Local and Global Error Estimates

  • Debora AmadoriEmail author
  • Laurent Gosse
Part of the SpringerBriefs in Mathematics book series (BRIEFSMATH)


In this chapter we analyze some simple examples, which suggest that the error quantification should take into account of the possible grow in time of the error. This observation provides a motivation for going beyond more classical local-in-time concepts of error (so-called Local Truncation Error).


Accuracy of schemes for balance laws Local truncation error 


  1. 1.
    B. Alpert, L. Greengard, T. Hagstrom, An integral evolution formula for the wave equation. J. Comput. Phys. 162(2), 536–543 (2000)Google Scholar
  2. 2.
    M. Arora, P.L. Roe, On postshock oscillations due to capturing schemes in unsteady flows. J. Comput. Phys. 130, 25–40 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    C. Berthon, C. Sarazin, R. Turpault, Space-time generalized Riemann problem solvers of order \(k\) for linear advection with unrestricted time step. J. Sci. Comput. 55, 268–308 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    F. Bouchut, Nonlinear stability of finite volume methods for hyperbolic conservation laws, and well-balanced schemes for sources, Frontiers in Mathematics Series (Birkhäuser, Boston, 2004). ISBN 3-7643-6665-6CrossRefzbMATHGoogle Scholar
  5. 5.
    A. Bressan, Hyperbolic systems of conservation laws—the one-dimensional Cauchy problem, vol. 20, Oxford Lecture Series in Mathematics and Its Applications (Oxford University Press, Oxford, 2000)zbMATHGoogle Scholar
  6. 6.
    M.J.P. Cullen, K.W. Morton, Analysis of evolutionary error in finite element and other methods. J. Comput. Phys. 34, 245–267 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    G. Efraimsson, G. Kreiss, A remark on numerical errors downstream of slightly viscous shocks. SIAM J. Numer. Anal. 36, 853–863 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    B. Engquist, B. Sjögreen, The convergence rate of finite difference schemes in the presence of shocks. SIAM J. Numer. Anal. 35, 2464–2485 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    H. Gilquin, Une famille de schémas numériques T.V.D. pour les lois de conservation hyperboliques. RAIRO—Model. Math. Anal. Numer. 20, 429–460 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    J.B. Goodman, R.J. LeVeque, A geometric approach to high resolution TVD schemes. SIAM J. Numer. Anal. 25, 268–284 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    L. Gosse, MUSCL reconstruction and Haar wavelets. Commun. Math. Sci. 13, 1501–1514 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Z. Haras, S. Ta’asan, Finite difference schemes for long-time integration. J. Comput. Phys. 114(2), 265–279 (1994)Google Scholar
  13. 13.
    S. Jin, J.-G. Liu, The effects of numerical viscosities. I. Slowly moving shocks. J. Comput. Phys. 126, 373–389 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    S. Lele, Compact finite difference schemes with spectral-like resolution. J. Comput. Phys. 103(1), 16–42 (1992)Google Scholar
  15. 15.
    R.J. LeVeque, Numerical Methods for Conservation Laws (Birkhauser, ETH Zurich, Basel, 1992)CrossRefzbMATHGoogle Scholar
  16. 16.
    R. Menikoff, Errors when shock waves interact due to numerical shock width. SIAM J. Sci. Comput. 15, 1227–1242 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    K.W. Morton, On the analysis of finite volume methods for evolutionary problems. SIAM J. Numer. Anal. 35, 2195–2222 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    S. Osher, Convergence of generalized MUSCL schemes. SIAM J. Numer. Anal. 22, 947–961 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    B. Popov, O. Trifonov, Order of convergence of second order schemes based on the MINMOD limiter. Math. Comput. 75, 1735–1753 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    J.M. Sanz-Serna, J.G. Verwer, Convergence analysis of one-step schemes in the method of lines. Appl. Math. Comput. 31, 183–196 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    J.M. Sanz-Serna, J.G. Verwer, Stability and convergence at the PDE/stiff ODE interface. Appl. Numer. Math. 5, 117–132 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    M. Siklosi, G. Efraimsson, Analysis of first order errors in shock calculations in two space dimensions. SIAM J. Numer. Anal. 43, 672–685 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    M. Siklosi, B. Batzorig, G. Kreiss, An investigation of the internal structure of shock profiles for shock capturing schemes. J. Comput. Appl. Math. 201, 8–29 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
  25. 25.
    B. Swartz, B. Wendroff, The relative efficiency of finite difference and finite element methods. I: hyperbolic problems and splines, SIAM J. Numer. Anal. 11, 979–993 (1974)Google Scholar
  26. 26.
    P.K. Sweby, High resolution schemes using flux limiters for hyperbolic conservation laws. SIAM J. Numer. Anal. 21, 995–1011 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    E.F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction, 3rd edn. (Springer, New York, 2009)CrossRefzbMATHGoogle Scholar
  28. 28.
    B. Van Leer, Towards the ultimate conservative difference scheme, V. A second order sequel to Godunov’s method. J. Comput. Phys. 32, 101–136 (1979)CrossRefGoogle Scholar
  29. 29.
    J.G. Verwer, Contractivity in locally one-dimensional splitting methods. Numer. Math. 44, 247–259 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    J.G. Verwer, J.M. Sanz-Serna, Convergence of method of lines approximations to partial differential equations. Computing 33, 297–313 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    R. Vichnevetsky, J.B. Bowles, Fourier analysis of numerical approximations of hyperbolic equations. SIAM J. Applied Math. 5 (1982)Google Scholar
  32. 32.
    Zaide, D.W.-M.: Numerical shock-wave anomalies. Ph.D. thesis, University of Michigan (2012)Google Scholar

Copyright information

© The Author(s) 2015

Authors and Affiliations

  1. 1.DISIMUniversità degli Studi dell’AquilaL’AquilaItaly
  2. 2.Istituto per le Applicazioni del CalcoloCNRRomeItaly

Personalised recommendations