Geodesic
On the practical accuracy of shock-capturing schemes
Matematičeskoe modelirovanie, Tome 25 (2013) no. 9, pp. 63-74.

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The paper suggests a method which allows to estimate the accuracy of translation of the Rankine–Hugoniot’s conditions through the shock wave front. The method concerns the calculation of the order of integral convergence of difference solution (as opposed to using its absolute value as in the norm $L_1$) on the intervals crossing the shock. In such integration the error arising in front of the shock due to its smearing, can be compensated by the similar error of the opposite sign after the front. Provided examples demonstrate that this approach allows to obtain the second order of integral convergence on the intervals crossing the shock for some of the classical high-order difference schemes.
Keywords: shock-capturing difference schemes, higher accuracy, discontinuous solutions, integral order of convergence. 
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O. A. Kovyrkina; V. V. Ostapenko. On the practical accuracy of shock-capturing schemes. Matematičeskoe modelirovanie, Tome 25 (2013) no. 9, pp. 63-74. http://geodesic.mathdoc.fr/item/MM_2013_25_9_a5/

[1] Rozhdestvenskii B. L., Yanenko N. N., Sistemy kvazilineinykh uravnenii, Nauka, M., 1978 | MR

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

[3] Kolgan V. P., “Primenenie operatorov sglazhivaniya v raznostnykh skhemakh vysokogo poryadka tochnosti”, Zh. vychisl. matem. i matem. fiz., 18:5 (1978), 1340–1345 | MR | Zbl

[4] Boris J. P., Book D. L., Hain K., “Flux-corrected transport. II: Generalizations of the method”, J. Comput. Phys., 18:3 (1975), 248–283 | DOI | MR | Zbl

[5] Van Leer B., “Toward the ultimate conservative difference scheme. V: A second-order sequel to Godunov's method”, J. Comp. Phys., 32:1 (1979), 101–136 | DOI | MR

[6] Harten A., “High resolution schemes for hyperbolic conservation laws”, J. Comp. Phys., 49 (1983), 357–393 | DOI | MR | Zbl

[7] Kurganov A., Tadmor E., “New High-Resolution Central Schemes for Nonlinear Conservation Laws and Convection-Diffusion Equations”, J. Comp. Phys., 160 (2000), 214–282 | MR

[8] Pinchukov V. I., Shu Ch.-V., Chislennye metody vysokikh poryadkov dlya zadach aerogidrodinamiki, Izd-vo SO RAN, Novosibirsk, 2000

[9] Ostapenko V. V., “O skhodimosti raznostnykh skhem za frontom nestatsionarnoi udarnoi volny”, Zh. vychisl. matem. i matem. fiz., 37:10 (1997), 1201–1212 | MR | Zbl

[10] Casper J., Carpenter M. H., “Computational consideration for the simulation of shock-induced sound”, SIAM J. Sci. Comput., 19:1 (1998), 813–828 | DOI | MR | Zbl

[11] Engquist B., Sjogreen B., “The convergence rate of finite difference schemes in the presence of shocks”, SIAM J. Numer. Anal., 35 (1998), 2464–2485 | DOI | MR | Zbl

[12] Kovyrkina O. A., Ostapenko V. V., “O skhodimosti raznostnykh skhem skvoznogo scheta”, Doklady Akademii nauk, 433:5 (2010), 599–603 | MR | Zbl

[13] MacCormack R. W., The effect of viscosity in hypervelosity impact cratering, AIAA paper, No 69-354, 1969

[14] Rusanov V. V., “Raznostnye skhemy tretego poryadka tochnosti dlya skvoznogo scheta razryvnykh reshenii”, Dokl. AN SSSR, 180:6 (1968), 1303–1305 | MR | Zbl

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

[16] Ostapenko V. V., “O konechno-raznostnoi approksimatsii uslovii Gyugonio na fronte udarnoi volny, rasprostranyayuscheisya s peremennoi skorostyu”, ZhVMiMF, 38:8 (1998), 1355–1367 | MR | Zbl