Geodesic
An explicit-iterative scheme for the time integration of the Navier--Stokes equations
Matematičeskoe modelirovanie, Tome 32 (2020) no. 4, pp. 57-74.

Voir la notice de l'article provenant de la source Math-Net.Ru

A new integration algorithm is proposed for the time-dependent Navier–Stokes equations of compressible flows. We propose operator splitting method to split the equations into convection and diffusion parts which are solved successively at each time step. The convection part is solved by the Godunov method, the diffusion part is solved by the explicit-iteration Chebyshev scheme which has no stability restriction on the time-step size. The resulting scheme ensures the fulfillment of the main conservation laws on arbitrary unstructured grids. The explicit nature of the calculations ensures the efficiency of the use of the scheme in various parallel technologies.
Keywords: numerical simulation, Navier–Stokes equations, splitting method, Chebyshev explicit-iterative scheme. 
@article{MM_2020_32_4_a4,
     author = {V. T. Zhukov and O. B. Feodoritova and N. D. Novikova and A. P. Duben},
     title = {An explicit-iterative scheme for the time integration of the {Navier--Stokes} equations},
     journal = {Matemati\v{c}eskoe modelirovanie},
     pages = {57--74},
     publisher = {mathdoc},
     volume = {32},
     number = {4},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MM_2020_32_4_a4/}
}
TY  - JOUR
AU  - V. T. Zhukov
AU  - O. B. Feodoritova
AU  - N. D. Novikova
AU  - A. P. Duben
TI  - An explicit-iterative scheme for the time integration of the Navier--Stokes equations
JO  - Matematičeskoe modelirovanie
PY  - 2020
SP  - 57
EP  - 74
VL  - 32
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MM_2020_32_4_a4/
LA  - ru
ID  - MM_2020_32_4_a4
ER  - 
%0 Journal Article
%A V. T. Zhukov
%A O. B. Feodoritova
%A N. D. Novikova
%A A. P. Duben
%T An explicit-iterative scheme for the time integration of the Navier--Stokes equations
%J Matematičeskoe modelirovanie
%D 2020
%P 57-74
%V 32
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MM_2020_32_4_a4/
%G ru
%F MM_2020_32_4_a4
V. T. Zhukov; O. B. Feodoritova; N. D. Novikova; A. P. Duben. An explicit-iterative scheme for the time integration of the Navier--Stokes equations. Matematičeskoe modelirovanie, Tome 32 (2020) no. 4, pp. 57-74. http://geodesic.mathdoc.fr/item/MM_2020_32_4_a4/

[1] L. G. Lojcyanskij, Mekhanika zhidkosti i gaza, Nauka, M., 1987, 840 pp.

[2] G. I. Marchuk, Metody rasshchepleniya, Nauka, M., 1988, 263 pp.

[3] V. T. ZHukov, YU. G. Rykov, O. B. Feodoritova, “Matematicheskaya model' techeniya mno-gokomponentnoj smesi gazov s uchetom vozmozhnosti vozniknoveniya zhidkoj fazy”, Preprinty IPM im. M.V. Keldysha, 2018, 183, 36 pp.

[4] I. M. Gel'fand, O. V. Lokucievskij, O raznostnyh skhemah dlya resheniya uravneniya teploprovodnosti, Fizmatgiz, M., 1962, 340 pp.

[5] V. T. ZHukov, Raznostnye skhemy lokal'nyh iteracij dlya parabolicheskih uravnenij, Preprint IPM im. M.V. Keldysha AN SSSR, No 183, 1986, 20 pp.

[6] V. T. ZHukov, “YAvno-iteracionnye skhemy dlya parabolicheskih uravnenij”, VANT. Seriya: Matematicheskoe modelirovanie fizicheskih processov, 1993, no. 4, 40–46

[7] V. T. Zhukov, “On explicit methods for the time integration of parabolic equations”, Math. Models Simul., 3:3 (2011), 311–332 | DOI | MR | Zbl

[8] V. T. Zhukov, A. V. Zabrodin, O. B. Feodoritova, “Features of the numerical simulation of the target of inertial thermonuclear synthesis in the approximation of heat conduction gas dynamics”, Comput. Math. Math. Phys., 34:12 (1994), 1591–1601 | MR | MR | Zbl | Zbl

[9] V. O. Lokucievskij, O. V. Lokucievskij, Primenenie chebyshevskih parametrov dlya chislennogo resheniya nekotoryh evolyucionnyh zadach, Preprint IPM im. M.V. Keldysha AN SSSR, No 99, 1984, 32 pp. | MR

[10] V. O. Lokucievskij, O. V. Lokucievskij, “O chislennom reshenii kraevyh zadach dlya uravnenij parabolicheskogo tipa”, Dokl. AN SSSR, 291:3 (1986), 540–544 | MR

[11] V. T. ZHukov, Chislennye eksperimenty po resheniyu uravneniya teploprovodnosti metodom lokal'nyh iteracij, Preprint IPM im. M.V. Keldysha AN SSSR, No 97, 1984, 22 pp.

[12] V. I. Lebedev, YAvnye raznostnye skhemy s peremennymi shagami po vremeni dlya resheniya zhestkih sistem uravnenij, Preprint OVM AN SSSR, No 177, 1987

[13] V. I. Lebedev, “Explicit difference schemes with time-variable steps for solving stiff systems of equations”, Sovjet. J. Numer.Analys. Math. Modelling, 4:2 (1989), 111–136 | MR

[14] V. I. Lebedev, “Kak reshat' yavnymi metodami zhestkie sistemy differencial'nyh uravnenij”, Vychislitel'nye processy i sistemy, 8, ed. G.I. Marchuk, Nauka, M., 1991, 237–291

[15] I. V. Abalakin, P. A. Bahvalov, A. V. Gorobec, A. P. Duben', T. K. Kozubskaya, “Parallel'nyj programmnyj kompleks NOISEtte dlya krupnomasshtabnyh raschetov zadach aerodinamiki i aeroakustiki”, Vychisl. metody i programmir., 13:3 (2012), 110–125

[16] P. Batten, D. M. Ingram, R. Saunders, D. M. Causon, “A time-splitting approach to solving the Navier-Stokes equations”, Computers Fluids, 25 (1993), 421–431 | DOI | MR

[17] V. M. Kovenya, P. V. Babintsev, “Application of Splitting Algorithms in the Method of Finite Volumes for Numerical Solution of the Navier–Stokes Equations”, J. Appl. Industr. Math., 12:3 (2018), 479–491 | DOI | MR

[18] A. S. Shvedov, V. T. Zhukov, “Explicit iterative difference schemes for parabolic equations”, Russian J. Numer. Anal. Math. Modelling, 13:2 (1998), 133–148 (In English) | DOI | MR | Zbl

[19] V. I. Lebedev, S. A. Finogenov, “Ordering of the iterative parameters in the cyclical Chebyshev iterative method”, Zh. vychisl. Mat. Mat. Fiz., 11:2 (1971), 425–438 | Zbl

[20] V. I. Polezhaev, Chislennoe reshenie uravnenij Navie-Stoksa dlya techeniya i teploobmena v zamknutoj dvumernoj oblasti, Diss. na soiskanie uchenoj stepeni kand. tekhnicheskih nauk, NIITP, M., 1967, 196 pp.