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A Simple Introduction to Error Estimation for Nonlinear Hyperbolic Conservation Laws

Some Ideas, Techniques, and Promising Results

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Part of the book series: Springer Series in Computational Mathematics ((SSCM,volume 26))

Abstract

In these notes, we present a simple introduction to the topic of a posteriori error estimation for nonlinear hyperbolic conservation laws. This is a topic of great practical interest which has been receiving increasing attention from many researchers in recent years. On the other hand, the highly complex character of its mathematics often obscures the main ideas behind the technical manipulations. Aware of this unfortunate situation, we have written these notes in an attempt to emphasize the ideas and simplify, as much as possible, the presentation of the techniques.

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Bibliography

  1. R. Abeyaratne and J.K Knowles, Implications of viscosity and strain gradient effects for the kinetics of propagating phase boundaries in solids, SIAM J. on Appl. Math. 51 (1991), 1205–1221.

    Article  MathSciNet  MATH  Google Scholar 

  2. M. Affouf and R. Caflish, A numerical study of riemann problem solutions and stability for a system of viscous conservation laws of mixed type, SIAM J. Appl. Math. (1991), 605–634.

    Google Scholar 

  3. R. Becker and R. Rannacher, A feed-back approach to error control in finite element methods: Basic analysis and examples, East-West J. Numer. Math., 4 (1996), 237–264.

    MathSciNet  MATH  Google Scholar 

  4. K. Bottcher and R. Rannacher, Adaptive error control in solving ordinary differential equations by the discontinuous Galerkin method, Preprint University of Heidelberg, 1996.

    Google Scholar 

  5. B. Cockburn, F. Coquel, and P LeFloch An error estimate for finite-volume methods for conservations laws, Math. Comp. 63 (1994), 77–103.

    Article  MathSciNet  MATH  Google Scholar 

  6. B. Cockburn and H. Gau, A posteriori error estimates for general numerical methods for scalar conservation laws, Mat. Aplic. Comp., 14 (1995), 37–47.

    MathSciNet  MATH  Google Scholar 

  7. B. Cockburn and P.-A. Gremaud, Error estimates for finite element methods for scalar conservation laws, SIAM J. Numer. Anal., 33 (1996), 522–554.

    Article  MathSciNet  MATH  Google Scholar 

  8. B. Cockburn and G. Gripenberg, Continuous dependence on the nonlinearities of solutions of degenerate parabolic equations, to appear in J. Diff. Equa.

    Google Scholar 

  9. B. Cockburn and J. X. Yang, A posteriori error estimates for nonlinear convection-diffusion equations, in preparation.

    Google Scholar 

  10. R. Furer F.-K. Hebeker and R. Rannacher, An adaptive finite element method for unsteady convection-dominated flows with stiff source terms, Preprint University of Heidelberg, 1996.

    Google Scholar 

  11. C. Johnson and A. Szepessy, Adaptive finite element methods for conservation laws based on a posteriori estimates, Comm. Pure Appl. Math., 48 (1995) 199–243.

    Article  MathSciNet  MATH  Google Scholar 

  12. B. J. Lucier, A stable adaptive scheme for hyperbolic conservation laws, SIAM J. Numer. Anal., 22 (1985), 180–203.

    Article  MathSciNet  MATH  Google Scholar 

  13. Lucier, A moving mesh numerical method for hyperbolic conservation laws, Math. Comp., 46 (1986), 59–69.

    Google Scholar 

  14. R. Sanders, On convergence of monotone finite difference schemes with variable spacing differencing, Math. Comp. 40 (1983), 91–106.

    Article  MathSciNet  MATH  Google Scholar 

  15. M. Slemrod, Dynamic phase transitions in a van der waals fluid, J. Diff. Equ. 52 (1984), 1–23.

    Article  MathSciNet  MATH  Google Scholar 

  16. M. Slemrod, Lax-Friedrichs and the viscosity-capillarity criterion, Physical Partial Differential Equations (J. Lightbourne and S. Rankin, eds.), Marcel Dekker, New York, 1984, pp. 75–84.

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  18. L. Truskinovsky, Equilibrium phase interfaces, Dokl. Akad. Nauk. SSSR 256 (1982), 306–310.

    Google Scholar 

  19. J.-P VilaConvergence and error estimates for finite volume schemes for general multidimensional scalar conservation laws, Model. Math. Anal. Numér., 28 (1994), 267–295.

    MathSciNet  MATH  Google Scholar 

  20. G. B. Witham, Linear and nonlinear waves, John Wiley amp; Sons (1974).

    Google Scholar 

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© 1999 Springer-Verlag Berlin Heidelberg

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Cockburn, B. (1999). A Simple Introduction to Error Estimation for Nonlinear Hyperbolic Conservation Laws. In: Ainsworth, M., Levesley, J., Marletta, M. (eds) The Graduate Student’s Guide to Numerical Analysis ’98. Springer Series in Computational Mathematics, vol 26. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-03972-4_1

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  • DOI: https://doi.org/10.1007/978-3-662-03972-4_1

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-08503-1

  • Online ISBN: 978-3-662-03972-4

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