Geodesic
On accuracy of MUSCL type scheme when calculating discontinuous solutions
Matematičeskoe modelirovanie, Tome 33 (2021) no. 1, pp. 105-121.

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

We considered the central-difference NT-scheme (Nessyahu-Tadmor scheme) with the second-order MUSCL reconstruction of flows. We studied the accuracy of the NT-scheme in calculations of shock waves propagating with a variable velocity. We showed that this scheme has approximately the first order of the local convergence in the domains of the influence of shock waves and the same order of integral convergence on the intervals, one of the boundaries of which is in the region of the influence of shock wave. As a result, the local accuracy of the NT-scheme is significantly reduced in these areas. Test calculations are presented that demonstrate these properties of the NT-scheme.
Keywords: NT scheme, WENO scheme, combined scheme, shock wave, accuracy of finite-difference scheme.
Mots-clés : MUSCL reconstruction 
@article{MM_2021_33_1_a7,
     author = {O. A. Kovyrkina and V. V. Ostapenko},
     title = {On accuracy of {MUSCL}  type scheme when calculating discontinuous solutions},
     journal = {Matemati\v{c}eskoe modelirovanie},
     pages = {105--121},
     publisher = {mathdoc},
     volume = {33},
     number = {1},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MM_2021_33_1_a7/}
}
TY  - JOUR
AU  - O. A. Kovyrkina
AU  - V. V. Ostapenko
TI  - On accuracy of MUSCL  type scheme when calculating discontinuous solutions
JO  - Matematičeskoe modelirovanie
PY  - 2021
SP  - 105
EP  - 121
VL  - 33
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MM_2021_33_1_a7/
LA  - ru
ID  - MM_2021_33_1_a7
ER  - 
%0 Journal Article
%A O. A. Kovyrkina
%A V. V. Ostapenko
%T On accuracy of MUSCL  type scheme when calculating discontinuous solutions
%J Matematičeskoe modelirovanie
%D 2021
%P 105-121
%V 33
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MM_2021_33_1_a7/
%G ru
%F MM_2021_33_1_a7
O. A. Kovyrkina; V. V. Ostapenko. On accuracy of MUSCL  type scheme when calculating discontinuous solutions. Matematičeskoe modelirovanie, Tome 33 (2021) no. 1, pp. 105-121. http://geodesic.mathdoc.fr/item/MM_2021_33_1_a7/

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

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

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

[4] A. Harten, S. Osher, “Uniformly high-order accurate nonoscillatory schemes”, SIAM J. Numer. Analys, 24:2 (1987), 279–309 | DOI | Zbl

[5] G. S. Jiang, C. W. Shu, “Efficient implementation of weighted ENO schemes”, J. Comp. Phys., 126 (1996), 202–228 | DOI | Zbl

[6] S. A. Karabasov, V. M. Goloviznin, “Compact accurately boundary-adjusting high-resolution technique for fluid dynamics”, J. Comp. Phys., 228 (2009), 7426–7451 | DOI | Zbl

[7] V. V. Ostapenko, “Approximation of conservation laws by high-resolution difference schemes”, USSR CM, 30:5 (1990), 91–100 | DOI | Zbl

[8] V. V. Ostapenko, “A method of increasing the order of the weak approximation of the conservationlaws on discontinuous solutions”, CM, 36:10 (1996), 1443–1451 | Zbl

[9] V. V. Ostapenko, “Construction of high-order accurate shock-capturing finite-difference schemes for unsteady shock waves”, Comp. Math. and Math. Phys., 40:12 (2000), 1784–1800 | Zbl

[10] V. V. Ostapenko, “Convergence of finite-difference schemes behind a shock front”, Comp. Math. and Math. Phys., 37:10 (1997), 1161–1172 | Zbl

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

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

[13] O. A. Kovyrkina, V. V. Ostapenko, “On the convergence of shock-capturing difference schemes”, Dokl. Math., 82:1 (2010), 599–603 | DOI | Zbl

[14] O. A. Kovyrkina, V. V. Ostapenko, “On the practical accuracy of shock-capturing schemes”, Math. Models Comp. Simul., 6:2 (2014), 183–191 | DOI | MR | Zbl

[15] N. A. Mikhailov, “The convergence order of WENO schemes behind a shock front”, Math. Models Comp. Simul., 7:5 (2015), 467–474 | DOI | MR | Zbl

[16] O. A. Kovyrkina, V. V. Ostapenko, “On monotonicity and accuracy of CABARET scheme calculating weak solutions with shocks”, Comp. Technologies, 23:2 (2018), 37–54

[17] H. Nessyahu, E. Tadmor, “Non-oscillatory central differencing for hyperbolic conservation laws”, J. Comp. Phys., 87:2 (1990), 408–463 | DOI | MR | Zbl

[18] A. Kurganov, E. Tadmor, “New high-resolution central schemes for nonlinear conservation laws and convection-diffusion equations”, J. Comp. Phys., 160:1 (2000), 241–282 | DOI | MR | Zbl

[19] V. V. Ostapenko, Hyperbolic systems of conservation laws and its application to the theory of shallow water, Second ed., NSU, Novosibirsk, 2014, 234 pp.

[20] V. V. Rusanov, “Difference schemes of third order accuracy for the through calculation of discontinuous solutions”, Dokl. Akad. Nauk SSSR, 180:6 (1968), 1303–1305 | MR | Zbl

[21] V. V. Ostapenko, “Finite-difference approximation of the Hugoniot conditions on a shock front propagating with variable velocity”, Comp. Math. and Math. Phys., 38:8 (1998), 1299–1311 | MR | Zbl

[22] O. A. Kovyrkina, V. V. Ostapenko, “On the construction of combined finite-difference schemes of high accuracy”, Dokl. Math., 97:1 (2018), 77–81 | DOI | DOI | MR | Zbl

[23] M. D. Bragin, B. V. Rogov, “On the accuracy of bicompact schemes as applied to computation of unsteady shock waves”, Comp. Math. Math. Phys., 58:9 (2018), 1435–1450 | DOI | DOI | MR | MR | Zbl

[24] O. A. Kovyrkina, V. V. Ostapenko, “Monotonicity of the CABARET scheme approximating a hyperbolic system of conservation laws”, Comp. Math. and Math. Phys., 58:9 (2018), 1435–1450 | DOI | DOI | MR | Zbl

[25] N. A. Zyuzina, O. A. Kovyrkina, V. V. Ostapenko, “Monotone finite-difference scheme preserving high accuracy in regions of shock influence”, Dokl. Math., 98:2 (2018), 506–510 | DOI | DOI | MR | MR | Zbl

[26] V. V. Ostapenko, N. A. Khandeeva, “The accuracy of finite-difference schemes calculating the interaction of shock waves”, Dokl. Phys., 64:4 (2019), 197–201 | DOI | DOI

[27] M. E. Ladonkina, O. A. Neklyudova, V. V. Ostapenko, V. F. Tishkin, “Combined DG scheme that maintains increased accuracy in shock wave areas”, Dokl. Math., 100:3 (2019), 519–523 | DOI | DOI | Zbl

[28] M. D. Bragin, B. V. Rogov, “Combined monotone bicompact scheme of higher order accuracy in domains of influence of nonstationary shock waves”, Doklady Math, 101:3 (2020), 239–243 | DOI | DOI

[29] B. Cockburn, “An introduction to the discontinuous Galerkin method for convection-dominated problems, advanced numerical approximation of nonlinear hyperbolic equations”, Lecture Notes in Mathematics, 1697, 1998, 151–268 | MR | Zbl

[30] A. Gelb, E. Tadmor, “Spectral reconstruction of piecewise smooth functions from their discrete data”, Math. Model. Numer. Anal., 36 (2002), 155–175 | DOI | MR | Zbl

[31] F. Arandiga, A. Baeza, R. Donat, “Vector cell-average multiresolution based on Hermite interpolation”, Adv. Comp. Math., 28 (2008), 1–22 | DOI | MR | Zbl

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

[33] J. Dewar, A. Kurganov, M. Leopold, “Pressure-based adaption indicator for compressible Euler equations”, Numer. Methods Partial Dierential Equations, 31 (2015), 1844–1874 | DOI | MR | Zbl