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Hyperbolic Problems: Theory, Numerics, Applications pp 873–882Cite as

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Error Estimates of Approximate Solutions for Nonlinear Scalar Conservation Laws

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Part of the book series: ISNM International Series of Numerical Mathematics ((ISNM,volume 141))

Abstract

There has been an enormous amount of work on error estimates for approximate solutions to scalar conservation laws. The methods of analysis include matching the traveling wave solutions, [8,24]; matching the Green function of the linearized problem [21]; weak W-1,1 convergence theory [32]; the Kruzkov-functional method [19]; and the energy-like method [34]. The results on error estimates include: For BV entropy solutions, an convergence rate in L1 obtained by Kuznetsov [19], Lucier [25] etc; For BV entropy solutions, an convergence rate in W-1,1 obtained by Tadmor-Nessyahu [27,32], Liu-Wang-Warnecke [22] etc; For piecewise smooth solutions an convergence rate in L1 obtained by Bakhvalov [1], Harabetian [9], Teng-Tang [39,42], Fan [7] etc; For piecewise smooth solutions an convergence rate in smooth region of the entropy solution obtained by Liu [21], Goodman-Xin [8], Engquist-Sjogreen [6], Tadmor-Tang [34,36] etc. In this paper, we will review some known results on error estimates and will discuss some recent development on the convergence rates. The corresponding methods in obtaining these results will be briefly illustrated.

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References

  1. N. S. Bakhvalov, Estimation of the error of numerical integration of a first-order quasilinear equation,Zh. Vychisl. Mat. i Mat. Fiz., 1 (1961), pp. 771–783; English transi. in USSR Comput. Math. and Math. Phys., 1 (1962), pp. 926–938.

    Google Scholar 

  2. F. Bouchut and B. Perthame, Kruzkov’s estimates for scalar conservation laws revisited, Trans. Amer. Math. Soc., 350 (1998), pp. 2847–2870.

    Article  MathSciNet  MATH  Google Scholar 

  3. A. Chalabi, On convergence of numerical schemes for hyperbolic conservation laws with stiff source terms, Math. Comp., 66 (1997), pp. 527–545.

    Article  MathSciNet  MATH  Google Scholar 

  4. B. Cockburn, Quasimonotone schemes for scalar conservation laws. I. II. III., SIAM J. Numer. Anal. 26 (1989) 1325–1341, 27 (1990) 247–258, 259–276.

    Google Scholar 

  5. B. Cockburn, F. Coquel and P. Lefloch, Convergence of finite volume methods for multidimensional conservation laws, SIAM J. Numer. Anal. 32 (1995), 687–705.

    Article  MathSciNet  MATH  Google Scholar 

  6. B. Engquist and B. Sjogreen, The convergence rate of finite difference schemes in the presence of shocks, Siam J. Numer. Anal., 35 (1998), pp. 2464–2485.

    Article  MathSciNet  MATH  Google Scholar 

  7. H. Fan, Existence of discrete traveling waves and error estimates for Godunov schemes of conservation laws, Math. Comp., 67 (1998), pp. 87–109.

    Article  MathSciNet  MATH  Google Scholar 

  8. J. Goodman and Z. Xin, Viscous limits for piecewise smooth solutions to systems of conservation laws, Arch. Rational Mech. Anal., 121 (1992), pp. 235–265.

    Article  MathSciNet  MATH  Google Scholar 

  9. E. Harabetian, Rarefactions and large time behavior for parabolic equations and monotone schemes, Comm. Math. Phys., 114 (1988), pp. 527–536.

    Article  MathSciNet  MATH  Google Scholar 

  10. A. Harten, The artificial compression method for computation of shocks and contact discontinuities, Comm. Pure Appl. Math., 30 (1977), pp. 611–638.

    Article  MathSciNet  MATH  Google Scholar 

  11. A. Harten, M. Hyman and P. D. Lax, On finite-difference approximations and entropy conditions for shocks (with Appendix by B. Keyfitz), Comp. Pure Appl. Math., 29 (1976), pp. 297–322.

    Article  MathSciNet  MATH  Google Scholar 

  12. H. Holden, K. H. Karlsen and N. H. Risebro, Operator splitting methods for generalized KdV equations. J. Comput. Phys., 153 (1999), pp. 203–222.

    Article  MathSciNet  MATH  Google Scholar 

  13. E. R. Jakobsen, K. H. Karlsen, and N. H. Risebro, On the convergence rate of operator splitting for Hamilton-Jacobi equations with source terms, Siam J. Numer. Anal., 39 (2001), pp. 499–518.

    Article  MathSciNet  MATH  Google Scholar 

  14. S. Jin and Z. Xin, The relaxing schemes for systems of conservation laws in arbitrary space dimensions, Comm. Pure. Appl. Math., 48 (1995), pp. 235–277.

    Article  MathSciNet  MATH  Google Scholar 

  15. C. Johnson and A. Szepessy, Convergence of a finite element methods for a nonlinear hyperbolic conservation law, Math. Comp. 49(1988), 427–444.

    Article  MathSciNet  Google Scholar 

  16. C. Johnson, A. Szepessy and P. Hansbo, On the convergence of shock-capturing streamline diffusion finite element methods for hyperbolic conservation laws, Math. Comp. 54 (1990), 107–129.

    Article  MathSciNet  MATH  Google Scholar 

  17. M. A. Katsoulakis and A. E. Tzavaras, Contractive relaxation systems and the scalar multidimensional conservation law, Comm. P.D.E., 22 (1997), 195–233.

    Article  MathSciNet  MATH  Google Scholar 

  18. D. KrÓner, S. Noelle and M. Rokyta, Convergence of higher order upwind finite volume schemes on unstructured grids for scalar conservation laws in several space dimensions, Numer. Math. 71 (1995) 527–560.

    Article  MathSciNet  Google Scholar 

  19. N. N. Kuznetsov, Accuracy of some approximate methods for computing the weak solutions of a first-order quasi-linear equation, USSR Comput. Math. and Math. Phys., 16 (1976), pp. 105–119.

    Article  Google Scholar 

  20. J. O. Langseth, A. Tveito and R. Winther, On the convergence of operator splitting applied to conservation laws with source terms, SIAM J. Numer. Anal., 33 (1996), pp. 843–863.

    Article  MathSciNet  MATH  Google Scholar 

  21. T.-P. Liu, Pointwise convergence to shock waves for viscous conservation laws, Comm. Pure Appl. Math., 50 (1997), pp. 1113–1182.

    Article  MathSciNet  MATH  Google Scholar 

  22. H. Liu, J. Wang and G. Warnecke, The Lip+-stability and error estimates for a relaxation scheme, SIAM J. Numer. Anal., 38 (2001), pp. 1154–1170.

    Article  MathSciNet  Google Scholar 

  23. H. Liu and G. Warnecke, Convergence rates for relaxation schemes approximating conservation laws, SIAM J. Numer. Anal., 37 (2000), pp. 1316–1337.

    Article  MathSciNet  MATH  Google Scholar 

  24. J. Liu and Z. Xin, L 1 - stability of stationary discrete shocks, Math. Comp., 60 (1993), pp. 233–244.

    MathSciNet  MATH  Google Scholar 

  25. B. J. Lucier, Error bounds for the methods of Glimm,Godunov and LeVeque, SIAM J. Numer. Anal., 22 (1985), pp. 1074–1081.

    Article  MathSciNet  MATH  Google Scholar 

  26. R. Natalini, Convergence to equilibrium for the relaxation approximations of conservation laws, Comm. Pure Appl. Math., 49 (1996), pp. 1–30.

    Article  MathSciNet  Google Scholar 

  27. H. Nessyahu and E. Tadmor, The convergence rate of approximate solutions for nonlinear scalar conservation laws, SIAM J. Numer. Anal., 29 (1992), pp. 1505–1519.

    Article  MathSciNet  MATH  Google Scholar 

  28. S. Osher and E. Tadmor, On the convergence of difference approximations to scalar conservation laws, Math. Comp. 50 (1988), pp. 19–51.

    Article  MathSciNet  MATH  Google Scholar 

  29. F. Sabac, The optimal convergence rate of monotone finite difference methods for hyperbolic conservation laws, SIAM J. Numer. Anal., 34 (1997), pp. 2306–2318.

    Article  MathSciNet  MATH  Google Scholar 

  30. H. J. Schroll, A. Tveito and R. Winther, An error bound for a finite difference scheme applied to a stiff system of conservation laws, SIAM J. Numer. Anal. 34 (1997), pp. 1152–1166.

    Article  MathSciNet  MATH  Google Scholar 

  31. C.-W. SBU, Numerical experiments on the accuracy of ENO and modified ENO schemes, J. Comput. Phys., 5 (1990), pp. 127–149.

    Google Scholar 

  32. E. Tadmor, Local error estimates for discontinuous solutions of nonlinear hyperbolic equations, SIAM J. Numer. Anal., 28 (1991), pp. 891–906.

    Article  MathSciNet  MATH  Google Scholar 

  33. E. Tadmor, Approximation solutions of nonlinear conservation laws, in Lecture Notes in Mathematics 1697 (A. Quarteroni, ed.), Springer, 1997, pp. 1–149.

    Google Scholar 

  34. E. TADMOR AND T. TANG, Pointwise convergence rate for scalar conservation laws with piecewise smooth solutions, SIAM J. Numer. Anal., 36 (1999), pp. 1739–1758.

    Article  MathSciNet  MATH  Google Scholar 

  35. E. Tadmor and T. Tang, The optimal convergence rate of finite difference solutions for nonlinear conservation laws, Proceedings of Seventh International Conference on Hyperbolic Problems (ETH Zurich, Switzerland), pp. 925–934, 1999.

    Google Scholar 

  36. E. Tadmor and T. Tang, Pointwise error estimates for relaxation approximations to conservation laws, SIAM J. Math. Anal., 32 (2001), pp. 870–886.

    Article  MathSciNet  Google Scholar 

  37. T. Tang and Z.-H. Teng, Error bounds for fractional step methods for conservation laws with source terms. SIAM J. Numer. Anal., 32 (1995), pp. 110–127.

    Article  MathSciNet  MATH  Google Scholar 

  38. T. Tang and Z.-H. Teng, The sharpness of Kuznetsov’s O(O) L l -error estimate for monotone difference schemes . Math. Comp. (1995), pp.581–589.

    Google Scholar 

  39. T. Tang and Z.-H. Teng, Viscosity methods for piecewise smooth solutions to scalar conservation laws, Math. Comp., 66 (1997), pp. 495–526.

    Article  MathSciNet  MATH  Google Scholar 

  40. T. Tang and Z.-H. Teng, On the regularity of approximate solutions to conservation laws with piecewise smooth solutions, SIAM J. Numer. Anal., 38 (2001), pp. 1483–1495.

    Article  MathSciNet  Google Scholar 

  41. T. Tang and Z.-H. Teng, Superconvergence of finite difference schemes for piece-wise smooth solutions with shocks,in preparation.

    Google Scholar 

  42. Z.-H. Teng, First-order L l -convergence for relaxation approximations to conservation laws, Comm Pure Appl. Math., 51 (1998), pp. 857–895.

    Article  MathSciNet  Google Scholar 

  43. Z. H. Teng, On the accuracy of fractional step method for conservation laws, SIAM J. Numer. Anal., 31 (1994), pp. 43–63.

    Article  MathSciNet  MATH  Google Scholar 

  44. Z.-H. TENG AND P. W. ZHANG, Optimal L l -rate of convergence for viscosity method and monotone scheme to piecewise constant solutions with shocks, SIAM J. Numer. Anal., 34 (1997), pp. 959–978.

    Article  MathSciNet  MATH  Google Scholar 

  45. W. C. WANG, On L’-convergence rate of viscous and numerical approximate solutions of genuinely nonlinear scalar conservation laws, SIAM J. Math. Anal., 30 (1998), pp. 38–52.

    Article  Google Scholar 

  46. M. Westdickenberg and S. Noelle, A new convergence proof for finite volume schemes using the kinetic formulation of conservation laws, SIAM J. Numer. Anal., 37 (2000), pp. 742–757.

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Dept of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong

    Tao Tang

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  1. Tao Tang
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Editors and Affiliations

  1. Max Planck Institute for Mathematics in the Sciences, Inselstrasse 22-26, Leipzig, 04103, Germany

    Heinrich Freistühler

  2. Institute of Analysis and Numerical Mathematics, Otto-von-Guericke-University, PSF 4120, Magdeburg, 39106, Germany

    Gerald Warnecke

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© 2001 Springer Basel AG

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Tang, T. (2001). Error Estimates of Approximate Solutions for Nonlinear Scalar Conservation Laws. In: Freistühler, H., Warnecke, G. (eds) Hyperbolic Problems: Theory, Numerics, Applications. ISNM International Series of Numerical Mathematics, vol 141. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8372-6_41

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  • DOI: https://doi.org/10.1007/978-3-0348-8372-6_41

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-9538-5

  • Online ISBN: 978-3-0348-8372-6

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