Hybrid schemes with high-order multioperators for computing discontinuous solutions

A. I. Tolstykh

Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 53 (2013) no. 9, pp. 1481-1502

Résumé

Results are presented concerning high-order multioperator schemes and their monotonized versions as applied to the computation of discontinuous solutions. Two types of hybrid schemes are considered. Solutions of several test problems, including those with extremely strong discontinuities, are presented. An example of solving the Navier–Stokes equations at low supersonic Mach numbers by applying multioperator schemes without monotonization is given.

Bibliographie

[1] Tolstykh A. I., “Multioperatornye skhemy proizvolnogo poryadka, ispolzuyuschie netsentrirovannye kompaktnye approksimatsii”, Dokl. AN, 366:3 (1999), 319–322 | MR | Zbl

[2] Tolstykh A. I., “O postroenii skhem zadannogo poryadka s lineinymi kombinatsiyami operatorov”, Zh. vychisl. matem. i matem. fiz., 40:8 (2000), 1206–1220 | MR | Zbl

[3] Woodward P., Colella P., “The numerical simulation of two-dimensional fluid flow with strong shocks”, J. Comput. Phys., 54 (1984), 115–173 | DOI | MR | Zbl

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

[5] Fedorenko R. P., “Primenenie raznostnykh skhem vysokoi tochnosti dlya chislennogo resheniya giperbolicheskikh uravnenii”, Zh. vychisl. matem. i matem. fiz., 2:6 (1962), 1122–1128 | MR | Zbl

[6] Petrov I. B., Kholodov A. S., “O regulyarizatsii razryvnykh chislennykh reshenii uravnenii giperbolicheskogo tipa”, Zh. vychisl. matem. i matem. fiz., 24:8 (1984), 1172–1188 | MR | Zbl

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

[8] Kholodov A. S., “Chislennye metody resheniya giperbolicheskikh uravnenii i sistem”, Entsiklopediya nizkotemperaturnoi plazmy, v. VII-1, Ch. 2, Yanus-K, M., 2008, 141–174

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

[10] Zalesac S. T., “Fully multidimensionalflux-corrected transport algorithms for fluids”, J. Comput. Phys., 31 (1979), 335–362 | DOI | MR

[11] Boris J. P., Book D. L., “Flux-corrected transport. I: SHASTA, a fluid transport algorithm that works”, J. Comput. Phys., 11 (1973), 38–69 | DOI | Zbl

[12] Tolstykh A. I., “Development of arbitrary-order multioperators-based schemes for parallel calculations. 1: High-er-than-fifth order approximations to convection terms”, J. Comput. Phys., 225 (2007), 2333–2353 | DOI | MR | Zbl

[13] Lipavskii M. V., Tolstykh A. I., Chigerëv E. N., “O skheme s multioperatornymi approksimatsiyami 9-go poryadka dlya parallelnykh vychislenii i o ee primenenii v zadachakh pryamogo chislennogo modelirovaniya”, Zh. vychisl. matem. i matem. fiz., 46:8 (2006), 1433–1452 | MR

[14] Savelev A. D., Tolstykh A. I., Shirobokov D. A., “O primenenii kompaktnykh i multioperatornykh skhem dlya chislennogo modelirovaniya akusticheskikh polei, vozbuzhdaemykh volnami neustoichivosti v sverkhzvukovykh struyakh”, Zh. vychisl. matem. i matem. fiz., 49:7 (2009), 1280–1294 | MR

[15] Tolstykh A. I., “O multioperatornom metode postroeniya approksimatsii i skhem proizvolno vysokogo poryadka”, Zh. vychisl. matem. i matem. fiz., 51:1 (2011), 56–73 | MR | Zbl

[16] Lax P. D., “Week solutions of nonlinear hyperbolic equations and their numerical cpmputations”, Commun. Pure and Appl. Math., 7:1 (1954), 159–193 | DOI | MR | Zbl

[17] Lax P. D., Wendroff B., “Systems of conservation laws”, Commun. Pure and Appl. Math., 13:2 (1960), 217–237 | DOI | MR | Zbl

[18] Ostapenko V. V., “O povyshenii poryadka slaboi approksimatsii zakonov sokhraneniya na razryvnykh resheniyakh”, Zh. vychisl. matem. i matem. fiz., 36:10 (1996), 146–157 | MR | Zbl

[19] Lipavskii M. V., Chigerëv E. N., Chislennoe issledovanie vikhrevykh dvizhenii zhidkostei s pomoschyu netsentrirovannykh kompaktnykh approksimatsii, VTs RAN, M., 2004

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

[21] Sod G. A., “A survey of several finite difference schemes for hyperbolic conservation laws”, J. Comput. Phys., 27 (1978), 1–31 | DOI | MR | Zbl

[22] Woodward P., Colella P., “The piecewise parabolic method (PPM) for gas-dynamical”, J. Comput. Phys., 54 (1984), 174–201 | DOI | MR | Zbl

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

[24] Liska R., Wendroff B., “Comparison of several difference schemes on 1D and 2D test problems for the Euker equations”, SIAM J. Sci. Comput., 26:3 (2003), 995–1017 | DOI | MR

[25] Elizarova T. G., Shilnikov E. V., “Popravka”, Zh. vychisl. matem. i matem. fiz., 50:4 (2010), 784 | MR

[26] Liu X.-D., Osher S., Chan T., “Weighted essentially non-oscillatory schemes”, J. Comput. Phys., 115 (1994), 200–212 | DOI | MR | Zbl

[27] Lipavskii M. V., Savelev A. D., Tolstykh A. I., Shirobokov D. A., “Multioperatornye skhemy do 18-go poryadka tochnosti s prilozheniyami k zadacham neustoichivosti i akustiki strui”, Uch. zapiski TsAGI, XLIII:3 (2012), 16–33

[28] Tolstykh A. I., Shirobokov D. A., “Fast calculations of screech using highly accurate multioperators-based schemes”, J. appl. acoustics, 74 (2013), 102–109 | DOI