vol. 8, no. 10, pp. 1121{1149.
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    ¬ ͇ ̃ÂÒڂ ̇ ̃‡Î ̧ÌÓ„Ó ÔË·ÎËÊÂÌˡ ËÒÔÓÎ ̧ÁÛ ̨ÚÒˇ Á̇ ̃ÂÌˡ, ÔÓÎÛ ̃ÂÌÌ ̊ ËÁ ÔË·ÎËÊÂÌˡ (15). ¬ ̃‡ÒÚÌÓÏ ÒÎÛ ̃‡ÂPr = 1(ÏÓ‰ÂÎ ̧ ¡√ [19]) ÙÛÌÍˆËˇf(S)Ì ÒÓ‰ÂÊËÚ Á̇ ̃ÂÌËÈ ‚ÂÍÚÓ‡ ÔÓÚÓÍ ÚÂÔ·, Ú‡Í ̃ÚÓ ÔÓÒΉÌË ÚË Û‡‚ÌÂÌˡ ÒËÒÚÂÏ ̊ (16) ÒÚ‡ÌÓ‚ˇÚÒˇ ËÁ· ̊ÚÓ ̃Ì ̊ÏË. ¬ ‰‡ÌÌÓÏ ÒÎÛ ̃‡Â ÒËÒÚÂχ (16) ÒÓ‚Ô‡‰‡ÂÚ Ò ÏÂÚÓ‰ÓÏ, Ô‰ÎÓÊÂÌÌ ̊Ï ‚
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    ‰Îˇ ¡√ -ÏÓ‰ÂÎË. ÉÎÂÌÌÓÈ ̃‡ÒÚ ̧ ̨ ÏÂÚÓ‰‡ Õ ̧ ̨ÚÓ̇ ˇ‚ΡÂÚÒˇ ‚ ̊ ̃ËÒÎÂÌË χÚˈ ̊ flÍÓ·ËMÒËÒÚÂÏ ̊ Û‡‚ÌÂÌËÈ (16), Ô‰ÒÚ‡‚Ρ ̨ ̆ÂÈ ËÁ Ò·ˇ ‰ËÒÍÂÚÌ ̊ ÒÛÏÏ ̊ ÓÚ ÔÓËÁ‚Ó‰Ì ̊ıf(S)ÔÓ Ï‡ÍÓÒÍÓÔË ̃ÂÒÍËÏ ÔÂÂÏÂÌÌ ̊Ï.

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Gusarov A.V., Smurov I. Gas-dynamic boundary conditions of evaporation and condensation: Numerical analysis of the Knudsen layer.Phys. Fluids, 2002, vol. 14, no. 12, pp. 4242{4255. DOI:10.1063/1.1516211
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    ¬ ͇ ̃ÂÒڂ ̇ ̃‡Î ̧ÌÓ„Ó ÔË·ÎËÊÂÌˡ ËÒÔÓÎ ̧ÁÛ ̨ÚÒˇ Á̇ ̃ÂÌˡ, ÔÓÎÛ ̃ÂÌÌ ̊ ËÁ ÔË·ÎËÊÂÌˡ (15). ¬ ̃‡ÒÚÌÓÏ ÒÎÛ ̃‡ÂPr = 1(ÏÓ‰ÂÎ ̧ ¡√ [19]) ÙÛÌÍˆËˇf(S)Ì ÒÓ‰ÂÊËÚ Á̇ ̃ÂÌËÈ ‚ÂÍÚÓ‡ ÔÓÚÓÍ ÚÂÔ·, Ú‡Í ̃ÚÓ ÔÓÒΉÌË ÚË Û‡‚ÌÂÌˡ ÒËÒÚÂÏ ̊ (16) ÒÚ‡ÌÓ‚ˇÚÒˇ ËÁ· ̊ÚÓ ̃Ì ̊ÏË. ¬ ‰‡ÌÌÓÏ ÒÎÛ ̃‡Â ÒËÒÚÂχ (16) ÒÓ‚Ô‡‰‡ÂÚ Ò ÏÂÚÓ‰ÓÏ, Ô‰ÎÓÊÂÌÌ ̊Ï ‚
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    ‰Îˇ ¡√ -ÏÓ‰ÂÎË. ÉÎÂÌÌÓÈ ̃‡ÒÚ ̧ ̨ ÏÂÚÓ‰‡ Õ ̧ ̨ÚÓ̇ ˇ‚ΡÂÚÒˇ ‚ ̊ ̃ËÒÎÂÌË χÚˈ ̊ flÍÓ·ËMÒËÒÚÂÏ ̊ Û‡‚ÌÂÌËÈ (16), Ô‰ÒÚ‡‚Ρ ̨ ̆ÂÈ ËÁ Ò·ˇ ‰ËÒÍÂÚÌ ̊ ÒÛÏÏ ̊ ÓÚ ÔÓËÁ‚Ó‰Ì ̊ıf(S)ÔÓ Ï‡ÍÓÒÍÓÔË ̃ÂÒÍËÏ ÔÂÂÏÂÌÌ ̊Ï.

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Men'shov I.S., Nakamura Y. An implicit advection upwind splitting scheme for hypersonic air flows in thermochemical nonequilibrium.A Collection of Technical Papers of 6thInt. Symp. on CFD, Lake Tahoe, Nevada, 1995. Vol. 2. 1995, p. 815.
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    √ÛÔÔËÛˇ ̃ÎÂÌ ̊ Ë ‚ ̊‰ÂΡˇ ‚ ˇ‚ÌÓÏ ‚ˉ ÍÓ ̋ÙÙˈËÂÌÚ ÔÂ‰∆fi, ÔÓÎÛ ̃‡ÂÏ di◦∆fi+ ∆t ∑ l cil◦∆fil= ∆tRni.(19) œˇÏÓ  ̄ÂÌË ÒËÒÚÂÏ ̊ Û‡‚ÌÂÌËÈ (19) ˇ‚ΡÂÚÒˇ ‰Ó‚ÓÎ ̧ÌÓ Á‡Ú‡ÚÌÓÈ ÓÔÂ‡ˆËÂÈ. œÓ ̋ÚÓÏÛ ÔÓËÁ‚Ó‰ËÚÒˇ ÔË·ÎËÊÂÌ̇ˇ Ù‡ÍÚÓËÁ‡ˆËˇ ‡ÁÂÊÂÌÌÓÈ Ï‡Úˈ ̊ ÒËÒÚÂÏ ̊ ̇ ÓÒÌÓ‚Â ÔÓ‰ıÓ‰‡, Ô‰ÎÓÊÂÌÌÓ„Ó ‚
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    . —ËÒÚÂχ  ̄‡ÂÚÒˇ ‚ ‰‚‡ ̋Ú‡Ô‡. —̇ ̃‡Î‡ ÔÓËÁ‚Ó‰ËÚÒˇ Ó·‡ÚÌ ̊È ıÓ‰ ÏÂÚÓ‰‡, Ë Ì‡ıÓ‰ˇÚÒˇ ÔÓÏÂÊÛÚÓ ̃Ì ̊ Á̇ ̃ÂÌˡ∆f∗i: di◦∆f∗i=− ∑ l:il<i cil◦∆f∗il+ ∆tRni, i=Nspace, . . . ,1.(20) œˇÏÓÈ ıÓ‰ ÏÂÚÓ‰‡ ‚ ̊‰‡ÂÚ ÓÍÓÌ ̃‡ÚÂÎ ̧Ì ̊ Á̇ ̃ÂÌˡ ‰Îˇ ÔË‡ ̆ÂÌˡ  ̄ÂÌˡ: di◦∆fi= ∆f ∗ i− ∑ l:il>i cil◦∆fil, i= 1, .

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Karypis G., Kumar V. Multilevel k-way partitioning scheme for irregular graphs.J. Parallel Distrib. Comput., 1998, vol. 48, pp. 96{129.
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    ˇ ̃ÂÈÍË, Ӊ̇ÍÓ, ÔË ËÒÔÓÎ ̧ÁÓ‚‡ÌËË ˆËÎË̉Ë ̃ÂÒÍÓÈ ÒËÒÚÂÏ ̊ ÍÓÓ‰ËÌ‡Ú ‚ ÒÂÚÍÛ ÏÓ„ÛÚ ‚ıÓ‰ËÚ ̧ ÔËÁÏ ̊. œÂÔÓˆÂÒÒÓ.œÂÔÓˆÂÒÒÓ Ô‡ÍÂÚ‡ ËÒÔÓÎ ̧ÁÛÂÚÒˇ ‰Îˇ ‡Á·ËÂÌˡ ÔÓÒÚ‡ÌÒÚ‚ÂÌÌ ̊ı ÒÂÚÓÍ Ì‡ ·ÎÓÍË ‰Îˇ Ôӂ‰ÂÌˡ Ô‡‡ÎÎÂÎ ̧Ì ̊ı ‚ ̊ ̃ËÒÎÂÌËÈ Ì‡ ÏÌÓ„Ó·ÎÓ ̃Ì ̊ı ÒÂÚ͇ı. –‡ÒÔ‰ÂÎÂÌË ˇ ̃ÂÂÍ ËÒıÓ‰ÌÓÈ Ó‰ÌÓ·ÎÓ ̃ÌÓÈ ÌÂÒÚÛÍÚÛËÓ‚‡ÌÌÓÈ ÒÂÚÍË ÏÂÊ‰Û ÔÓˆÂÒÒÓ‡ÏË ÔÓËÁ‚Ó‰ËÚÒˇ Ò ÔÓÏÓ ̆ ̧ ̨ ÔÓ„‡ÏÏ ̊ Metis
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    . »ÒÔÓÎ ̧ÁÛˇ ̋ÚÓ ‡Á·ËÂÌËÂ, ÔÂÔÓˆÂÒÒÓ ‡ÒÔ‰ÂΡÂÚ ˇ ̃ÂÈÍË ÒÂÚÍË ÏÂÊ‰Û ÔÓˆÂÒÒÓ‡ÏË, ÔÓ‚Ó‰ËÚ ÔÂÂÌÛÏÂ‡ˆË ̨ ‚ÒÂı ̋ÎÂÏÂÌÚÓ‚ ÒÂÚÍË (‚Â ̄ËÌ, „‡ÌÂÈ Ë ˇ‰Â), Ó·ÂÒÔ ̃Ë‚‡ÂÚ ÌÂÓ·ıÓ‰ËÏÓ ‰Îˇ Ôӂ‰ÂÌˡ ‡Ò ̃ÂÚÓ‚ ÔÂÂÍ ̊ÚË ÏÂÊ‰Û ·ÎÓ͇ÏË Ë ÒÓÁ‰‡ÂÚ ÒÔËÒÍË ˇ ̃ÂÂÍ ‰Îˇ Ó·ÏÂ̇ ‰‡ÌÌ ̊ÏË ÏÂÊ‰Û ‚ÒÂÏË ÔÓˆÂÒÒÓ‡ÏË.

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Dumbser M., Kaser M., Titarev V.A., Toro E.F. Quadrature-free non-oscillatory finite volume schemes on unstructured meshes for nonlinear hyperbolic systems.Journal of Computational Physics, 2007, vol. 226, iss. 1, pp. 204{243. DOI:10.1016/j.jcp.2007.04.004
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    ÂÏÂÌË. ƒÎˇ ÍËÌÂÚË ̃ÂÒÍÓ„Ó Û‡‚ÌÂÌˡ ‚ÓÁÏÓÊÌ ̊ ‰‚‡ ÔÓ‰ıÓ‰‡ Í ÔÓÒÚÓÂÌË ̨ Ô‡‡ÎÎÂÎ ̧ÌÓÈ ‚ÂÒËË ̃ËÒÎÂÌÌÓ„Ó ÏÂÚÓ‰‡. œÂ‚ ̊È ÔÓ‰ıÓ‰ ˇ‚ΡÂÚÒˇ Ú‡‰ËˆËÓÌÌ ̊Ï Ë Ô‰ÔÓ·„‡ÂÚ ‡Á·ËÂÌË ÔÓÒÚ‡ÌÒÚ‚ÂÌÌÓÈ ÒÂÚÍË Ì‡ ·ÎÓÍË Ò ÌÂÍÓÚÓ ̊Ï ÔÂÂÍ ̊ÚËÂÏ Ò Û ̃ÂÚÓÏ ̄‡·ÎÓ̇ ‡ÁÌÓÒÚÌÓÈ ÒıÂÏ ̊. –‡ÎËÁ‡ˆËˇ ˇ‚ÌÓÈ ‡ÁÌÓÒÚÌÓÈ ÒıÂÏ ̊ ̇ ÏÌÓ„Ó·ÎÓ ̃Ì ̊ı ÌÂÒÚÛÍÚÛËÓ‚‡ÌÌ ̊ı ÒÂÚ͇ı ıÓÓ ̄Ó ÓÔË҇̇ ‚ ÎËÚÂ‡ÚÛÂ, ÒÏ., ̇ÔËÏÂ,
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    . œˇÏÓ ӷӷ ̆ÂÌË Ì¡‚ÌÓ„Ó ‡Î„ÓËÚχ (12) ̇ ÏÌÓ„Ó·ÎÓ ̃Ì ̊ ÒÂÚÍË ÒÚ‡ÎÍË‚‡ÂÚÒˇ Ò ÒÂ ̧ÂÁÌ ̊ÏË ÚÛ‰ÌÓÒÚˇÏË Í‡Í Ò ‡Î„ÓËÚÏË ̃ÂÒÍÓÈ ÚÓ ̃ÍË ÁÂÌˡ, Ú‡Í Ë ‚ Ó·ÂÒÔ ̃ÂÌËË ıÓÓ ̄ÂÈ Ï‡Ò ̄Ú‡·ËÛÂÏÓÒÚË ÏÂÚÓ‰‡ ÔÓ ̃ËÒÎÛ ÔÓˆÂÒÒÓÓ‚.

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Aristov V.V., Zabelok C.A. [A deterministic method for solving the Boltzmann equation with parallel computations].Zhurnal vychislitel'noy matematiki i matematicheskoy fiziki, 2002, vol. 42, no. 3, pp. 425{437. (English translation:Computational Mathematics and Mathematical Physics, 2002, vol. 42, iss. 3, pp. 406{418).
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    ÂÁ Ó·ÏÂ̇ ‰‡ÌÌ ̊ÏË ÏÂÊ‰Û ·ÎÓ͇ÏË Ì‡ ̋Ú‡Ô  ̄ÂÌˡ ÎËÌÂÈÌÓÈ ÒËÒÚÂÏ ̊ (20), (21). œÓÎÛ ̃‡ ̨ ̆ËÈÒˇ ÔË ̋ÚÓÏ Ô‡‡ÎÎÂÎ ̧Ì ̊È ÏÌÓ„Ó·ÎÓ ̃Ì ̊È ‡Î„ÓËÚÏ  ̄ÂÌˡ Á‡‰‡ ̃Ë Ì ˇ‚ΡÂÚÒˇ ÒÚÓ„Ó ̋Í‚Ë‚‡ÎÂÌÚÌ ̊Ï Ó‰ÌÓ·ÎÓ ̃ÌÓÏÛ ÏÂÚÓ‰Û, ÌÓ ÒıÓ‰ËÚÒˇ Í ÚÓÏÛ Ê ÒÚ‡ˆËÓ̇ÌÓÏÛ  ̄ÂÌË ̨. ¬Ó ‚ÚÓÓÏ ÔÓ‰ıÓ‰Â, ‚ÔÂ‚ ̊ ÔËÏÂÌÂÌÌÓÏ ‰Îˇ  ̄ÂÌˡ ÚÓ ̃ÌÓ„Ó Û‡‚ÌÂÌˡ ¡ÓÎ ̧ˆÏ‡Ì‡ Ò ÔÓÏÓ ̆ ̧ ̨ ˇ‚ÌÓÈ ‡ÁÌÓÒÚÌÓÈ ÒıÂÏ ̊ ‚ ‡·ÓÚÂ
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    , ËÒÔÓÎ ̧ÁÛÂÚÒˇ ‡Á·ËÂÌË ÒÍÓÓÒÚÌÓÈ ÒÂÚÍË. œË ̋ÚÓÏ Í‡Ê‰ÓÏÛ ÔÓˆÂÒÒÓÛ ÒÚ‡‚ËÚÒˇ ‚ ÒÓÓÚ‚ÂÚÒÚ‚Ë ̇·Ó ÚÓ ̃ÂÍ ÒÍÓÓÒÚÌÓ„Ó ÔÓÒÚ‡ÌÒÚ‚‡, ‰Îˇ ÍÓÚÓÓ„Ó ÒÚÓËÚÒˇ  ̄ÂÌË ÍËÌÂÚË ̃ÂÒÍÓ„Ó Û‡‚ÌÂÌˡ Ò ‡ÁÌÓÒÚÌÓÈ ÒıÂÏ ̊.

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Dumbser M., Titarev V.A., Utyuzhnikov S.V. [Implicit multiblock method for solving a kinetic equation on unstructured meshes].Zhurnal vychislitel'noy matematiki i matematicheskoy fiziki, 2013, vol. 53, no. 5, pp. 762{782. DOI:10.7868/S0044466913050141(English translation: Computational Mathematics and Mathematical Physics, 2013, vol. 53, iss. 5, pp. 601{615. DOI:10.1134/S0965542513050126).
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    ̧ÌÓ Ú·ӂ‡ÎÒˇ Ó·ÏÂÌ ‰‡ÌÌ ̊ÏË Ì‡ „‡Ìˇı „‡ÌË ̃Ì ̊ı ˇ ̃ÂÂÍ, ̃ÚÓ ÒÛ ̆ÂÒÚ‚ÂÌÌÓ ÛÒÎÓÊÌˇÎÓ ÔÓ„‡ÏÏÌ ̊È ÍÓ‰. ƒÎˇ ‚ÚÓÓ„Ó ÔÓ‰ıÓ‰‡ ‡ÎËÁÓ‚‡Ì‡ „˷͇ˇ Ôӈ‰Û‡ Ó·ÏÂ̇ ‰‡ÌÌ ̊ÏË Ì‡ ̋Ú‡Ô ‡Ò ̃ÂÚ ̊ χÍÓÒÍÓÔË ̃ÂÒÍËı ‚ÂÎË ̃ËÌ: MPIAllReduce ‚ ̊Á ̊‚‡ÂÚÒˇ ÚÓÎ ̧ÍÓ ‰Îˇ ÚÂı ˇ ̃ÂÂÍ, ‚ ÍÓÚÓ ̊ı ËÚÂ‡ˆËÓÌÌ ̊È ÔÓˆÂÒÒ (16) Ì ÒÓ ̄ÂÎÒˇ. —‡‚ÌÂÌË ‰‚Ûı ÔÓ‰ıÓ‰Ó‚ Í Ó„‡ÌËÁ‡ˆËË Ô‡‡ÎÎÂÎ ̧Ì ̊ı ‚ ̊ ̃ËÒÎÂÌËÈ Ôӂ‰ÂÌÓ ‚
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    [18,38]
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    . œÓ͇Á‡Ì‡ ıÓÓ ̄‡ˇ Ï‡Ò ̄Ú‡·ËÛÂÏÓÒÚ ̧ ÍÓ‰‡ ̇ 4-ı ˇ‰ÂÌ ̊ı ÔÓˆÂÒÒÓ‡ı Intel Xeon ÔË ËÒÔÓÎ ̧ÁÓ‚‡ÌËË ÔË ËÒÔÓÎ ̧ÁÓ‚‡ÌËË ‰Ó 512 ‚ ̊ ̃ËÒÎËÚÂÎ ̧Ì ̊ı ˇ‰Â. 4. œËÎÓÊÂÌˡ — ÔÓÏÓ ̆ ̧ ̨ ‚ ̊ ̃ËÒÎËÚÂÎ ̧ÌÓ„Ó Ô‡ÍÂÚ‡ <ÕÂÒ‚ÂÚ‡È-3ƒ> ÏÓ„ÛÚ ÏÓ‰ÂÎËÓ‚‡Ú ̧Òˇ Í‡Í ÔÓÒÚ ̊ Ú ̃ÂÌˡ ‡ÁÂÊÂÌÌÓ„Ó „‡Á‡, Ú‡Í Ë Ú ̃ÂÌˡ ‚ ÒÎÓÊÌ ̊ı „ÂÓÏÂÚË ̃ÂÒÍËı ӷ·ÒÚˇı.

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Titarev V.A., Shakhov E.M. [Nonisothermal gas flow in a long channel analyzed on the basis of the kinetic S-Model].Zhurnal vychislitel'noy matematiki i matematicheskoy fiziki, 2010, vol. 50, no. 12, pp. 2246{2260. (English translation:Computational Mathematics and Mathematical Physics, 2010, vol. 50, iss. 12, pp. 2131{2144. DOI:10.1134/S0965542510120146).
Total in-text references: 1
  1. In-text reference with the coordinate start=33112
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    ̇ÒÚÓˇ ̆ÂÏÛ ‚ÂÏÂÌË ‰ÓÒÚ‡ÚÓ ̃ÌÓ ÔÓÎÌÓ ‡Á‡·ÓÚ‡Ì ̊ ÏÂÚÓ‰ ̊ ‡Ò ̃ÂÚ‡ Ú ̃ÂÌËÈ ‚ ÚÛ·‡ı Ë Í‡Ì‡Î‡ı ·ÂÒÍÓÌ ̃ÌÓÈ ‰ÎËÌ ̊ Ë ÔÓËÁ‚ÓÎ ̧ÌÓ„Ó ÔÓÔÂ ̃ÌÓ„Ó Ò ̃ÂÌˡ ̇ ÓÒÌÓ‚Â ÎË̇ËÁÓ‚‡ÌÌ ̊ı ÍËÌÂÚË ̃ÂÒÍËı ÏÓ‰ÂÎÂÈ
    Exact
    [39]
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    . Œ‰ÌÓÈ ËÁ ‡ÍÚۇΠ̧Ì ̊ı ÒÓ‚ÂÏÂÌÌ ̊ı Á‡‰‡ ̃ ˇ‚ΡÂÚÒˇ ÌÂÎËÌÂÈ̇ˇ Á‡‰‡ ̃‡ Ó· ËÒÚ ̃ÂÌËË ‚ ‚‡ÍÛÛÏ. ¬ ÒÚ‡Ú ̧ [40] ÏÂÚÓ‰ÓÏ ÃÓÌÚÂ- ‡ÎÓ ËÁÛ ̃ÂÌ ̊ ÓÒÓ·ÂÌÌÓÒÚË Ú‡ÍÓ„Ó Ú ̃ÂÌˡ ‰Îˇ ÓÚÌÓÒËÚÂÎ ̧ÌÓ ÍÓÓÚÍËı ÍÛ„Î ̊ı ÚÛ·.

40
Varoutis S., Valougeorgis D., Sazhin O., Sharipov F. Rarefied gas flow through short tubes into vacuum.Journal of Vacuum Science and Technology A, 2008, vol. 26, no. 1, pp. 228{238. DOI10.1116/1.2830639
Total in-text references: 4
  1. In-text reference with the coordinate start=33222
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    ̇ÒÚÓˇ ̆ÂÏÛ ‚ÂÏÂÌË ‰ÓÒÚ‡ÚÓ ̃ÌÓ ÔÓÎÌÓ ‡Á‡·ÓÚ‡Ì ̊ ÏÂÚÓ‰ ̊ ‡Ò ̃ÂÚ‡ Ú ̃ÂÌËÈ ‚ ÚÛ·‡ı Ë Í‡Ì‡Î‡ı ·ÂÒÍÓÌ ̃ÌÓÈ ‰ÎËÌ ̊ Ë ÔÓËÁ‚ÓÎ ̧ÌÓ„Ó ÔÓÔÂ ̃ÌÓ„Ó Ò ̃ÂÌˡ ̇ ÓÒÌÓ‚Â ÎË̇ËÁÓ‚‡ÌÌ ̊ı ÍËÌÂÚË ̃ÂÒÍËı ÏÓ‰ÂÎÂÈ [39]. Œ‰ÌÓÈ ËÁ ‡ÍÚۇΠ̧Ì ̊ı ÒÓ‚ÂÏÂÌÌ ̊ı Á‡‰‡ ̃ ˇ‚ΡÂÚÒˇ ÌÂÎËÌÂÈ̇ˇ Á‡‰‡ ̃‡ Ó· ËÒÚ ̃ÂÌËË ‚ ‚‡ÍÛÛÏ. ¬ ÒÚ‡Ú ̧Â
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    [40]
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    ÏÂÚÓ‰ÓÏ ÃÓÌÚÂ- ‡ÎÓ ËÁÛ ̃ÂÌ ̊ ÓÒÓ·ÂÌÌÓÒÚË Ú‡ÍÓ„Ó Ú ̃ÂÌˡ ‰Îˇ ÓÚÌÓÒËÚÂÎ ̧ÌÓ ÍÓÓÚÍËı ÍÛ„Î ̊ı ÚÛ·. — Û‚ÂÎË ̃ÂÌËÂÏ ÓÚÌÓÒËÚÂÎ ̧ÌÓÈ ‰ÎËÌ ̊ ÚÛ· ̊ ÔÓ·ÎÂχ ÛÒÎÓÊÌˇÂÚÒˇ Ë Ú·ÛÂÚ ‚Ò ·ÓÎ ̧ ̄Ëı Ë ·ÓÎ ̧ ̄Ëı Á‡Ú‡Ú Ò ̃ÂÚÌÓ„Ó ‚ÂÏÂÌË.

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    ¬ ̉‡‚ÌÂÈ ‡·ÓÚ [41] ËÒÚ ̃ÂÌË „‡Á‡ ̃ÂÂÁ ÍÓÓÚÍÛ ̨ ÍÛ„ÎÛ ̨ ÚÛ·Û Ô‰ÎÓÊÂÌÓ ‚ ‚ˉ ӉÌÓÈ ËÁ Òڇ̉‡ÚÌ ̊ı ‚ÂËÙË͇ˆËÓÌÌ ̊ı Á‡‰‡ ̃, ̇ ÍÓÚÓÓÈ ÏÓÊÌÓ Ò‡‚ÌË‚‡Ú ̧ ÂÁÛÎ ̧Ú‡Ú ̊ Ò ̃ÂÚ‡ ÔÓ ‡ÁÌ ̊Ï ÏÂÚÓ‰‡Ï Ë ÍÓÏÔÎÂÍÒ‡Ï ÔÓ„‡ÏÏ. ƒÂڇΠ̧̇ˇ ÔÓÒÚ‡Ìӂ͇ Á‡‰‡ ̃Ë Ó Ú ̃ÂÌËË „‡Á‡ ̃ÂÂÁ ÚÛ·Û ÍÓÌ ̃ÌÓÈ ‰ÎËÌ ̊ ËÁÎÓÊÂ̇ ‚ ‡·ÓÚ‡ı
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    [27,40,43]
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    . –‡ÒÒχÚË‚‡ÂÚÒˇ ÒÚ‡ˆËÓ̇ÌÓ Ú ̃ÂÌË ‡ÁÂÊÂÌÌÓ„Ó „‡Á‡ ËÁ ÂÁÂ‚Û‡‡ 1‚ ÂÁÂ‚Û‡2 ̃ÂÂÁ ÚÛ·Û ‰ÎËÌ ̊LË ÍÛ„Ó‚Ó„Ó ÔÓÔÂ ̃ÌÓ„Ó Ò ̃ÂÌˡ Ò ‡‰ËÛÒÓÏ ‚ıÓ‰ÌÓ„Ó Ú ̃ÂÌˡa. ¬ Ó· ̆ÂÏ ÒÎÛ ̃‡Â ‡‰ËÛÒ ÚÛ· ̊ ÏÓÊÂÚ ÏÂÌˇÚ ̧Òˇ ÔÓ ‰ÎËÌÂ.

  3. In-text reference with the coordinate start=36785
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    “‡ÍËÏ Ó·‡ÁÓÏ, ̃ËÒÎÓ ÛÁÎÓ‚ ‚ ̄ÂÒÚËÏÂÌÓÈ ‡Ò ̃ÂÚÌÓÈ ÒÂÚÍ ÏÓÊÂÚ ‰ÓÒÚË„‡Ú ̧4.5×109. –ËÒ. 2.œÓÒÚ‡ÌÒÚ‚ÂÌÌ ̊ ÒÂÚÍË ‰Îˇ Ú ̃ÂÌˡ ‚ ÍÓÓÚÍÓÈ ÍÛ„ÎÓÈ ÚÛ·Â ¬ Ú‡·Î.1Ô˂‰ÂÌ ̊ ÂÁÛÎ ̧Ú‡Ú ̊ ‡Ò ̃ÂÚÓ‚ Ô‡ÍÂÚÓÏ <ÕÂÒ‚ÂÚ‡È-3ƒ>, ÏÂÚÓ‰ÓÏ DSMC
    Exact
    [40]
    Suffix
    Ë ̋ÍÒÔÂËÏÂÌڇΠ̧Ì ̊ ‰‡ÌÌ ̊ [42]. ƒË‡Ô‡ÁÓÌ ËÁÏÂÌÂÌˡ Ô‡‡ÏÂÚ‡ ‡ÁÂÊÂÌÌÓÒÚË0≤δ≤ ≤500ÒÓÓÚ‚ÂÚÒÚ‚ÛÂÚ ÔÂÂıÓ‰Û ÓÚ Ò‚Ó·Ó‰ÌÓÏÓÎÂÍÛΡÌÓ„Ó ÂÊËχ Ú ̃ÂÌˡ ‰Ó Ô‡ÍÚË ̃ÂÒÍË ÍÓÌÚËÌۇΠ̧ÌÓ„Ó. ¬Ë‰Ì‡ · ̊ÒÚ‡ˇ ÒıÓ‰ËÏÓÒÚ ̧ ‡Ò ̃ÂÚÓ‚ ‡ÒıÓ‰‡ χÒÒ ̊ ÔÓ ÔÓÒÚ‡ÌÒÚ‚ÂÌÌÓÈ ÒÂÚÍÂ. œËÂÏÎÂχˇ ÚÓ ̃ÌÓÒÚ ̧ ‰Îˇ ‚ÒÂıδÓ·ÂÒÔ ̃Â̇ ÛÊ ̇ ‚ÚÓÓÈ, ÓÚÌÓÒËÚÂÎ ̧ÌÓ „Û·ÓÈ, ÒÂÚÍÂ.

  4. In-text reference with the coordinate start=37498
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    –‡ÒıÓʉÂÌËÂ Ò ̋ÍÒÔÂËÏÂÌڇΠ̧Ì ̊ÏË ‰‡ÌÌ ̊ÏË ÒÓÒÚ‡‚ΡÂÚ2{3% ÔË ÓˆÂÌÍÂ Ó ̄Ë·ÍË ËÁÏÂÂÌËÈ ‚2%. “ ‡ · Î Ë ˆ ‡ 1 —ıÓ‰ËÏÓÒÚ ̧ ÔÓ ÔÓÒÚ‡ÌÒÚ‚ÂÌÌÓÈ ÒÂÚÍ ‰Îˇ ÌÓÏËÓ‚‡ÌÌÓ„Ó ‡ÒıÓ‰‡ χÒÒ ̊ δ <ÕÂÒ‚ÂÚ‡È-3ƒ> DSMC
    Exact
    [40]
    Suffix
    ›ÍÒÔÂËÏÂÌÚ [42] (i)(ii)(iii) 0.00.6660.6700.6720.672 0.10.6780.6830.6840.6800.675 1.00.7580.7660.7680.7540.743 10.01.0351.0611.0661.0621.06 100.01.2901.3511.3671.3581.33 200.01.3311.4061.4251.412 500.01.3311.4541.4741.449 œ  Ë Ï Â ̃ ‡ Ì Ë Â .

41
Sharipov F. Benchmark problems in rarefied gas dynamics.Vacuum. Spec. iss. \Vacuum Gas Dynamics. Theory, experiments and practical applications", 2012, vol. 86, no. 11, pp. 1697{ 1700. DOI:10.1016/j.vacuum.2012.02.048
Total in-text references: 1
  1. In-text reference with the coordinate start=33471
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    ¬ ÒÚ‡Ú ̧ [40] ÏÂÚÓ‰ÓÏ ÃÓÌÚÂ- ‡ÎÓ ËÁÛ ̃ÂÌ ̊ ÓÒÓ·ÂÌÌÓÒÚË Ú‡ÍÓ„Ó Ú ̃ÂÌˡ ‰Îˇ ÓÚÌÓÒËÚÂÎ ̧ÌÓ ÍÓÓÚÍËı ÍÛ„Î ̊ı ÚÛ·. — Û‚ÂÎË ̃ÂÌËÂÏ ÓÚÌÓÒËÚÂÎ ̧ÌÓÈ ‰ÎËÌ ̊ ÚÛ· ̊ ÔÓ·ÎÂχ ÛÒÎÓÊÌˇÂÚÒˇ Ë Ú·ÛÂÚ ‚Ò ·ÓÎ ̧ ̄Ëı Ë ·ÓÎ ̧ ̄Ëı Á‡Ú‡Ú Ò ̃ÂÚÌÓ„Ó ‚ÂÏÂÌË. ¬ ̉‡‚ÌÂÈ ‡·ÓÚÂ
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    [41]
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    ËÒÚ ̃ÂÌË „‡Á‡ ̃ÂÂÁ ÍÓÓÚÍÛ ̨ ÍÛ„ÎÛ ̨ ÚÛ·Û Ô‰ÎÓÊÂÌÓ ‚ ‚ˉ ӉÌÓÈ ËÁ Òڇ̉‡ÚÌ ̊ı ‚ÂËÙË͇ˆËÓÌÌ ̊ı Á‡‰‡ ̃, ̇ ÍÓÚÓÓÈ ÏÓÊÌÓ Ò‡‚ÌË‚‡Ú ̧ ÂÁÛÎ ̧Ú‡Ú ̊ Ò ̃ÂÚ‡ ÔÓ ‡ÁÌ ̊Ï ÏÂÚÓ‰‡Ï Ë ÍÓÏÔÎÂÍÒ‡Ï ÔÓ„‡ÏÏ. ƒÂڇΠ̧̇ˇ ÔÓÒÚ‡Ìӂ͇ Á‡‰‡ ̃Ë Ó Ú ̃ÂÌËË „‡Á‡ ̃ÂÂÁ ÚÛ·Û ÍÓÌ ̃ÌÓÈ ‰ÎËÌ ̊ ËÁÎÓÊÂ̇ ‚ ‡·ÓÚ‡ı [27,40,43].

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Fujimoto T., Usami M. Rarefied gas flow through a circular orifice and short tubes.Journal of Fluids Engineering, 1984, vol. 106, no. 4, pp. 367{373. DOI:10.1115/1.3243132
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    “‡ÍËÏ Ó·‡ÁÓÏ, ̃ËÒÎÓ ÛÁÎÓ‚ ‚ ̄ÂÒÚËÏÂÌÓÈ ‡Ò ̃ÂÚÌÓÈ ÒÂÚÍ ÏÓÊÂÚ ‰ÓÒÚË„‡Ú ̧4.5×109. –ËÒ. 2.œÓÒÚ‡ÌÒÚ‚ÂÌÌ ̊ ÒÂÚÍË ‰Îˇ Ú ̃ÂÌˡ ‚ ÍÓÓÚÍÓÈ ÍÛ„ÎÓÈ ÚÛ·Â ¬ Ú‡·Î.1Ô˂‰ÂÌ ̊ ÂÁÛÎ ̧Ú‡Ú ̊ ‡Ò ̃ÂÚÓ‚ Ô‡ÍÂÚÓÏ <ÕÂÒ‚ÂÚ‡È-3ƒ>, ÏÂÚÓ‰ÓÏ DSMC [40] Ë ̋ÍÒÔÂËÏÂÌڇΠ̧Ì ̊ ‰‡ÌÌ ̊Â
    Exact
    [42]
    Suffix
    . ƒË‡Ô‡ÁÓÌ ËÁÏÂÌÂÌˡ Ô‡‡ÏÂÚ‡ ‡ÁÂÊÂÌÌÓÒÚË0≤δ≤ ≤500ÒÓÓÚ‚ÂÚÒÚ‚ÛÂÚ ÔÂÂıÓ‰Û ÓÚ Ò‚Ó·Ó‰ÌÓÏÓÎÂÍÛΡÌÓ„Ó ÂÊËχ Ú ̃ÂÌˡ ‰Ó Ô‡ÍÚË ̃ÂÒÍË ÍÓÌÚËÌۇΠ̧ÌÓ„Ó. ¬Ë‰Ì‡ · ̊ÒÚ‡ˇ ÒıÓ‰ËÏÓÒÚ ̧ ‡Ò ̃ÂÚÓ‚ ‡ÒıÓ‰‡ χÒÒ ̊ ÔÓ ÔÓÒÚ‡ÌÒÚ‚ÂÌÌÓÈ ÒÂÚÍÂ. œËÂÏÎÂχˇ ÚÓ ̃ÌÓÒÚ ̧ ‰Îˇ ‚ÒÂıδÓ·ÂÒÔ ̃Â̇ ÛÊ ̇ ‚ÚÓÓÈ, ÓÚÌÓÒËÚÂÎ ̧ÌÓ „Û·ÓÈ, ÒÂÚÍÂ.

  2. In-text reference with the coordinate start=37514
    Prefix
    –‡ÒıÓʉÂÌËÂ Ò ̋ÍÒÔÂËÏÂÌڇΠ̧Ì ̊ÏË ‰‡ÌÌ ̊ÏË ÒÓÒÚ‡‚ΡÂÚ2{3% ÔË ÓˆÂÌÍÂ Ó ̄Ë·ÍË ËÁÏÂÂÌËÈ ‚2%. “ ‡ · Î Ë ˆ ‡ 1 —ıÓ‰ËÏÓÒÚ ̧ ÔÓ ÔÓÒÚ‡ÌÒÚ‚ÂÌÌÓÈ ÒÂÚÍ ‰Îˇ ÌÓÏËÓ‚‡ÌÌÓ„Ó ‡ÒıÓ‰‡ χÒÒ ̊ δ <ÕÂÒ‚ÂÚ‡È-3ƒ> DSMC [40]›ÍÒÔÂËÏÂÌÚ
    Exact
    [42]
    Suffix
    (i)(ii)(iii) 0.00.6660.6700.6720.672 0.10.6780.6830.6840.6800.675 1.00.7580.7660.7680.7540.743 10.01.0351.0611.0661.0621.06 100.01.2901.3511.3671.3581.33 200.01.3311.4061.4251.412 500.01.3311.4541.4741.449 œ  Ë Ï Â ̃ ‡ Ì Ë Â .

43
Titarev V.A., Shakhov E.M. [End effects in rarefied gas outflow into vacuum through a long tube].Izvestiya RAN. Mekhanika Zhidkosti i Gaza, 2013, no. 5, pp. 146{158. (English translation:Fluid Dynamics, 2013, vol. 48, iss. 5, pp. 697{707. DOI:10.1134/S001546281305013X).
Total in-text references: 2
  1. In-text reference with the coordinate start=33773
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    ¬ ̉‡‚ÌÂÈ ‡·ÓÚ [41] ËÒÚ ̃ÂÌË „‡Á‡ ̃ÂÂÁ ÍÓÓÚÍÛ ̨ ÍÛ„ÎÛ ̨ ÚÛ·Û Ô‰ÎÓÊÂÌÓ ‚ ‚ˉ ӉÌÓÈ ËÁ Òڇ̉‡ÚÌ ̊ı ‚ÂËÙË͇ˆËÓÌÌ ̊ı Á‡‰‡ ̃, ̇ ÍÓÚÓÓÈ ÏÓÊÌÓ Ò‡‚ÌË‚‡Ú ̧ ÂÁÛÎ ̧Ú‡Ú ̊ Ò ̃ÂÚ‡ ÔÓ ‡ÁÌ ̊Ï ÏÂÚÓ‰‡Ï Ë ÍÓÏÔÎÂÍÒ‡Ï ÔÓ„‡ÏÏ. ƒÂڇΠ̧̇ˇ ÔÓÒÚ‡Ìӂ͇ Á‡‰‡ ̃Ë Ó Ú ̃ÂÌËË „‡Á‡ ̃ÂÂÁ ÚÛ·Û ÍÓÌ ̃ÌÓÈ ‰ÎËÌ ̊ ËÁÎÓÊÂ̇ ‚ ‡·ÓÚ‡ı
    Exact
    [27,40,43]
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    . –‡ÒÒχÚË‚‡ÂÚÒˇ ÒÚ‡ˆËÓ̇ÌÓ Ú ̃ÂÌË ‡ÁÂÊÂÌÌÓ„Ó „‡Á‡ ËÁ ÂÁÂ‚Û‡‡ 1‚ ÂÁÂ‚Û‡2 ̃ÂÂÁ ÚÛ·Û ‰ÎËÌ ̊LË ÍÛ„Ó‚Ó„Ó ÔÓÔÂ ̃ÌÓ„Ó Ò ̃ÂÌˡ Ò ‡‰ËÛÒÓÏ ‚ıÓ‰ÌÓ„Ó Ú ̃ÂÌˡa. ¬ Ó· ̆ÂÏ ÒÎÛ ̃‡Â ‡‰ËÛÒ ÚÛ· ̊ ÏÓÊÂÚ ÏÂÌˇÚ ̧Òˇ ÔÓ ‰ÎËÌÂ.

  2. In-text reference with the coordinate start=35343
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    . œË ‡Ì‡ÎËÁ ÂÁÛÎ ̧Ú‡ÚÓ‚ Û‰Ó·Ì ÔÓÎ ̧ÁÓ‚‡Ú ̧Òˇ ‚ÂÎË ̃ËÌÓÈQ, Ô‰ÒÚ‡‚Ρ ̨ ̆ÂÈ ÒÓ·ÓÈ ‡ÒıÓ‰ ̇M„‡Á‡ ̃ÂÂÁ Ò ̃ÂÌË ÚÛ· ̊, ÌÓÏËÓ‚‡ÌÌ ̊È Ì‡ ‚ÂÎË ̃ËÌÛ ‡ÒıÓ‰‡ ̇M отв 0ÔË Ò‚Ó·Ó‰ÌÓ-ÏÓÎÂÍÛΡÌÓÏ ËÒÚ ̃ÂÌËË ‚ ‚‡ÍÛÛÏ ̃ÂÂÁ ÓÚ‚ÂÒÚË [44,45]: Q= ̇M / ̇M отв 0, M ̇отв 0= √ π/2.(22) ŒÒÌÓ‚Ì ̊Ï ˇ‚ΡÂÚÒˇ ÒÎÛ ̃‡È ÍÛ„ÎÓÈ ÚÛ· ̊ ÔÓÒÚÓˇÌÌÓ„Ó ‡‰ËÛÒ‡, ‡‚ÌÓ„Ó ‡‰ËÛÒÛ ‚ıÓ‰ÌÓ„Ó Ò ̃ÂÌˡa. ¬ ˆËÍΠ‡·ÓÚ
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    [17,27,43,46]
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     ̄ÂÌË Á‡‰‡ ̃Ë · ̊ÎÓ ÔÓÒÚÓÂÌÓ Ô‡ÍÂÚÓÏ <ÕÂÒ‚ÂÚ‡È-3ƒ> ‰Îˇ χÎÓ„Ó ÓÚÌÓ ̄ÂÌˡ ‰‡‚ÎÂÌËÈ|p10/p20|−1-1(ÎË̇ËÁÓ‚‡Ì̇ˇ Á‡‰‡ ̃‡), ‡ Ú‡ÍÊ ‰‚Ûı ÌÂÎËÌÂÈÌ ̊ı ÂÊËÏÓ‚p10/p20= 2(ÛÏÂÂÌÌ ̊È ÔÂÂÔ‡‰ ‰‡‚ÎÂÌËÈ) Ëp10/p20=∞ (Ú ̃ÂÌË ‚ ‚‡ÍÛÛÏ).

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Shakhov E.M. [The axisymmetric nonlinear steady flow of a rarefied gas in a pipe of circular cross-section].Zhurnal vychislitel'noy matematiki i matematicheskoy fiziki, 1996, vol. 36, no. 8, pp. 169{179. (English translation:Computational Mathematics and Mathematical Physics, 1996, vol. 36, no. 8, pp. 1123{1131.).
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    ŒÒÌÓ‚ÌÓÈ ‡Ò ̃ÂÚÌÓÈ ‚ÂÎË ̃ËÌÓÈ ˇ‚ΡÂÚÒˇ ‡ÒıÓ‰ χÒÒ ̊ ̇M ̃ÂÂÁ Ò ̃ÂÌË ÚÛ· ̊, ÓÚÌÂÒÂÌÌ ̊È Ímn10β1a2. œË ‡Ì‡ÎËÁ ÂÁÛÎ ̧Ú‡ÚÓ‚ Û‰Ó·Ì ÔÓÎ ̧ÁÓ‚‡Ú ̧Òˇ ‚ÂÎË ̃ËÌÓÈQ, Ô‰ÒÚ‡‚Ρ ̨ ̆ÂÈ ÒÓ·ÓÈ ‡ÒıÓ‰ ̇M„‡Á‡ ̃ÂÂÁ Ò ̃ÂÌË ÚÛ· ̊, ÌÓÏËÓ‚‡ÌÌ ̊È Ì‡ ‚ÂÎË ̃ËÌÛ ‡ÒıÓ‰‡ ̇M отв 0ÔË Ò‚Ó·Ó‰ÌÓ-ÏÓÎÂÍÛΡÌÓÏ ËÒÚ ̃ÂÌËË ‚ ‚‡ÍÛÛÏ ̃ÂÂÁ ÓÚ‚ÂÒÚËÂ
    Exact
    [44,45]
    Suffix
    : Q= ̇M / ̇M отв 0, M ̇отв 0= √ π/2.(22) ŒÒÌÓ‚Ì ̊Ï ˇ‚ΡÂÚÒˇ ÒÎÛ ̃‡È ÍÛ„ÎÓÈ ÚÛ· ̊ ÔÓÒÚÓˇÌÌÓ„Ó ‡‰ËÛÒ‡, ‡‚ÌÓ„Ó ‡‰ËÛÒÛ ‚ıÓ‰ÌÓ„Ó Ò ̃ÂÌˡa. ¬ ˆËÍΠ‡·ÓÚ [17,27,43,46]  ̄ÂÌË Á‡‰‡ ̃Ë · ̊ÎÓ ÔÓÒÚÓÂÌÓ Ô‡ÍÂÚÓÏ <ÕÂÒ‚ÂÚ‡È-3ƒ> ‰Îˇ χÎÓ„Ó ÓÚÌÓ ̄ÂÌˡ ‰‡‚ÎÂÌËÈ|p10/p20|−1-1(ÎË̇ËÁÓ‚‡Ì̇ˇ Á‡‰‡ ̃‡), ‡ Ú‡ÍÊ ‰‚Ûı ÌÂÎËÌÂÈÌ ̊ı ÂÊËÏÓ‚p10/p20= 2(ÛÏÂÂÌÌ ̊È ÔÂÂÔ‡‰ ‰‡‚ÎÂÌËÈ) Ëp10/p20=∞ (Ú ̃ÂÌË ‚ ‚‡ÍÛÛÏ).

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Varoutis S., Valougeorgis D., Sharipov F. Simulation of gas flow through tubes of finite length over the whole range of rarefaction for various pressure drop ratios.Journal of Vacuum Science and Technology A, 2009, vol. 27, no. 6, pp. 1377{1391. DOI:10.1116/1.3248273
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    ŒÒÌÓ‚ÌÓÈ ‡Ò ̃ÂÚÌÓÈ ‚ÂÎË ̃ËÌÓÈ ˇ‚ΡÂÚÒˇ ‡ÒıÓ‰ χÒÒ ̊ ̇M ̃ÂÂÁ Ò ̃ÂÌË ÚÛ· ̊, ÓÚÌÂÒÂÌÌ ̊È Ímn10β1a2. œË ‡Ì‡ÎËÁ ÂÁÛÎ ̧Ú‡ÚÓ‚ Û‰Ó·Ì ÔÓÎ ̧ÁÓ‚‡Ú ̧Òˇ ‚ÂÎË ̃ËÌÓÈQ, Ô‰ÒÚ‡‚Ρ ̨ ̆ÂÈ ÒÓ·ÓÈ ‡ÒıÓ‰ ̇M„‡Á‡ ̃ÂÂÁ Ò ̃ÂÌË ÚÛ· ̊, ÌÓÏËÓ‚‡ÌÌ ̊È Ì‡ ‚ÂÎË ̃ËÌÛ ‡ÒıÓ‰‡ ̇M отв 0ÔË Ò‚Ó·Ó‰ÌÓ-ÏÓÎÂÍÛΡÌÓÏ ËÒÚ ̃ÂÌËË ‚ ‚‡ÍÛÛÏ ̃ÂÂÁ ÓÚ‚ÂÒÚËÂ
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    [44,45]
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    : Q= ̇M / ̇M отв 0, M ̇отв 0= √ π/2.(22) ŒÒÌÓ‚Ì ̊Ï ˇ‚ΡÂÚÒˇ ÒÎÛ ̃‡È ÍÛ„ÎÓÈ ÚÛ· ̊ ÔÓÒÚÓˇÌÌÓ„Ó ‡‰ËÛÒ‡, ‡‚ÌÓ„Ó ‡‰ËÛÒÛ ‚ıÓ‰ÌÓ„Ó Ò ̃ÂÌˡa. ¬ ˆËÍΠ‡·ÓÚ [17,27,43,46]  ̄ÂÌË Á‡‰‡ ̃Ë · ̊ÎÓ ÔÓÒÚÓÂÌÓ Ô‡ÍÂÚÓÏ <ÕÂÒ‚ÂÚ‡È-3ƒ> ‰Îˇ χÎÓ„Ó ÓÚÌÓ ̄ÂÌˡ ‰‡‚ÎÂÌËÈ|p10/p20|−1-1(ÎË̇ËÁÓ‚‡Ì̇ˇ Á‡‰‡ ̃‡), ‡ Ú‡ÍÊ ‰‚Ûı ÌÂÎËÌÂÈÌ ̊ı ÂÊËÏÓ‚p10/p20= 2(ÛÏÂÂÌÌ ̊È ÔÂÂÔ‡‰ ‰‡‚ÎÂÌËÈ) Ëp10/p20=∞ (Ú ̃ÂÌË ‚ ‚‡ÍÛÛÏ).

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Titarev V.A. Rarefied gas flow in a circular pipe of finite length.Vacuum, 2013, vol. 94, pp. 92{103. DOI:10.1016/j.vacuum.2013.01.012
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    . œË ‡Ì‡ÎËÁ ÂÁÛÎ ̧Ú‡ÚÓ‚ Û‰Ó·Ì ÔÓÎ ̧ÁÓ‚‡Ú ̧Òˇ ‚ÂÎË ̃ËÌÓÈQ, Ô‰ÒÚ‡‚Ρ ̨ ̆ÂÈ ÒÓ·ÓÈ ‡ÒıÓ‰ ̇M„‡Á‡ ̃ÂÂÁ Ò ̃ÂÌË ÚÛ· ̊, ÌÓÏËÓ‚‡ÌÌ ̊È Ì‡ ‚ÂÎË ̃ËÌÛ ‡ÒıÓ‰‡ ̇M отв 0ÔË Ò‚Ó·Ó‰ÌÓ-ÏÓÎÂÍÛΡÌÓÏ ËÒÚ ̃ÂÌËË ‚ ‚‡ÍÛÛÏ ̃ÂÂÁ ÓÚ‚ÂÒÚË [44,45]: Q= ̇M / ̇M отв 0, M ̇отв 0= √ π/2.(22) ŒÒÌÓ‚Ì ̊Ï ˇ‚ΡÂÚÒˇ ÒÎÛ ̃‡È ÍÛ„ÎÓÈ ÚÛ· ̊ ÔÓÒÚÓˇÌÌÓ„Ó ‡‰ËÛÒ‡, ‡‚ÌÓ„Ó ‡‰ËÛÒÛ ‚ıÓ‰ÌÓ„Ó Ò ̃ÂÌˡa. ¬ ˆËÍΠ‡·ÓÚ
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    [17,27,43,46]
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     ̄ÂÌË Á‡‰‡ ̃Ë · ̊ÎÓ ÔÓÒÚÓÂÌÓ Ô‡ÍÂÚÓÏ <ÕÂÒ‚ÂÚ‡È-3ƒ> ‰Îˇ χÎÓ„Ó ÓÚÌÓ ̄ÂÌˡ ‰‡‚ÎÂÌËÈ|p10/p20|−1-1(ÎË̇ËÁÓ‚‡Ì̇ˇ Á‡‰‡ ̃‡), ‡ Ú‡ÍÊ ‰‚Ûı ÌÂÎËÌÂÈÌ ̊ı ÂÊËÏÓ‚p10/p20= 2(ÛÏÂÂÌÌ ̊È ÔÂÂÔ‡‰ ‰‡‚ÎÂÌËÈ) Ëp10/p20=∞ (Ú ̃ÂÌË ‚ ‚‡ÍÛÛÏ).

47
Titarev V.A., Shakhov E.M., Utyuzhnikov S.V. Rarefied gas flow through a diverging conical pipe into vacuum.Vacuum, 2014, vol. 101, pp. 10{17. DOI:10.1016/j.vacuum.2013.07.030
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    ˇ‚ΡÂÚÒˇ ÒÎÛ ̃‡ÈL/a= 10, ‰Îˇ ÍÓÚÓÓ„Ó ÂÒÚ ̧  ̄ÂÌË ÏÂÚÓ‰ÓÏ DSMC. ¬ ˆËÚËÛÂÏ ̊ı ‡·ÓÚ‡ ÔÓ͇Á‡ÌÓ ÓÚÎË ̃ÌÓ ÒÓ‚Ô‡‰ÂÌË ÂÁÛÎ ̧Ú‡ÚÓ‚ Ò ̃ÂÚ‡ ÔÓS-ÏÓ‰ÂÎ ̧ÌÓÏÛ ÍËÌÂÚË ̃ÂÒÍÓÏÛ Û‡‚ÌÂÌË ̨ Ë ÏÂÚÓ‰‡ DMSC; ‡ÒıÓʉÂÌË ÏÂÊ‰Û ‡Ò ̃ÂÚ‡ÏË ÒÓÒÚ‡‚ΡÂÚ Ì ·ÓΠ1%. œÓÏËÏÓ ÚÛ· ̊ ÔÓÒÚÓˇÌÌÓ„Ó ‡‰ËÛÒ‡,  ̄ÂÌˡ Á‡‰‡ ̃Ë ÔÓÒÚÓÂÌÓ ‰Îˇ ÍÓÌË ̃ÂÒÍÓÈ ÚÛ· ̊ ‰Îˇ ‡ÁÎË ̃Ì ̊ı Á̇ ̃ÂÌËÈ ‡‰ËÛÒÓ‚ ‚ıÓ‰ÌÓ„Ó Ë ‚ ̊ıÓ‰ÌÓ„Ó Ò ̃ÂÌËÈ
    Exact
    [47]
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    , ‡ Ú‡ÍÊ ‰Îˇ ÒÓÒÚ‡‚ÌÓÈ ÍÛ„ÎÓÈ ÚÛ· ̊ [48], ÒÓÒÚÓˇ ̆ÂÈ ËÁ ‰‚Ûı ̃‡ÒÚÂÈ Ó‰Ë̇ÍÓ‚ÓÈ ‰ÎËÌ ̊ ÔÓÒÚÓˇÌÌÓ„Ó ‡‰ËÛÒ‡; ÔË ̋ÚÓÏ ÓÚÌÓ ̄ÂÌË ‡‰ËÛÒÓ‚ ̃‡ÒÚÂÈ ‡‚ÌÓ2. Շ˷ÓΠËÌÚÂÂÒÌ ̊Ï Ò ÚÓ ̃ÍË ÁÂÌˡ ‚ÓÁÌË͇ ̨ ̆ÂÈ Í‡ÚËÌ ̊ Ú ̃ÂÌˡ ˇ‚ΡÂÚÒˇ ÒÎÛ ̃‡È ‡Ò ̄Ëˇ ̨ ̆ÂÈÒˇ ÒÓÒÚ‡‚ÌÓÈ ÚÛ· ̊, ‚ ÍÓÚÓÓÈ ‚ÚÓ‡ˇ ÒÂÍˆËˇ, ÔËÏ ͇̊ ̨ ̆‡ˇ Í ‚‡ÍÛÛÏÌÓÈ Í‡ÏÂÂ, ËÏÂÂÚ ‚‰‚Ó ·ÓÎ ̧ ̄ËÈ ‡‰ËÛÒ ÔÓ Ò‡‚ÌÂÌË ̨ Ò ÔÂ‚ÓÈ.

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Titarev V.A., Shakhov E.V. Rarefied gas flow into vacuum through a pipe composed of two circular sections of different radii.Vacuum, 2014. DOI:10.1016/j.vacuum.2014.02.019(in press).
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  1. In-text reference with the coordinate start=38481
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     ̄ÂÌË ÏÂÚÓ‰ÓÏ DSMC. ¬ ˆËÚËÛÂÏ ̊ı ‡·ÓÚ‡ ÔÓ͇Á‡ÌÓ ÓÚÎË ̃ÌÓ ÒÓ‚Ô‡‰ÂÌË ÂÁÛÎ ̧Ú‡ÚÓ‚ Ò ̃ÂÚ‡ ÔÓS-ÏÓ‰ÂÎ ̧ÌÓÏÛ ÍËÌÂÚË ̃ÂÒÍÓÏÛ Û‡‚ÌÂÌË ̨ Ë ÏÂÚÓ‰‡ DMSC; ‡ÒıÓʉÂÌË ÏÂÊ‰Û ‡Ò ̃ÂÚ‡ÏË ÒÓÒÚ‡‚ΡÂÚ Ì ·ÓΠ1%. œÓÏËÏÓ ÚÛ· ̊ ÔÓÒÚÓˇÌÌÓ„Ó ‡‰ËÛÒ‡,  ̄ÂÌˡ Á‡‰‡ ̃Ë ÔÓÒÚÓÂÌÓ ‰Îˇ ÍÓÌË ̃ÂÒÍÓÈ ÚÛ· ̊ ‰Îˇ ‡ÁÎË ̃Ì ̊ı Á̇ ̃ÂÌËÈ ‡‰ËÛÒÓ‚ ‚ıÓ‰ÌÓ„Ó Ë ‚ ̊ıÓ‰ÌÓ„Ó Ò ̃ÂÌËÈ [47], ‡ Ú‡ÍÊ ‰Îˇ ÒÓÒÚ‡‚ÌÓÈ ÍÛ„ÎÓÈ ÚÛ· ̊
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    , ÒÓÒÚÓˇ ̆ÂÈ ËÁ ‰‚Ûı ̃‡ÒÚÂÈ Ó‰Ë̇ÍÓ‚ÓÈ ‰ÎËÌ ̊ ÔÓÒÚÓˇÌÌÓ„Ó ‡‰ËÛÒ‡; ÔË ̋ÚÓÏ ÓÚÌÓ ̄ÂÌË ‡‰ËÛÒÓ‚ ̃‡ÒÚÂÈ ‡‚ÌÓ2. Շ˷ÓΠËÌÚÂÂÒÌ ̊Ï Ò ÚÓ ̃ÍË ÁÂÌˡ ‚ÓÁÌË͇ ̨ ̆ÂÈ Í‡ÚËÌ ̊ Ú ̃ÂÌˡ ˇ‚ΡÂÚÒˇ ÒÎÛ ̃‡È ‡Ò ̄Ëˇ ̨ ̆ÂÈÒˇ ÒÓÒÚ‡‚ÌÓÈ ÚÛ· ̊, ‚ ÍÓÚÓÓÈ ‚ÚÓ‡ˇ ÒÂÍˆËˇ, ÔËÏ ͇̊ ̨ ̆‡ˇ Í ‚‡ÍÛÛÏÌÓÈ Í‡ÏÂÂ, ËÏÂÂÚ ‚‰‚Ó ·ÓÎ ̧ ̄ËÈ ‡‰ËÛÒ ÔÓ Ò‡‚ÌÂÌË ̨ Ò ÔÂ‚ÓÈ.

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Titarev V.A. Direct numerical solution of model kinetic equations for flows in arbitrary threedimensional geometries.Proc. 28thInt. Symp. on Rarefied Gas Dynamics 2012. AIP Conference Proceedings 1501, Amer. Inst. Physics, 2012, pp. 262{271. DOI:10.1063/1.4769518
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    — ÓÒÚÓÏ Á̇ ̃ÂÌˡ Ô‡‡ÏÂÚ‡ ‡ÁÂÊÂÌÌÓÒÚË (ÛÏÂÌ ̧ ̄ÂÌˡ ̃ËÒ· ÌÛ‰ÒÂ̇) ÒÚÛÍÚÛ‡ Û‰‡ÌÓÈ ‚ÓÎÌ ̊ ÒÚ‡ÌÓ‚ËÚÒˇ ·ÓΠ‚ ̊‡ÊÂÌÌÓÈ Ë ÔÓÁËˆËˇ χıÓ‚ÒÍÓ„Ó ‰ËÒ͇ ÛÒڇ̇‚ÎË‚‡ÂÚÒˇ ÓÍÓÎÓz≈9. «‡‰‡ ̃Ë ‚Ì ̄ÌÂ„Ó Ó·ÚÂ͇Ìˡ ÚÂÎ ÒÎÓÊÌÓÈ ÙÓÏ ̊.¬ ͇ ̃ÂÒÚ‚Â ËÎÎ ̨ÒÚ‡ˆËË ÔËÏÂÌÂÌˡ Ô‡ÍÂÚ‡ ÔÓ„‡ÏÏ Í Á‡‰‡ ̃‡Ï ‚Ì ̄ÌÂÈ ‡ ̋Ó‰Ë̇ÏËÍË ÔÓÒÚÓËÏ ÔÓΠڠ̃ÂÌˡ ‚ÓÍÛ„ ‰‚Ûı „ÂÓÏÂÚË ̃ÂÒÍËı ÏÓ‰ÂÎÂÈ
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    . œÂ‚‡ˇ ÏÓ‰ÂÎ ̧ Ô‰ÒÚ‡‚ΡÂÚ ÒÓ·ÓÈ ÏÓ‰ÂÎ ̧ ‡ÍÂÚ ̊ Ò ‰‚ÛÏˇ Í ̊Î ̧ˇÏË Ë ̃ÂÚ ̊ ̧Ïˇ ÒÚ‡·ËÎËÁ‡ÚÓ‡ÏË. √ÂÓÏÂÚˡ ÏÓ‰ÂÎË ÔÓ‰Ó·ÌÓ ÓÔË҇̇ ‚ [50]. ƒË‡ÏÂÚ ‡ÍÂÚ ̊ Ë Á̇ ̃ÂÌˡ ‰‡‚ÎÂÌˡ Ë ÚÂÏÔÂ‡ÚÛ ̊ ‚ ̇·Â„‡ ̨ ̆ÂÏ ÔÓÚÓÍ ‚ ̊·Ë‡ ̨ÚÒˇ ‚ ͇ ̃ÂÒÚ‚Â Ï‡Ò ̄Ú‡·Ó‚ ‰ÎËÌ ̊l∗,p∗,T∗.

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Garanzha V.A., Kudryavtseva L.N., Utyuzhnikov S.V. Variational method for untangling and optimization of spatial meshes.J. Comp. and Appl. Math., 2014, vol. 269, pp. 24{41.
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    ‡ÊÂÌÌÓÈ Ë ÔÓÁËˆËˇ χıÓ‚ÒÍÓ„Ó ‰ËÒ͇ ÛÒڇ̇‚ÎË‚‡ÂÚÒˇ ÓÍÓÎÓz≈9. «‡‰‡ ̃Ë ‚Ì ̄ÌÂ„Ó Ó·ÚÂ͇Ìˡ ÚÂÎ ÒÎÓÊÌÓÈ ÙÓÏ ̊.¬ ͇ ̃ÂÒÚ‚Â ËÎÎ ̨ÒÚ‡ˆËË ÔËÏÂÌÂÌˡ Ô‡ÍÂÚ‡ ÔÓ„‡ÏÏ Í Á‡‰‡ ̃‡Ï ‚Ì ̄ÌÂÈ ‡ ̋Ó‰Ë̇ÏËÍË ÔÓÒÚÓËÏ ÔÓΠڠ̃ÂÌˡ ‚ÓÍÛ„ ‰‚Ûı „ÂÓÏÂÚË ̃ÂÒÍËı ÏÓ‰ÂÎÂÈ [18,49]. œÂ‚‡ˇ ÏÓ‰ÂÎ ̧ Ô‰ÒÚ‡‚ΡÂÚ ÒÓ·ÓÈ ÏÓ‰ÂÎ ̧ ‡ÍÂÚ ̊ Ò ‰‚ÛÏˇ Í ̊Î ̧ˇÏË Ë ̃ÂÚ ̊ ̧Ïˇ ÒÚ‡·ËÎËÁ‡ÚÓ‡ÏË. √ÂÓÏÂÚˡ ÏÓ‰ÂÎË ÔÓ‰Ó·ÌÓ ÓÔË҇̇ ‚
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    [50]
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    . ƒË‡ÏÂÚ ‡ÍÂÚ ̊ Ë Á̇ ̃ÂÌˡ ‰‡‚ÎÂÌˡ Ë ÚÂÏÔÂ‡ÚÛ ̊ ‚ ̇·Â„‡ ̨ ̆ÂÏ ÔÓÚÓÍ ‚ ̊·Ë‡ ̨ÚÒˇ ‚ ͇ ̃ÂÒÚ‚Â Ï‡Ò ̄Ú‡·Ó‚ ‰ÎËÌ ̊l∗,p∗,T∗. “ÂÏÔÂ‡ÚÛ‡ ÔÓ‚ÂıÌÓÒÚË ÔËÌËÏ‡Î‡Ò ̧ ‡‚ÌÓÈ ÚÂÏÔÂ‡ÚÛ ̇·Â„‡ ̨ ̆Â„Ó ÔÓÚÓ͇ (ıÓÎÓ‰ÌÓ ÚÂÎÓ).

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Vaganov A.V., Drozdov S.M., Kosykh A.P., Nersesov G.G., Chelysheva I.F., Yumashev V.L. [Numerical simulation of aerodymics winged re-entry space vehicle].Uchenye zapiski TsAGI, 2009, vol. 40, no. 2, pp. 3{15. (English translation:TsAGI Science Journal, 2009, vol. 40, iss. 2, pp. 131{149. DOI:10.1615/TsAGISciJ.v40.i2.10).
Total in-text references: 1
  1. In-text reference with the coordinate start=42303
    Prefix
    —ΉÛÂÚ Ú‡Í Ê ÓÚÏÂÚËÚ ̧, ̃ÚÓ Ì·ÓÎ ̧ ̄‡ˇ ÔÓÚÂˇ ÒËÏÏÂÚËË Ú ̃ÂÌˡ ‚ ÔÎÓÒÍÓÒÚËy{zˇ‚ΡÂÚÒˇ ÒΉÒÚ‚ËÂÏ ËÒÔÓÎ ̧ÁÓ‚‡ÌË ‰ÓÒÚ‡ÚÓ ̃ÌÓ „Û·ÓÈ Ë ÌÂÒËÏÏÂÚË ̃ÌÓÈ ÚÂÚ‡ ̋‰‡Î ̧ÌÓÈ ÒÂÚÍË. ¬ÚÓ‡ˇ ÏÓ‰ÂÎ ̧ Ô‰ÒÚ‡‚ΡÂÚÒˇ ÒÓ·ÓÈ ¬ÓÁ‰Û ̄ÌÓ- ÓÒÏË ̃ÂÒÍËÈ ¿ÔÔ‡‡Ú (¬ ¿) ÷¿√»
    Exact
    [51,52]
    Suffix
    . ÃÓ‰ÂÎ ̧ ¬ ¿ ËÏÂÂÚ ÒÎÓÊÌÛ ̨ ÙÓÏÛ Ë ÒÓÒÚÓËÚ ËÁ Ù ̨ÁÂÎˇÊ‡ Ò Á‡ÚÛÔÎÂÌÌ ̊Ï ÌÓÒÍÓÏ, ‰‚Ûı Í ̊Î ̧‚, ‚ÂÚË͇Π̧ÌÓ„Ó ÍËΡ Ë ̆ËÚ͇. Œ· ̆‡ˇ ‰ÎË̇ Ú· ÒÓ ̆ËÚÍÓÏ |10Ï. Œ· ̆‡ˇ ‰ÎË̇ ¬ ¿ Ë Á̇ ̃ÂÌˡ ‰‡‚ÎÂÌˡ Ë ÚÂÏÔÂ‡ÚÛ ̊ ‚ ̇·Â„‡ ̨ ̆ÂÏ ÔÓÚÓÍ ‚ ̊·Ë‡ ̨ÚÒˇ ‚ ͇ ̃ÂÒÚ‚Â Ï‡Ò ̄Ú‡·Ó‚ ‰ÎËÌ ̊l∗,p∗,T∗.

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Vaganov A.V., Drozdov S.M., Zadonskiy S.M., Chelysheva I.F., Kosykh A.P., Nersesov G.G., Yumashev V.L. [Study of aerodynamics of the aerospace vehicle with a deflected balancing flap].Uchenye zapiski TsAGI, 2009, vol. 40, no. 5, pp. 3{15. (English translation:TsAGI Science Journal, 2009, vol. 40, iss. 5, pp. 517{534. DOI:10.1615/TsAGISciJ.v40.i5.10).
Total in-text references: 1
  1. In-text reference with the coordinate start=42303
    Prefix
    —ΉÛÂÚ Ú‡Í Ê ÓÚÏÂÚËÚ ̧, ̃ÚÓ Ì·ÓÎ ̧ ̄‡ˇ ÔÓÚÂˇ ÒËÏÏÂÚËË Ú ̃ÂÌˡ ‚ ÔÎÓÒÍÓÒÚËy{zˇ‚ΡÂÚÒˇ ÒΉÒÚ‚ËÂÏ ËÒÔÓÎ ̧ÁÓ‚‡ÌË ‰ÓÒÚ‡ÚÓ ̃ÌÓ „Û·ÓÈ Ë ÌÂÒËÏÏÂÚË ̃ÌÓÈ ÚÂÚ‡ ̋‰‡Î ̧ÌÓÈ ÒÂÚÍË. ¬ÚÓ‡ˇ ÏÓ‰ÂÎ ̧ Ô‰ÒÚ‡‚ΡÂÚÒˇ ÒÓ·ÓÈ ¬ÓÁ‰Û ̄ÌÓ- ÓÒÏË ̃ÂÒÍËÈ ¿ÔÔ‡‡Ú (¬ ¿) ÷¿√»
    Exact
    [51,52]
    Suffix
    . ÃÓ‰ÂÎ ̧ ¬ ¿ ËÏÂÂÚ ÒÎÓÊÌÛ ̨ ÙÓÏÛ Ë ÒÓÒÚÓËÚ ËÁ Ù ̨ÁÂÎˇÊ‡ Ò Á‡ÚÛÔÎÂÌÌ ̊Ï ÌÓÒÍÓÏ, ‰‚Ûı Í ̊Î ̧‚, ‚ÂÚË͇Π̧ÌÓ„Ó ÍËΡ Ë ̆ËÚ͇. Œ· ̆‡ˇ ‰ÎË̇ Ú· ÒÓ ̆ËÚÍÓÏ |10Ï. Œ· ̆‡ˇ ‰ÎË̇ ¬ ¿ Ë Á̇ ̃ÂÌˡ ‰‡‚ÎÂÌˡ Ë ÚÂÏÔÂ‡ÚÛ ̊ ‚ ̇·Â„‡ ̨ ̆ÂÏ ÔÓÚÓÍ ‚ ̊·Ë‡ ̨ÚÒˇ ‚ ͇ ̃ÂÒÚ‚Â Ï‡Ò ̄Ú‡·Ó‚ ‰ÎËÌ ̊l∗,p∗,T∗.