Geodesic
Symmetric difference schemes of componentwise splitting and equivalent predictor-corrector scheme based on the Godunov method as applied to multidimensional gasdynamic simulation
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 49 (2009) no. 11, pp. 1970-1987 
O. A. Makotra; N. Ya. Moiseev; I. Yu. Silant'eva; T. V. Topchii; N. L. Frolova. Symmetric difference schemes of componentwise splitting and equivalent predictor-corrector scheme based on the Godunov method as applied to multidimensional gasdynamic simulation. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 49 (2009) no. 11, pp. 1970-1987. http://geodesic.mathdoc.fr/item/ZVMMF_2009_49_11_a5/
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     title = {Symmetric difference schemes of componentwise splitting and equivalent predictor-corrector scheme based on the {Godunov} method as applied to multidimensional gasdynamic simulation},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {1970--1987},
     year = {2009},
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     number = {11},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2009_49_11_a5/}
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Voir la notice de l'article provenant de la source Math-Net.Ru

An approach is described for improving the accuracy of numerical solutions to multidimensional gasdynamic problems produced by Godunov's schemes. The basic idea behind the approach is to construct symmetric difference schemes based on splitting with respect to spatial variables with the subsequent transformation into equivalent predictor-corrector schemes. It is shown that the computation of “large” values by solving the one-dimensional Riemann problem at the interface of two neighboring cells leads to approximation errors in Godunov’s schemes. It is proposed to reconstruct large values so as to eliminate this source of errors. The time integration step in the modified schemes is consistent with that in the one-dimensional schemes and, on spatially uniform meshes, is 2 and 3 times larger than that in Godunov's classical schemes for two- and three-dimensional problems, respectively. The numerical results obtained for test problems confirm the improvement of the accuracy of solutions produced by the modified schemes.

[1] Godunov S. K., “Raznostnyi metod chislennogo rascheta razryvnykh reshenii uravnenii gidrodinamiki”, Matem. sb., 47:3 (1959), 271–306 | MR | Zbl

[2] Godunov S. K., Zabrodin A. B., Ivanov M. Ya. i dr., Chislennoe reshenie mnogomernykh zadach gazovoi dinamiki, Nauka, M., 1976 | MR | Zbl

[3] Kulikovskii A. G., Pogorelov N. V., Semenov A. Yu., Matematicheskie voprosy chislennogo resheniya giperbolicheskikh sistem uravnenii, Fizmatlit, M., 2001 | MR

[4] Godunov S. K., Zabrodin A. B., Prokopov G. P., “Raznostnaya skhema dlya dvumernykh nestatsionarnykh zadach gazovoi dinamiki i raschet obtekaniya s otoshedshei udarnoi volnoi”, Zh. vychisl. matem. i matem. fiz., 1:6 (1961), 1020–1050 | MR | Zbl

[5] Rozhdestvenskii B. L., Yanenko H. N., Sistemy kvazilineinykh uravnenii i ikh primenenie k gazovoi dinamike, Nauka, M., 1978

[6] Abgrall R., “Approximation du probleme de Riemann vraiment multidimensionneles equations d'Euler par une methode de type Roe, I: La linearization”, Cr. Acad. Sci. Ser. I, 319 (1994), 499–504 ; “II: Solution du pro,leme de Riemann fpproche”, 625–629 | MR | Zbl | MR | Zbl

[7] LeVeque R. J., “Wave propagation algorithms for multidimensional hyperbolic systems”, J. Comput. Phys., 131 (1997), 327 | DOI | Zbl

[8] Gilquin H., Laurens J., Roiser C., “Multi-dimensional Riemann problems for linear hyperbolic systems”, Notes Number. Fluid Mech., 43 (1993), 284 | MR | Zbl

[9] Brio M., Zakharian A. R., Webb G. M., “Two-dimensional Riemann solver for Euler equations of gas dynamics”, J. Comput. Phys., 167 (2001), 177–195 | DOI | Zbl

[10] Vasilev E. I., “$W$-modifikatsiya metoda S. K. Godunova i ee primenenie dlya dvumernykh nestatsionarnykh techenii zapylennogo gaza”, Zh. vychisl. matem. i matem. fiz., 36:1 (1996), 122–135 | MR

[11] Colella P., “Multidimensional upwind methods for hyperbolic conservation laws”, J. Comput. Phys., 87 (1990), 171–200 | DOI | MR | Zbl

[12] Moiseev H. Ya., Silanteva I. Yu., “Raznostnye skhemy proizvolnogo poryadka approksimatsii dlya resheniya lineinykh uravnenii perenosa s postoyannymi koeffitsientami metodom Godunova s antidiffuziei”, Zh. vychisl. matem. i matem. fiz., 48:7 (2008), 1282–1293

[13] Peaceman D. W., Rachford H. H., Jr., “The numerical solution of parabolic and elliptic differential equations”, J. Soc. Industr. Appl. Math., 3:1 (1955), 28–42 | DOI | MR

[14] Douglas J., Jr., “On the numerical integration of $u_{xx}+u_{yy}=u_t$ by implicit methods”, J. Soc. Industr. Appl. Math., 3:1 (1955), 42–65 | DOI | MR | Zbl

[15] Bagrinovskii K. A., Godunov S. K., “Raznostnye skhemy dlya mnogomernykh zadach”, Dokl. AN SSSR, 115 (1957), 431–433 | MR

[16] Yanenko H. H., Metod drobnykh shagov resheniya mnogomernykh zadach matematicheskoi fiziki, Nauka, Novosibirsk, 1967

[17] Marchuk G. I., Metody rasschepleniya, Nauka, M., 1988 | MR

[18] Kovenya V. M., Tarnavskii G. A., Chernyi S. G., Primenenie metoda rasschepleniya v zadachakh aerodinamiki, Nauka, SO, Novosibirsk, 1990

[19] Godunov S. K., Vospominaniya o raznostnykh skhemakh, Nauchn. kniga, Novosibirsk, 1997

[20] Godunov S. K., Zabrodin A. B., “O raznostnykh skhemakh vtorogo poryadka tochnosti dlya mnogomernykh zadach”, Zh. vychisl. matem. i matem. fiz., 2:4 (1962), 706–708 | MR | Zbl

[21] Samarskii A. A., “O printsipe additivnosti dlya postroeniya ekonomichnykh raznostnykh skhem”, Dokl. AN SSSR, 165:6 (1965)

[22] Shokin Yu. I., Metod differentsialnogo priblizheniya, Nauka, Novosibirsk, 1979 | MR

[23] Dyakonov V. P., Maple 8 v matematike, fizike i obrazovanii, SOLON-Press, M., 2003

[24] Moiseev N. Ya., “Ob odnoi modifikatsii raznostnoi skhemy Godunova”, VANT. Ser. Metodiki i programmy chisl. resheniya zadach matem. fiz., 1986, no. 3, 35–43 | MR