Minimal dissipation hybrid bicompact schemes for hyperbolic equations

M. D. Bragin   ; B. V. Rogov

Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 56 (2016) no. 6, pp. 958-972

Résumé

New monotonicity-preserving hybrid schemes are proposed for multidimensional hyperbolic equations. They are convex combinations of high-order accurate central bicompact schemes and upwind schemes of first-order accuracy in time and space. The weighting coefficients in these combinations depend on the local difference between the solutions produced by the high- and low-order accurate schemes at the current space-time point. The bicompact schemes are third-order accurate in time, while having the fourth order of accuracy and the first difference order in space. At every time level, they can be solved by marching in each spatial variable without using spatial splitting. The upwind schemes have minimal dissipation among all monotone schemes constructed on a minimum space-time stencil. The hybrid schemes constructed has been successfully tested as applied to a number of two-dimensional gas dynamics benchmark problems.

Bibliographie

[1] Lax P. D., Hyperbolic systems of conservation laws and the mathematical theory of shock waves, SIAM, 1973 | MR | Zbl

[2] Vázquez-Cendón E., Hidalgo A., Navarro P. G., Cea L., Numerical methods for hyperbolic equations, Taylor Francis Group, 2013

[3] Lele S. K., “Compact finite difference schemes with spectral-like resolution”, J. Comput. Phys., 103:1 (1992), 16–42 | DOI | MR | Zbl

[4] Xuliaug Liu, Shuhai Zhang, Hanxin Zhang, Chi-Wang Shu, “A new class of central compact schemes with spectral-like resolution. I: Linear schemes”, J. Comput. Phys., 248 (2013), 235–256 | DOI | MR

[5] Belotserkovskii O. M., Byrkin A. P., Mazurov A. P., Tolstykh A. I., “Raznostnyi metod povyshennoi tochnosti dlya rascheta techenii vyazkogo gaza”, Zh. vychisl. matem. i matem. fiz., 22:6 (1982), 1480–1490 | MR | Zbl

[6] Tolstykh A. I., Kompaktnye raznostnye skhemy i ikh primenenie v zadachakh aerogidrodinamiki, Nauka, M., 1990 | MR

[7] Mikhailovskaya M. N., Rogov B. V., “Monotonnye kompaktnye skhemy beguschego scheta dlya sistem uravnenii giperbolicheskogo tipa”, Zh. vychisl. matem. i matem. fiz., 52:4 (2012), 672–695 | MR | Zbl

[8] Rogov B. V., Mikhailovskaya M. N., “O skhodimosti kompaktnykh raznostnykh skhem”, Matem. modelirovanie, 20:1 (2008), 99–116 | MR | Zbl

[9] Sengupta T. K., Ganerwal G., Dipankar A., “High accuracy compact schemes and Gibbs' phenomenon”, J. Sci. Comput., 21:3 (2004), 253–268 | DOI | MR | Zbl

[10] Lax P. D., “Gibbs phenomena”, J. Sci. Comput., 28:2/3 (2006), 445–449 | DOI | MR | Zbl

[11] Godunov S. K., “Raznostnyi metod chislennogo rascheta razryvnykh reshenii uravnenii gidrodinamiki”, Matem. sb., 47(89):3 (1959), 271–306 | MR | Zbl

[12] Cockburn B., Shu C.-W., “Nonlinearly stable compact schemes for shock calculations”, SIAM J. Numer. Anal., 31:3 (1994), 607–627 | DOI | MR | Zbl

[13] Yee H. C., “Explicit and implicit multidimensional compact high-resolution shock-capturing methods: Formulation”, J. Comput. Phys., 131:1 (1997), 216–232 | DOI | Zbl

[14] Ostapenko V. V., “Simmetrichnye kompaktnye skhemy s iskusstvennymi vyazkostyami povyshennogo poryadka divergentnosti”, Zh. vychisl. matem. i matem. fiz., 42:7 (2002), 1019–1038 | MR | Zbl

[15] Fiorina B., Lele S. K., “An artificial nonlinear diffusivity method for supersonic reacting flows with shocks”, J. Comput. Phys., 222 (2006), 246–264 | DOI | MR

[16] Kawai S., Lele S. K., “Localized artificial diffusivity scheme for discontinuity capturing on curvilinear meshes”, J. Comput. Phys., 227:22 (2008), 9498–9526 | DOI | MR | Zbl

[17] Kurganov A., Liu Y., “New adaptive artificial viscosity method for hyperbolic systems of conservation laws”, J. Comput. Phys., 231:24 (2012), 8114–8132 | DOI | MR | Zbl

[18] Ekaterinaris J. A., “Implicit, high-resolution, compact schemes for gas dynamics and aeroacoustics”, J. Comput. Phys., 156:2 (1999), 272–299 | DOI | MR | Zbl

[19] Yee H. C., Sandham N. D., Djomehri M. J., “Low-dissipative high-order shock-capturing methods using characteristic-based filters”, J. Comput. Phys., 150:1 (1999), 199–238 | DOI | MR | Zbl

[20] Yee H. C., Sjögreen B., “Adaptive filtering and limiting in compact high order methods for multiscale gas dynamics and MHD systems”, Comput. Fluids, 37:5 (2008), 593–619 | DOI | MR | Zbl

[21] Darian H. M., Esfahanian V., Hejranfar K., “A shock-detecting sensor for filtering of high-order compact finite difference schemes”, J. Comput. Phys., 230:3 (2011), 494–514 | DOI | MR | Zbl

[22] Adams N. A., Shariff K., “A high-resolution hybrid compact-ENO scheme for shock-turbulence interaction problems”, J. Comput. Phys., 127:1 (1996), 27–51 | DOI | MR | Zbl

[23] Pirozzoli S., “Conservative hybrid compact-WENO schemes for shock-turbulence interaction”, J. Comput. Phys., 178:1 (2002), 81–117 | DOI | MR | Zbl

[24] Ren Y.-X., Liu M., Zhang H., “A characteristic-wise hybrid compact-WENO scheme for solving hyperbolic conservation laws”, J. Comput. Phys., 192:2 (2003), 365–386 | DOI | MR | Zbl

[25] Shen Y., Yang G., “Hybrid finite compact-WENO schemes for shock calculation”, Int. J. Numer. Meth. Fluids, 53:4 (2007), 531–560 | DOI | MR | Zbl

[26] Deng X., Maekawa H., “Compact high-order accurate nonlinear schemes”, J. Comput. Phys., 130:1 (1997), 77–91 | DOI | MR | Zbl

[27] Deng X., Zhang H., “Developing high-order weighted compact nonlinear schemes”, J. Comput. Phys., 165:1 (2000), 22–44 | DOI | MR | Zbl

[28] Jiang L., Shan H., Liu C., “Weighted compact scheme for shock capturing”, Int. J. Comput. Fluid Dyn., 15:2 (2001), 147–155 | DOI | MR | Zbl

[29] Zhang S., Jiang S., Shu C.-W., “Development of nonlinear weighted compact schemes with increasingly higher order accuracy”, J. Comput. Phys., 227:15 (2008), 7294–7321 | DOI | MR | Zbl

[30] Ghosh D., Baeder J. D., “Compact reconstruction schemes with weighted ENO limiting for hyperbolic conservation laws”, SIAM J. Sci. Comput., 34:3 (2012), A1678–A1706 | DOI | MR | Zbl

[31] Guo Y., Xiong T., Shi Y., “A positivity-preserving high order finite volume compact-WENO scheme for compressible Euler equations”, J. Comput. Phys., 274 (2014), 505–523 | DOI | MR

[32] Huankun Fu, Zhengjie Wang, Yonghua Yan, Chaoqun Liu, “Modified weighted compact scheme with global weights for shock capturing”, Comput. Fluids, 96 (2014), 165–176 | DOI | MR

[33] Rogov B. V., Mikhailovskaya M. N., “Monotonnaya vysokotochnaya kompaktnaya skhema beguschego scheta dlya kvazilineinykh uravnenii giperbolicheskogo tipa”, Matem. modelirovanie, 23:12 (2011), 65–78 | Zbl

[34] Rogov B. V., “Monotonnaya bikompaktnaya skhema dlya kvazilineinykh uravnenii giperbolicheskogo tipa”, Dokl. AN, 446:5 (2012), 504–509 | Zbl

[35] Rogov B. V., “Vysokotochnaya monotonnaya kompaktnaya skhema beguschego scheta dlya mnogomernykh uravnenii giperbolicheskogo tipa”, Zh. vychisl. matem. i matem. fiz., 53:2 (2013), 264–274 | DOI | MR | Zbl

[36] Chikitkin A. V., Rogov B. V., Utyuzhnikov S. V., “High-order accurate monotone compact rumming scheme for multidimensional hyperdolic equations”, Appl. Numer. Math., 93 (2015), 150–163 | DOI | MR | Zbl

[37] Grudnitskii V. G., Prokhorchuk Yu. A., “Odin priem postroeniya raznostnykh skhem s proizvolnym poryadkom approksimatsii differentsialnykh uravnenii v chastnykh proizvodnykh”, Dokl. AN, 234:6 (1977), 1249–1252 | MR | Zbl

[38] Belotserkovskii O. M., Grudnitskii V. G., Prokhorchuk Yu. A., “Raznostnaya skhema vtorogo poryadka tochnosti na minimalnom shablone dlya giperbolicheskikh uravnenii”, Zh. vychisl. matem. i matem. fiz., 23:1 (1983), 119–126 | MR | Zbl

[39] Wornom S. F., “Application of two-point implicit central-difference methods to hyperbolic systems”, Comput. Fluids, 20:3 (1991), 321–331 | DOI | MR | Zbl

[40] Tolstykh A. I., “O gibridnykh skhemakh s multioperatorami vysokogo poryadka dlya scheta razryvnykh reshenii”, Zh. vychisl. matem. i matem. fiz., 53:9 (2013), 1481–1502 | DOI | Zbl

[41] Rogov B. V., Mikhailovskaya M. N., “Monotonnye bikompaktnye skhemy dlya lineinogo uravneniya perenosa”, Dokl. AN, 436:5 (2011), 600–605 | Zbl

[42] Toro E. F., Riemann solvers and numerical methods for fluid dynamics: a practical introduction, Springer, 2009 | MR | Zbl

[43] Liska R., Wendroff B., “Comparison of several difference schemes on 1D and 2D test problems for the Euler equations”, SIAM J. Sci. Comput., 25:3 (2003), 995–1017 | DOI | MR | Zbl

[44] Bragin M. D., Rogov B. V., “Gibridnye skhemy beguschego scheta dlya uravnenii giperbolicheskogo tipa na osnove protivopotochnykh i biokompaktnykh simmetrichnykh skhem”, Zh. vychisl. matem. i matem. fiz., 55:7 (2015), 1196–1207 | DOI | Zbl

[45] Shokin Yu. I., Yanenko H. H., Metod differentsialnogo priblizheniya. Primenenie k gazovoi dinamike, Izd-vo SO RAN, Novosibirsk, 1985 | MR

[46] Guermond J.-L., Pasquetti R., Popov B., “Entropy viscosity method for nonlinear conservation laws”, J. Comput. Phys., 230 (2011), 4248–4267 | DOI | MR | Zbl

[47] Friedrichs K. O., Hyers D. H., “Symmetric hyperbolic linear differential equations”, Commun. Pure Appl. Math., 7:2 (1954), 345–392 | DOI | MR | Zbl

[48] Kholodov A. S., Kholodov Ya. A., “O kriteriyakh monotonnosti raznostnykh skhem dlya uravnenii giperbolicheskogo tipa”, Zh. vychisl. matem. i matem. fiz., 46:9 (2006), 1638–1667 | MR