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Combined Numerical Schemes

  • GENERAL NUMERICAL METHODS
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Abstract

A survey of works concerning high-order accurate numerical methods designed for shock-capturing computations of discontinuous solutions to hyperbolic systems of conservation laws is presented. The basic problems arising in the theory of such methods are formulated, and approaches to their solution are proposed. Primary attention is given to fundamentally new shock-capturing methods (known as combined schemes) that monotonically localize shock fronts, while preserving high accuracy in shock influence areas. Test computations are presented that demonstrate the significant advantages of combined schemes over standard NFC ones when applied to computing discontinuous solutions with shock waves.

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REFERENCES

  1. S. K. Godunov, “Difference method for computing discontinuous solutions of fluid dynamics equations,” Mat. Sb. 47 (3), 271–306 (1959).

    MathSciNet  MATH  Google Scholar 

  2. B. Van Leer, “Toward the ultimate conservative difference scheme: V. A second-order sequel to Godunov’s method,” J. Comput. Phys. 32 (1), 101–136 (1979). https://doi.org/10.1016/0021-9991(79)90145-1

    Article  MATH  Google Scholar 

  3. A. Harten, “High resolution schemes for hyperbolic conservation laws,” J. Comput. Phys. 49 (3), 357–393 (1983). https://doi.org/10.1016/0021-9991(83)90136-5

    Article  MathSciNet  MATH  Google Scholar 

  4. A. Harten and S. Osher, “Uniformly high-order accurate nonoscillatory schemes,” SIAM J. Numer. Anal. 24 (2), 279–309 (1987). https://doi.org/10.1007/978-3-642-60543-7_11

    Article  MathSciNet  MATH  Google Scholar 

  5. H. Nessyahu and E. Tadmor, “Non-oscillatory central differencing for hyperbolic conservation laws,” J. Comput. Phys. 87 (2), 408–463 (1990). https://doi.org/10.1016/0021-9991(90)90260-8

    Article  MathSciNet  MATH  Google Scholar 

  6. X.-D. Liu, S. Osher, and T. Chan, “Weighted essentially nonoscillatory schemes,” J. Comput. Phys. 115 (1), 200–212 (1994). https://doi.org/10.1006/jcph.1994.1187

    Article  MathSciNet  MATH  Google Scholar 

  7. G. S. Jiang and C. W. Shu, “Efficient implementation of weighted ENO schemes,” J. Comput. Phys. 126 (1), 202–228 (1996). https://doi.org/10.1006/jcph.1996.0130

    Article  MathSciNet  MATH  Google Scholar 

  8. B. Cockburn, “An introduction to the discontinuous Galerkin method for convection-dominated problems, advanced numerical approximation of nonlinear hyperbolic equations,” Lect. Notes Math. 1697, 150–268 (1998). https://doi.org/10.1007/BFb0096353

    Article  Google Scholar 

  9. S. A. Karabasov and V. M. Goloviznin, “Compact accurately boundary-adjusting high-resolution technique for fluid dynamics,” J. Comput. Phys. 228 (19), 7426–7451 (2009). https://doi.org/10.1016/j.jcp.2009.06.037

    Article  MathSciNet  MATH  Google Scholar 

  10. B. V. Rogov and M. N. Mikhailovskaya, “Monotonic bicompact schemes for linear transport equations,” Math. Models Comput. Simul. 4 (1), 92–100 (2012). https://doi.org/10.1134/S2070048212010103

    Article  MathSciNet  MATH  Google Scholar 

  11. M. D. Bragin and B. V. Rogov, “Conservative limiting method for high-order bicompact schemes as applied to systems of hyperbolic equations,” Appl. Numer. Math. 151, 229–245 (2020). https://doi.org/10.1016/j.apnum.2020.01.005

    Article  MathSciNet  MATH  Google Scholar 

  12. V. V. Ostapenko, “Convergence of finite-difference schemes behind a shock front,” Comput. Math. Math. Phys. 37 (10), 1161–1172 (1997).

    MathSciNet  MATH  Google Scholar 

  13. J. Casper and M. H. Carpenter, “Computational consideration for the simulation of shock-induced sound,” SIAM J. Sci. Comput. 19 (1), 813–828 (1998). https://doi.org/10.1137/S1064827595294101

    Article  MathSciNet  MATH  Google Scholar 

  14. B. Engquist and B. Sjogreen, “The convergence rate of finite difference schemes in the presence of shocks,” SIAM J. Numer. Anal. 35, 2464–2485 (1998). https://www.jstor.org/stable/2587267

    Article  MathSciNet  MATH  Google Scholar 

  15. V. V. Ostapenko, “Construction of high-order accurate shock-capturing finite-difference schemes for unsteady shock waves,” Comput. Math. Math. Phys. 40 (12), 1784–1800 (2000).

    MathSciNet  MATH  Google Scholar 

  16. O. A. Kovyrkina and V. V. Ostapenko, “On the convergence of shock-capturing difference schemes,” Dokl. Math. 82 (1), 599–603 (2010). https://doi.org/10.1134/S1064562410040265

    Article  MathSciNet  MATH  Google Scholar 

  17. N. A. Mikhailov, “The convergence order of WENO schemes behind a shock front,” Math. Models Comput. Simul. 7, 467–474 (2015). https://doi.org/10.1134/S2070048215050075

    Article  MathSciNet  Google Scholar 

  18. M. E. Ladonkina, O. A. Neklyudova, V. V. Ostapenko, and V. F. Tishkin, “On the accuracy of the discontinuous Galerkin method in calculation of shock waves,” Comput. Math. Math. Phys. 58 (8), 1344–1353 (2018). https://doi.org/10.1134/S0965542518080122

    Article  MathSciNet  MATH  Google Scholar 

  19. O. A. Kovyrkina and V. V. Ostapenko, “Monotonicity and accuracy of the KABARE scheme as applied to computation of generalized solutions with shock waves,” Vychisl. Tekhnol. 23 (2), 37–54 (2018).

    Google Scholar 

  20. O. A. Kovyrkina and V. V. Ostapenko, “Accuracy of MUSCL-type schemes in shock wave calculations,” Dokl. Math. 101 (3), 209–213 (2020). https://doi.org/10.1134/S1064562420030126

    Article  MathSciNet  MATH  Google Scholar 

  21. M. D. Bragin and B. V. Rogov, “On the accuracy of bicompact schemes as applied to computation of unsteady shock waves,” Comput. Math. Math. Phys. 60 (5), 1344–1353 (2020). https://doi.org/10.1134/S0965542520050061

    Article  MathSciNet  MATH  Google Scholar 

  22. V. V. Ostapenko, “Finite-difference approximation of the Hugoniot conditions on a shock front propagating with variable velocity,” Comput. Math. Math. Phys. 38 (8), 1299–1311 (1998).

    MathSciNet  MATH  Google Scholar 

  23. V. V. Rusanov, “Third-order accurate shock-capturing schemes for computing discontinuous solutions,” Dokl. Akad. Nauk SSSR 180 (6), 1303–1305 (1968).

    MathSciNet  Google Scholar 

  24. A. Gelb and E. Tadmor, “Adaptive edge detectors for piecewise smooth data based on the minmod limiter,” J. Sci. Comput. 28, 279–306 (2006). https://doi.org/10.1007/s10915-006-9088-6

    Article  MathSciNet  MATH  Google Scholar 

  25. F. Arandiga, A. Baeza, and R. Donat, “Vector cell-average multiresolution based on Hermite interpolation,” Adv. Comput. Math. 28, 1–22 (2008). https://doi.org/10.1007/s10444-005-9007-7

    Article  MathSciNet  MATH  Google Scholar 

  26. J. L. Guermond, R. Pasquetti, and B. Popov, “Entropy viscosity method for nonlinear conservation laws,” J. Comput. Phys. 230, 4248–4267 (2011). https://doi.org/10.1016/j.jcp.2010.11.043

    Article  MathSciNet  MATH  Google Scholar 

  27. J. Dewar, A. Kurganov, and M. Leopold, “Pressure-based adaption indicator for compressible Euler equations,” Numer. Methods Partial Differ. Equations 31, 1844–1874 (2015). https://doi.org/10.1002/num.21970

    Article  MathSciNet  MATH  Google Scholar 

  28. M. D. Bragin and B. V. Rogov, “Minimal dissipation hybrid bicompact schemes for hyperbolic equations,” Comput. Math. Math. Phys. 56 (6), 947–961 (2016). https://doi.org/10.1134/S0965542516060099

    Article  MathSciNet  MATH  Google Scholar 

  29. O. A. Kovyrkina and V. V. Ostapenko, “On the construction of combined finite-difference schemes of high accuracy,” Dokl. Math. 97 (1), 77–81 (2018). https://doi.org/10.1134/S1064562418010246

    Article  MathSciNet  MATH  Google Scholar 

  30. O. A. Kovyrkina and V. V. Ostapenko, “Monotonicity of the CABARET scheme approximating a hyperbolic system of conservation laws,” Comput. Math. Math. Phys. 58 (9), 1435–1450 (2018). https://doi.org/10.1134/S0965542518090129

    Article  MathSciNet  MATH  Google Scholar 

  31. N. A. Zyuzina, O. A. Kovyrkina, and V. V. Ostapenko, “Monotone finite-difference scheme preserving high accuracy in regions of shock influence,” Dokl. Math. 98 (2), 506–510 (2018). https://doi.org/10.1134/S1064562418060315

    Article  MathSciNet  MATH  Google Scholar 

  32. M. E. Ladonkina, O. A. Neklyudova, V. V. Ostapenko, and V. F. Tishkin, “Combined DG scheme that maintains increased accuracy in shock wave areas,” Dokl. Math. 100 (3), 519–523 (2019). https://doi.org/10.1134/S106456241906005X

    Article  MathSciNet  MATH  Google Scholar 

  33. M. D. Bragin and B. V. Rogov, “Combined monotone bicompact scheme of higher order accuracy in domains of influence of nonstationary shock waves,” Dokl. Math. 101 (3), 239–243 (2020). https://doi.org/10.1134/S1064562420020076

    Article  MathSciNet  MATH  Google Scholar 

  34. I. Faragó, A. Havasi, and Z. Zlatev, “Efficient implementation of stable Richardson extrapolation algorithms,” Comput. Math. Appl. 60 (8), 2309–2325 (2010). https://doi.org/10.1016/j.camwa.2010.08.025

    Article  MathSciNet  MATH  Google Scholar 

  35. P. D. Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves (Soc. Ind. Appl. Math., Philadelphia, 1972).

    Google Scholar 

  36. B. L. Rozhdestvenskii and N. N. Yanenko, Systems of Quasilinear Equations and Their Applications to Gas Dynamics (Nauka, Moscow, 1978; Am. Math. Soc., Providence, 1983).

  37. V. V. Ostapenko and N. A. Khandeeva, “Justification of the method of integral convergence for studying the accuracy of difference schemes,” Math. Models Comput. Simul. 13, 1028–1037 (2021). https://doi.org/10.1134/S207004822106017X

    Article  MathSciNet  MATH  Google Scholar 

  38. J. J. Stoker, Water Waves (Wiley, New York, 1957).

    MATH  Google Scholar 

  39. S. Z. Burstein and A. A. Mirin, “Third order difference methods for hyperbolic equations,” J. Comput. Phys. 5 (3), 547–571 (1970). https://doi.org/10.1016/0021-9991(70)90080-X

    Article  MathSciNet  MATH  Google Scholar 

  40. P. Lax and B. Wendroff, “Systems of conservation laws,” Commun. Pure Appl. Math. 13, 217–237 (1960). https://doi.org/10.1002/cpa.3160130205

    Article  MATH  Google Scholar 

  41. R. W. MacCormack, “The effect of viscosity in hypervelocity impact cratering,” AIAA, No. 169, 69–354 (1969). https://doi.org/10.2514/6.1969-354

  42. V. V. Ostapenko, “Approximation of conservation laws by high-resolution difference schemes,” Comput. Math. Math. Phys. 30 (5), 91–100 (1990). https://doi.org/10.1016/0041-5553(90)90165-O

    Article  MathSciNet  MATH  Google Scholar 

  43. V. V. Ostapenko, “Equivalent definitions of conservative finite-difference schemes,” Comput. Math. Math. Phys. 29 (4), 100–110 (1989). https://doi.org/10.1016/0041-5553(89)90124-9

    Article  MathSciNet  MATH  Google Scholar 

  44. V. V. Ostapenko, “A method of increasing the order of the weak approximation of the laws of conservation on discontinuous solutions,” Comput. Math. Math. Phys. 36 (10), 1443–1451 (1996).

    MathSciNet  MATH  Google Scholar 

  45. V. V. Ostapenko, “Approximation of Hugoniot’s conditions by explicit conservative difference schemes for nonstationary shock waves,” Sib. Zh. Vychisl. Mat. 1 (1), 77–88 (1998).

    MathSciNet  MATH  Google Scholar 

  46. V. V. Ostapenko, “On a local fulfilment of conservation laws at the “smoothed” shock wave front,” Mat. Mod. 2 (7), 129–138 (1990).

    MATH  Google Scholar 

  47. R. Hirsh, “Higher order accurate difference solutions of a fluid mechanics problems by a compact differencing technique,” J. Comput. Phys. 19 (1), 90–109 (1975). https://doi.org/10.1016/0021-9991(75)90118-7

    Article  MathSciNet  MATH  Google Scholar 

  48. A. E. Berger, J. M. Solomon, M. Ciment, S. H. Leventhal, and B. C. Weinberg, “Generalized OCI schemes for boundary layer problems,” Math. Comput. 35 (6), 695–731 (1980). https://doi.org/10.1090/S0025-5718-1980-0572850-8

    Article  MathSciNet  Google Scholar 

  49. O. M. Belotserkovskii, A. P. Byrkin, A. P. Mazurov, and A. I. Tolstykh, “High-order accuracy difference method for computing viscous gas flows,” USSR Comput. Math. Math. Phys. 22 (6), 206–216 (1982). https://doi.org/10.1016/0041-5553(82)90110-0

    Article  MATH  Google Scholar 

  50. A. I. Tolstykh, Compact Finite Difference Schemes and Application in Aerodynamic Problems (Nauka, Moscow, 1990) [in Russian].

    Google Scholar 

  51. V. V. Ostapenko, “Symmetric compact schemes with higher order conservative artificial viscosities,” Comput. Math. Math. Phys. 42 (7), 980–999 (2002).

    MathSciNet  Google Scholar 

  52. A. Iserles, “Generalized leapfrog methods,” IMA J. Numer. Anal. 6 (4), 381–392 (1986). https://doi.org/10.1093/imanum/6.4.381

    Article  MathSciNet  MATH  Google Scholar 

  53. V. M. Goloviznin and A. A. Samarskii, “Finite approximation of convective transport with a space splitting of time derivative,” Mat. Model. 10 (1), 86–100 (1998).

    MathSciNet  MATH  Google Scholar 

  54. V. M. Goloviznin and A. A. Samarskii, “Some properties of the difference scheme CABARET,” Mat. Model. 10 (1), 101–116 (1998).

    MathSciNet  MATH  Google Scholar 

  55. V. M. Goloviznin, M. A. Zaitsev, S. F. Karabasov, and I. A. Korotkin, New CFD Algorithms for Multiprocessor Computer Systems (Mosk. Gos. Univ., Moscow, 2013) [in Russian].

    Google Scholar 

  56. S. A. Karabasov and V. M. Goloviznin, “New efficient high-resolution method for nonlinear problems in aeroacoustics,” AIAA J. 45 (12), 2861–2871 (2007). https://doi.org/10.2514/1.29796

    Article  Google Scholar 

  57. S. A. Karabasov, P. S. Berloff, and V. M. Goloviznin, “Cabaret in the ocean gyres,” Ocean Model. 30 (2), 155–168 (2009). https://doi.org/10.1016/j.ocemod.2009.06.009

    Article  Google Scholar 

  58. V. M. Goloviznin and V. A. Isakov, “Balance-characteristic scheme as applied to the shallow water equations over a rough bottom,” Comput. Math. Math. Phys. 57 (7), 1140–1157 (2017). https://doi.org/10.1134/S0965542517070089

    Article  MathSciNet  MATH  Google Scholar 

  59. O. A. Kovyrkina and V. V. Ostapenko, “On monotonicity of two-layer in time CABARET scheme,” Math. Models Comput. Simul. 5, 180–189 (2013). https://doi.org/10.1134/S2070048213020051

    Article  MathSciNet  Google Scholar 

  60. O. A. Kovyrkina and V. V. Ostapenko, “Monotonicity of the CABARET scheme approximating a hyperbolic equation with a sign-changing characteristic field,” Comput. Math. Math. Phys. 56 (5), 783–801 (2016). https://doi.org/10.1134/S0965542516050122

    Article  MathSciNet  MATH  Google Scholar 

  61. O. A. Kovyrkina and V. V. Ostapenko, “On the monotonicity of the CABARET scheme in the multidimensional case,” Dokl. Math. 91 (3), 323–328 (2015). https://doi.org/10.1134/S1064562415030217

    Article  MathSciNet  MATH  Google Scholar 

  62. N. A. Zyuzina and V. V. Ostapenko, “On the monotonicity of the CABARET scheme approximating a scalar conservation law with a convex flux,” Dokl. Math. 93 (1), 69–73 (2016). https://doi.org/10.1134/S1064562416010282

    Article  MathSciNet  MATH  Google Scholar 

  63. N. A. Zyuzina and V. V. Ostapenko, “Monotone approximation of a scalar conservation law based on the C-ABARET scheme in the case of a sign-changing characteristic field,” Dokl. Math. 94 (2), 538–542 (2016). https://doi.org/10.1134/S1064562416050185

    Article  MathSciNet  MATH  Google Scholar 

  64. V. V. Ostapenko and A. A. Cherevko, “Application of the CABARET scheme for calculation of discontinuous solutions of the scalar conservation law with nonconvex flux,” Dokl. Phys. 62 (10), 470–474 (2017). https://doi.org/10.1134/S1028335817100056

    Article  Google Scholar 

  65. N. A. Zyuzina, V. V. Ostapenko, and E. I. Polunina, “A splitting method for a CABARET scheme approximating a nonuniform scalar conservation law,” Numer. Anal. Appl. 11, 146–157 (2018). https://doi.org/10.1134/S1995423918020052

    Article  MathSciNet  MATH  Google Scholar 

  66. W. H. Reed and T. R. Hill, “Triangular mesh methods for the neutron transport equation,” Los Alamos Scientific Laboratory Report LA-UR-73-79, USA (1973). https://www.osti.gov/servlets/purl/4491151

  67. D. N. Arnold, F. Brezzi, B. Cockburn, and L. D. Marini, “Unified analysis of discontinuous Galerkin methods for elliptic problems,” SIAM J. Numer. Anal. 39 (5), 1749–1779 (2002). https://doi.org/10.1137/S0036142901384162

    Article  MathSciNet  MATH  Google Scholar 

  68. J. Peraire and P. O. Persson, “High-order discontinuous Galerkin methods for CFD,” Adv. CFD: Adaptive High-Order Methods Comput. Fluid Dyn. 2, 119–152 (2011). https://doi.org/10.1142/9789814313193_0005

    Article  MATH  Google Scholar 

  69. M. E. Ladonkina, O. A. Neklyudova, and V. F. Tishkin, “Application of the RKDG method for gas dynamics problems,” Math. Models Comput. Simul. 6, 397–407 (2014). https://doi.org/10.1134/S207004821404005X

    Article  MathSciNet  MATH  Google Scholar 

  70. C. W. Shu, “High order WENO and DG methods for time-dependent convection-dominated PDEs: A brief survey of several recent developments,” J. Comput. Phys. 316, 598–613 (2016). https://doi.org/10.1016/j.jcp.2016.04.030

    Article  MathSciNet  MATH  Google Scholar 

  71. A. V. Volkov, “Features of the application of the Galerkin method to the three-dimensional Navier–Stokes equations on unstructured hexahedral meshes,” Uch. Zap. TsAGI 40 (6), 41–59 (2009). https://www.elibrary.ru/item.asp?id=13065602

  72. K. Yasue, M. Furudate, N. Ohnishi, and K. Sawada, “Implicit discontinuous Galerkin method for RANS simulation utilizing pointwise relaxation algorithm,” Commun. Comput. Phys. 7 (3), 510–533 (2010). https://doi.org/10.4208/cicp.2009.09.055

    Article  MathSciNet  MATH  Google Scholar 

  73. M. Dumbser, “Arbitrary high order P N P M  schemes on unstructured meshes for the compressible Navier–Stokes equations,” Comput. Fluid. 39 (1), 60–76 (2010). https://doi.org/10.1016/j.compfluid.2009.07.003

    Article  MATH  Google Scholar 

  74. M. M. Krasnov, P. A. Kuchugov, M. E. Ladonkina, and V. F. Tishkin, “Discontinuous Galerkin method on three-dimensional tetrahedral grids: Using the operator programming method,” Math. Models Comput. Simul. 9, 529–543 (2017). https://doi.org/10.1134/S2070048217050064

    Article  MathSciNet  MATH  Google Scholar 

  75. H. Luo, J. D. Baum, and R. Löhner, “Fast p-multigrid discontinuous Galerkin method for compressible flow at all speeds,” AIAA J. 46 (3), 635–652 (2008). https://doi.org/10.2514/1.28314

    Article  Google Scholar 

  76. H. Luo, J. D. Baum, and R. Löhner, “A Hermite WENO-based limiter for discontinuous Galerkin method on unstructured grids,” J. Comput. Phys. 225, 686–713 (2007). https://doi.org/10.1016/j.jcp.2006.12.017

    Article  MathSciNet  MATH  Google Scholar 

  77. J. Zhu, X. Zhong, C. W. Shu, and J. Qiu, “Runge–Kutta discontinuous Galerkin method using a new type of WENO limiters on unstructured meshes,” J. Comput. Phys. 248, 200–220 (2013). https://doi.org/10.1016/j.jcp.2013.04.012

    Article  MathSciNet  MATH  Google Scholar 

  78. A. V. Volkov and S. V. Lyapunov, “Monotonization of the finite element method in gas dynamics problems,” Uch. Zap. TsAGI 40 (4), 15–28 (2009). https://www.elibrary.ru/item.asp?id=12904664

  79. L. Krivodonova, J. Xin, J. F. Remacle, N. Chevogeon, and J. Flaherty, “Shock detection and limiting with discontinuous Galerkin methods for hyperbolic conservation laws,” Appl. Numer. Math. 48 (3), 323–338 (2004). https://doi.org/10.1016/j.apnum.2003.11.002

    Article  MathSciNet  MATH  Google Scholar 

  80. L. Krivodonova, “Limiters for high-order discontinuous Galerkin methods,” J. Comput. Phys. 226 (1), 879–896 (2007). https://doi.org/10.1016/j.jcp.2007.05.011

    Article  MathSciNet  MATH  Google Scholar 

  81. M. E. Ladonkina, O. A. Neklyudova, and V. F. Tishkin, “Constructing a limiter based on averaging the solutions for the discontinuous Galerkin method,” Math. Models Comput. Simul. 11, 61–73 (2019). https://doi.org/10.1134/S2070048219010101

    Article  MathSciNet  Google Scholar 

  82. P. Mocz, M. Vogelsberger, D. Sijacki, R. Pakmor, and L. Hernquist, “A discontinuous Galerkin method for solving the fluid and MHD equations in astrophysical simulations,” Mon. Not. R. Astron. Soc. 437 (1), 397–414 (2014). https://doi.org/10.1093/mnras/stt1890

    Article  Google Scholar 

  83. A. Klockner, T. Warburton, and J. S. Hesthaven, “Nodal discontinuous Galerkin methods on graphics processors,” J. Comput. Phys. 228 (21), 7863–7882 (2009). https://doi.org/10.1016/j.jcp.2009.06.041

    Article  MathSciNet  MATH  Google Scholar 

  84. J. Chan, Z. Wang, A. Modave, J. Remacle, and T. Warburton, “GPU-accelerated discontinuous Galerkin methods on hybrid meshes,” J. Comput. Phys. 318, 142–168 (2016). https://doi.org/10.1016/j.jcp.2016.04.003

    Article  MathSciNet  MATH  Google Scholar 

  85. M. M. Krasnov and M. E. Ladonkina, “Discontinuous Galerkin method on three-dimensional tetrahedral grids. The use of template metaprogramming of the C++ language,” Program. Computer Software 43, 172–183 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  86. B. V. Rogov, “Dispersive and dissipative properties of the fully discrete bicompact schemes of the fourth order of spatial approximation for hyperbolic equations,” Appl. Numer. Math. 139, 136–155 (2019). https://doi.org/10.1016/j.apnum.2019.01.008

    Article  MathSciNet  MATH  Google Scholar 

  87. A. V. Chikitkin and B. V. Rogov, “Family of central bicompact schemes with spectral resolution property for hyperbolic equations,” Appl. Numer. Math. 142, 151–170 (2019). https://doi.org/10.1016/j.apnum.2019.03.007

    Article  MathSciNet  MATH  Google Scholar 

  88. B. V. Rogov and M. N. Mikhailovskaya, “On the convergence of compact difference schemes,” Math. Models Comput. Simul. 1 (1), 91–104 (2009). https://doi.org/10.1134/S2070048209010104

    Article  MathSciNet  Google Scholar 

  89. M. N. Mikhailovskaya and B. V. Rogov, “Monotone compact running schemes for systems of hyperbolic equations,” Comput. Math. Math. Phys. 52 (4), 578–600 (2012). https://doi.org/10.1134/S0965542512040124

    Article  MathSciNet  MATH  Google Scholar 

  90. B. V. Rogov, “High-order accurate monotone compact running scheme for multidimensional hyperbolic equations,” Comput. Math. Math. Phys. 53 (2), 205–214 (2013). https://doi.org/10.1134/S0965542513020097

    Article  MathSciNet  MATH  Google Scholar 

  91. A. V. Chikitkin, B. V. Rogov, and S. V. Utyuzhnikov, “High-order accurate monotone compact running scheme for multidimensional hyperbolic equations,” Appl. Numer. Math. 93, 150–163 (2015). https://doi.org/10.1016/j.apnum.2014.02.008

    Article  MathSciNet  MATH  Google Scholar 

  92. M. D. Bragin and B. V. Rogov, “High‑order bicompact schemes for numerical modeling of multispecies multi-reaction gas flows,” Math. Models Comput. Simul. 13, 106–115 (2021). https://doi.org/10.1134/S2070048221010063

    Article  MathSciNet  Google Scholar 

  93. M. D. Bragin and B. V. Rogov, “Bicompact schemes for the multidimensional convection–diffusion equation,” Comput. Math. Math. Phys. 61 (4), 607–624 (2021). https://doi.org/10.1134/S0965542521040023

    Article  MathSciNet  MATH  Google Scholar 

  94. M. D. Bragin and B. V. Rogov, “Accuracy of bicompact schemes in the problem of Taylor–Green vortex decay,” Comput. Math. Math. Phys. 61 (11), 1723–1742 (2021). https://doi.org/10.1134/S0965542521110051

    Article  MathSciNet  MATH  Google Scholar 

  95. M. D. Bragin and B. V. Rogov, “On exact dimensional splitting for a multidimensional scalar quasilinear hyperbolic conservation law,” Dokl. Math. 94 (1), 382–386 (2016). https://doi.org/10.1134/S1064562416040086

    Article  MathSciNet  MATH  Google Scholar 

  96. M. D. Bragin and B. V. Rogov, “Iterative approximate factorization for difference operators of high-order bicompact schemes for multidimensional nonhomogeneous hyperbolic systems,” Dokl. Math. 95 (2), 140–143 (2017). https://doi.org/10.1134/S1064562417020107

    Article  MathSciNet  MATH  Google Scholar 

  97. M. D. Bragin and B. V. Rogov, “Combined multidimensional bicompact scheme with higher order accuracy in domains of influence of nonstationary shock waves,” Dokl. Math. 102 (2), 360–363 (2020). https://doi.org/10.1134/S1064562420050282

    Article  MathSciNet  MATH  Google Scholar 

  98. R. Alexander, “Diagonally implicit Runge–Kutta methods for stiff O.D.E.’s,” SIAM J. Numer. Anal. 14 (6), 1006–1021 (1977). https://www.jstor.org/stable/2156678

    Article  MathSciNet  MATH  Google Scholar 

  99. V. V. Ostapenko and N. A. Khandeeva, “The accuracy of finite-difference schemes calculating the interaction of shock waves,” Dokl. Phys. 64 (4), 197–201 (2019). https://doi.org/10.1134/S1028335819040128

    Article  Google Scholar 

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Funding

The reported study was funded in part by the Russian Foundation for Basic Research and the National Natural Science Foundation of China (project no. 21-51-53012) (Sections 4–7, 9) and by the Russian Science Foundation (project no. 21-11-00198) (Sections 8, 10).

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Authors and Affiliations

  1. Federal Research Center Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, 125047, Moscow, Russia

    M. D. Bragin, M. E. Ladonkina & V. F. Tishkin

  2. Moscow Institute of Physics and Technology (National Research University), 141700, Dolgoprudnyi, Moscow oblast, Russia

    M. D. Bragin

  3. Lavrentyev Institute of Hydrodynamics, Siberian Branch, Russian Academy of Sciences, 630090, Novosibirsk, Russia

    O. A. Kovyrkina, V. V. Ostapenko & N. A. Khandeeva

Authors
  1. M. D. Bragin
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  2. O. A. Kovyrkina
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  3. M. E. Ladonkina
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  4. V. V. Ostapenko
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  5. V. F. Tishkin
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  6. N. A. Khandeeva
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Corresponding authors

Correspondence to M. D. Bragin, O. A. Kovyrkina, M. E. Ladonkina, V. V. Ostapenko, V. F. Tishkin or N. A. Khandeeva.

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To the blessed memory of Boris Vadimovich Rogov

Translated by I. Ruzanova

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Bragin, M.D., Kovyrkina, O.A., Ladonkina, M.E. et al. Combined Numerical Schemes. Comput. Math. and Math. Phys. 62, 1743–1781 (2022). https://doi.org/10.1134/S0965542522100025

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  • DOI: https://doi.org/10.1134/S0965542522100025

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