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On the Integral Convergence of Numerical Schemes Calculating Gas-Dynamic Shock Waves

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Abstract

A comparative analysis of the accuracy of shock-capturing schemes, such as the RBM (Rusanov–Burstein–Mirin), CWA (Compact high order Weak Approximation), and A-WENO (Alternative Weighted Essentially Non-Oscillatory) schemes is carried out by numerically solving a gas-dynamic Cauchy problem with smooth periodic initial data. It is shown that, in the presence of shock waves, the RBM and CWA schemes (which do not involve nonlinear flux correction) have a higher order of integral convergence, which provides significantly higher accuracy to these schemes (compared to A-WENO) in areas of shock wave influence, despite the noticeable nonphysical oscillations at their fronts. This makes it possible to use the RBM and CWA schemes as basic ones in constructing combined schemes that monotonically localize shock fronts and, at the same time, maintain higher order accuracy in shock influence areas.

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REFERENCES

  1. S. K. Godunov, “A difference method for numerical calculation of discontinuous solutions of the equations of hydrodynamics,” Mat. Sb. 47 (3), 271–306 (1959).

    MathSciNet  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  ADS  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  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  5. 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 

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

    Article  ADS  MathSciNet  Google Scholar 

  7. S. Karni, A. Kurganov, and G. Petrova, “A smoothness indicator for adaptive algorithms for hyperbolic systems,” J. Comput. Phys. 178, 323–341 (2002). https://doi.org/10.1006/jcph.2002.7024

    Article  ADS  MathSciNet  CAS  Google Scholar 

  8. A. G. Kulikovskii, N. V. Pogorelov, and A. Yu. Semenov, Mathematical Aspects of Numerical Solution of Hyperbolic Systems (Fizmatlit, Moscow, 2001; Chapman and Hall/CRC, London, 2001).

  9. R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems (Cambridge Univ. Press, Cambridge, 2002). https://doi.org/10.1017/CBO9780511791253

    Book  Google Scholar 

  10. E. F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction (Springer, Berlin, 2009). https://doi.org/10.1007/b79761

    Book  Google Scholar 

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

    MathSciNet  Google Scholar 

  12. J. Casper and M. H. Carpenter, “Computational consideration for the simulation of shock-induced sound,” SIAM J. Sci. Comput. 19 (3), 813–828 (1998). https://doi.org/10.1137/S1064827595294101

    Article  MathSciNet  Google Scholar 

  13. S. Chu, O. A. Kovyrkina, A. Kurganov, and V. V. Ostapenko, “Experimental convergence rate study for three shock-capturing schemes and development of highly accurate combined schemes,” Numer. Methods Part. Differ. Equations 5, 4317–4346 (2023). https://doi.org/10.1002/num.23053

    Article  MathSciNet  Google Scholar 

  14. O. A. Kovyrkina and V. V. Ostapenko, “On the practical accuracy of shock-capturing schemes,” Math. Models Comput. Simul. 6 (2), 183–191 (2014). https://doi.org/10.1134/S2070048214020069

    Article  MathSciNet  Google Scholar 

  15. N. A. Mikhailov, “The convergence order of WENO schemes behind a shock front,” Math. Models Comput. Simul. 7 (5), 467–474 (2015). https://doi.org/10.1134/S2070048215050075

    Article  MathSciNet  Google Scholar 

  16. 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  Google Scholar 

  17. 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  Google Scholar 

  18. 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  Google Scholar 

  19. 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), 864–878 (2020). https://doi.org/10.1134/S0965542520050061

    Article  MathSciNet  Google Scholar 

  20. O. A. Kovyrkina, A. A. Kurganov, and V. V. Ostapenko, “Comparative analysis of the accuracy of three different schemes in the calculation of shock waves,” Math. Models Comput. Simul. 15 (3), 401–414 (2023). https://doi.org/10.1134/S2070048223030092

    Article  MathSciNet  Google Scholar 

  21. M. D. Bragin, O. A. Kovyrkina, M. E. Ladonkina, V. V. Ostapenko, V. F. Tishkin, and N. A. Khandeeva, “Combined numerical schemes,” Comput. Math. Math. Phys. 62 (11), 1743–1781 (2022). https://doi.org/10.1134/S0965542522100025

    Article  MathSciNet  Google Scholar 

  22. 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 

  23. 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  ADS  MathSciNet  Google Scholar 

  24. 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  Google Scholar 

  25. B.-S. Wang, W. S. Don, A. Kurganov, and Y. Liu, “Fifth-order A-WENO schemes based on the adaptive diffusion central-upwind Rankine–Hugoniot fluxes,” Commun. Appl. Math. Comput. 5, 295–314 (2023). https://doi.org/10.1007/s42967-021-00161-2

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Funding

The research in Sections 4–7 was supported by the Russian Science Foundation, project no. 22-11-00060.

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Authors and Affiliations

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

    V. V. Ostapenko, E. I. Polunina & N. A. Khandeeva

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  1. V. V. Ostapenko
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  2. E. I. Polunina
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  3. N. A. Khandeeva
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Correspondence to V. V. Ostapenko.

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Translated by I. Ruzanova

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Ostapenko, V.V., Polunina, E.I. & Khandeeva, N.A. On the Integral Convergence of Numerical Schemes Calculating Gas-Dynamic Shock Waves. Dokl. Math. 108, 374–381 (2023). https://doi.org/10.1134/S1064562423701260

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