On accuracy of MUSCL type scheme when calculating discontinuous solutions. (Russian. English summary) Zbl 1457.76110
Summary: 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.
MSC:
| 76M20 | Finite difference methods applied to problems in fluid mechanics |
| 76L05 | Shock waves and blast waves in fluid mechanics |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
References:
| [1] | S. K. Godunov, “Raznostnyi metod chislennogo rascheta razryvnych reshenii uravnenii gidrodinamiki”, Matem. Sbornik, 47:3 (1959), 271-306 · Zbl 0171.46204 |
| [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 · Zbl 1364.65223 · doi:10.1016/0021-9991(79)90145-1 |
| [3] | A. Harten, “High resolution schemes for hyperbolic conservation laws”, J. Comp. Phys., 49 (1983), 357-393 · Zbl 0565.65050 · doi:10.1016/0021-9991(83)90136-5 |
| [4] | A. Harten, S. Osher, “Uniformly high-order accurate nonoscillatory schemes”, SIAM J. Numer. Analys, 24:2 (1987), 279-309 · Zbl 0627.65102 · doi:10.1007/978-3-642-60543-7\_11 |
| [5] | G. S. Jiang, C. W. Shu, “Efficient implementation of weighted ENO schemes”, J. Comp. Phys., 126 (1996), 202-228 · Zbl 0877.65065 · doi:10.1006/jcph.1996.0130 |
| [6] | S. A. Karabasov, V. M. Goloviznin, “Compact accurately boundary-adjusting high-resolution technique for fluid dynamics”, J. Comp. Phys., 228 (2009), 7426-7451 · Zbl 1172.76034 · doi:10.1016/ |
| [7] | V. V. Ostapenko, “Approximation of conservation laws by high-resolution difference schemes”, USSR CM&MP, 30:5 (1990), 91-100 · Zbl 0739.65077 · doi:10.1016/0041-5553(90)90165-O |
| [8] | V. V. Ostapenko, “A method of increasing the order of the weak approximation of the conservationlaws on discontinuous solutions”, CM&MP, 36:10 (1996), 1443-1451 · Zbl 0911.65079 |
| [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 1001.76069 |
| [10] | V. V. Ostapenko, “Convergence of finite-difference schemes behind a shock front”, Comp. Math. and Math. Phys., 37:10 (1997), 1161-1172 · Zbl 1122.76355 |
| [11] | J. Casper, M. H. Carpenter, “Computational consideration for the simulation of shock-induced sound”, SIAM J. Sci. Comp., 19:1 (1998), 813-828 · Zbl 0918.76045 · doi:10.1137/S1064827595294101 |
| [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 · Zbl 0922.76254 · doi:10.1137/S0036142997317584 |
| [13] | O. A. Kovyrkina, V. V. Ostapenko, “On the convergence of shock-capturing difference schemes”, Dokl. Math., 82:1 (2010), 599-603 · Zbl 1204.65105 · doi:10.1134/S1064562410040265 |
| [14] | O. A. Kovyrkina, V. V. Ostapenko, “On the practical accuracy of shock-capturing schemes”, Math. Models & Comp. Simul., 6:2 (2014), 183-191 · Zbl 1357.65121 · doi:10.1134/S2070048214020069 |
| [15] | N. A. Mikhailov, “The convergence order of WENO schemes behind a shock front”, Math. Models & Comp. Simul., 7:5 (2015), 467-474 · Zbl 1340.76048 · doi:10.1134/S2070048215050075 |
| [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 · Zbl 0697.65068 · doi:10.1016/0021-9991(90)90260-8 |
| [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 · Zbl 0987.65085 · doi:10.1006/jcph.2000.6459 |
| [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 · Zbl 0179.22202 |
| [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 · Zbl 1086.76552 |
| [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 · Zbl 1393.65014 · doi:10.1134/S1064562418010246 |
| [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 · Zbl 1460.76643 · doi:10.1134/S0965542518090129 |
| [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 · Zbl 1448.35332 · doi:10.1134/S0965542518090129 |
| [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 · Zbl 1407.65137 · doi:10.1134/S2070048219010186 |
| [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:10.1134/S1028335819040128 |
| [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 · Zbl 1444.76085 · doi:10.1134/S106456241906005X |
| [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 · Zbl 07424598 · doi:10.1134/S1064562420020076 |
| [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 · Zbl 0927.65120 |
| [30] | A. Gelb, E. Tadmor, “Spectral reconstruction of piecewise smooth functions from their discrete data”, Math. Model. Numer. Anal., 36 (2002), 155-175 · Zbl 1056.42001 · doi:10.1051/m2an:2002008 |
| [31] | F. Arandiga, A. Baeza, R. Donat, “Vector cell-average multiresolution based on Hermite interpolation”, Adv. Comp. Math., 28 (2008), 1-22 · Zbl 1131.65007 · doi:10.1007/s10444-005-9007-7 |
| [32] | J. L. Guermond, R. Pasquetti, B. Popov, “Entropy viscosity method for nonlinear conservation laws”, J. Comput. Phys., 230 (2011), 4248-4267 · Zbl 1220.65134 · doi:10.1016/j.jcp.2010.11.043 |
| [33] | J. Dewar, A. Kurganov, M. Leopold, “Pressure-based adaption indicator for compressible Euler equations”, Numer. Methods Partial Dierential Equations, 31 (2015), 1844-1874 · Zbl 1331.76078 · doi:10.1002/num.21970 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
