Advertisement

Numerical Algorithms

, Volume 31, Issue 1–4, pp 5–25 | Cite as

Efficient Collocation Schemes for Singular Boundary Value Problems

  • Winfried Auzinger
  • Othmar Koch
  • Ewa Weinmüller
Article

Abstract

We discuss an error estimation procedure for the global error of collocation schemes applied to solve singular boundary value problems with a singularity of the first kind. This a posteriori estimate of the global error was proposed by Stetter in 1978 and is based on the idea of Defect Correction, originally due to Zadunaisky. Here, we present a new, carefully designed modification of this error estimate which not only results in less computational work but also appears to perform satisfactorily for singular problems. We give a full analytical justification for the asymptotical correctness of the error estimate when it is applied to a general nonlinear regular problem. For the singular case, we are presently only able to provide computational evidence for the full convergence order, the related analysis is still work in progress. This global estimate is the basis for a grid selection routine in which the grid is modified with the aim to equidistribute the global error. This procedure yields meshes suitable for an efficient numerical solution. Most importantly, we observe that the grid is refined in a way reflecting only the behavior of the solution and remains unaffected by the unsmooth direction field close to the singular point.

ordinary differential equations singularity of the first kind boundary value problems numerical solution collocation methods global error estimation mesh selection 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    U. Ascher, J. Christiansen and R.D. Russell, A collocation solver for mixed order systems of boundary values problems, Math. Comp. 33 (1978) 659–679.Google Scholar
  2. [2]
    U. Ascher, J. Christiansen and R.D. Russell, Collocation software for boundary value ODEs, ACM Trans. Math. Software 7(2) (1981) 209–222.Google Scholar
  3. [3]
    U. Ascher, R.M.M. Mattheij and R.D. Russell, Numerical Solution of Boundary Value Problems for Ordinary Differential Equations (Prentice-Hall, Englewood Cliffs, NJ, 1988).Google Scholar
  4. [4]
    W. Auzinger, O. Koch, P. Kofler and E. Weinmüller, The application of shooting to singular boundary value problems, Technical Report Nr. 126/99, Inst. for Appl. Math. and Numer. Anal., Vienna University of Technology, Austria (1999); available at http://fsmat.at/¸othmar/research.html.Google Scholar
  5. [5]
    W. Auzinger, O. Koch, W. Polster and E. Weinmüller, Ein Algorithmus zur Gittersteuerung bei Kollokationsverfahren für singuläre Randwertprobleme, Technical Report, ANUM Preprint Nr. 21/01, Inst. for Appl. Math. and Numer. Anal., Vienna University of Technology, Austria (2001); available at http://fsmat.at/¸othmar/research.html.Google Scholar
  6. [6]
    E. Badralexe and A.J. Freeman, Eigenvalue equation for a general periodic potential and its multipole expansion solution, Phys. Rev. B 37(3) (1988) 1067–1084.Google Scholar
  7. [7]
    L. Bauer, E.L. Reiss and H.B. Keller, Axisymmetric buckling of hollow spheres and hemispheres, Comm. Pure Appl. Math. 23 (1970) 529–568.Google Scholar
  8. [8]
    T.W. Carr and T. Erneux, Understanding the bifurcation to traveling waves in a class-b laser using a degenerate Ginzburg-Landau equation, Phys. Rev. A 50 (1994) 4219–4227.Google Scholar
  9. [9]
    C.Y. Chan and Y.C. Hon, A constructive solution for a generalized Thomas-Fermi theory of ionized atoms, Quart. Appl. Math. 45 (1987) 591–599.Google Scholar
  10. [10]
    C. de Boor and B. Swartz, Collocation at Gaussian points, SIAM J. Numer. Anal. 10 (1973) 582–606.Google Scholar
  11. [11]
    M. Drmota, R. Scheidl, H. Troger and E. Weinmüller, On the imperfection sensitivity of complete spherical shells, Comp. Mech. 2 (1987) 63–74.Google Scholar
  12. [12]
    R. Fazio, A novel approach to the numerical solution of boundary value problems on infinite intervals, SIAM J. Numer. Anal. 33 (1996) 1473–1483.Google Scholar
  13. [13]
    R. Frank, Schätzungen des globalen Diskretisierungsfehlers bei Runge-Kutta-Methoden, ISNM 27 (1975) 45–70.Google Scholar
  14. [14]
    R. Frank, The method of Iterated defect correction and its application to two-point boundary value problems, Part I, Numer. Math. 25 (1976) 409–419.Google Scholar
  15. [15]
    R. Frank and C. Ñberhuber, Iterated Defect Correction for differential equations, Part I: Theoretical results, Computing 20 (1978) 207–228.Google Scholar
  16. [16]
    M. Gräff and E. Weinmüller, Schätzungen des lokalen Diskretisierungsfehlers bei singulären Anfangswertproblemen, Technical Report Nr. 66/86, Inst. for Appl. Math. and Numer. Anal., Vienna University of Technology, Austria (1986).Google Scholar
  17. [17]
    E. Hairer, S.P. N¸rsett and G. Wanner, Solving Ordinary Differential Equations I (Springer, Berlin, 1987).Google Scholar
  18. [18]
    F.R. de Hoog and R. Weiss, Difference methods for boundary value problems with a singularity of the first kind, SIAM J. Numer. Anal. 13 (1976) 775–813.Google Scholar
  19. [19]
    F.R. de Hoog and R. Weiss, The application of linear multistep methods to singular initial value problems, Math. Comp. 32 (1977) 676–690.Google Scholar
  20. [20]
    F.R. de Hoog and R. Weiss, Collocation methods for singular boundary value problems, SIAM J. Numer. Anal. 15 (1978) 198–217.Google Scholar
  21. [21]
    F.R. de Hoog and R. Weiss, The application of Runge-Kutta schemes to singular initial value problems, Math. Comp. 44 (1985) 93–103.Google Scholar
  22. [22]
    O. Koch, P. Kofler and E. Weinmüller, Initial value problems for systems of ordinary first and second order differential equations with a singularity of the first kind, Analysis 21 (2001) 373–389.Google Scholar
  23. [23]
    O. Koch and E. Weinmüller, Acceleration techniques for singular initial value problems, in: Problems in Modern Applied Mathematics, ed. N. Mastorakis (WSES Press, 2000) pp. 6–11.Google Scholar
  24. [24]
    O. Koch and E. Weinmüller, The convergence of shooting methods for singular boundary value problems, to appear in Math. Comp.Google Scholar
  25. [25]
    O. Koch and E. Weinmüller, Iterated Defect Correction for the solution of singular initial value problems, SIAM J. Numer. Anal. 38 (2001) 1784–1799.Google Scholar
  26. [26]
    P. Kofler, Theorie und numerische Lösung singulärer Anfangswertprobleme gewöhnlicher Differentialgleichungen mit der Singularität erster Art, Ph.D. thesis, Inst. for Appl. Math. and Numer. Anal., Vienna University of Technology, Austria (1998).Google Scholar
  27. [27]
    X. Liu, A note on the Sturmian Theorem for singular boundary value problems, J. Math. Anal. Appl. 237 (1999) 393–403.Google Scholar
  28. [28]
    R. März and E. Weinmüller, Solvability of boundary value problems for systems of singular differential-algebraic equations, SIAM J. Math. Anal. 24 (1993) 200–215.Google Scholar
  29. [29]
    G. Moore, Computation and parametrization of periodic and connecting orbits, IMA J. Numer. Anal. 15 (1995) 245–263.Google Scholar
  30. [30]
    S.V. Parter, M.L. Stein and P.R. Stein, On the multiplicity of solutions of a differential equation arising in chemical reactor theory, Technical Report Nr. 194, Department of Computer Sciences, University of Wisconsin (1973).Google Scholar
  31. [31]
    R.D. Russell and J. Christiansen, Adaptive mesh selection strategies for solving boundary value problems, SIAM J. Numer. Anal. 15(1) (1978) 59–80.Google Scholar
  32. [32]
    H.J. Stetter, Analysis of Discretization Methods for Ordinary Differential Equations (Springer, Berlin, 1973).Google Scholar
  33. [33]
    H.J. Stetter, The defect correction principle and discretization methods, Numer. Math. 29 (1978) 425–443.Google Scholar
  34. [34]
    E. Weinmüller, Collocation for singular boundary value problems of second order, SIAM J. Numer. Anal. 23 (1986) 1062–1095.Google Scholar
  35. [35]
    P.E. Zadunaisky, On the estimation of errors propagated in the numerical integration of ODEs, Numer. Math. 27 (1976) 21–39.Google Scholar

Copyright information

© Kluwer Academic Publishers 2002

Authors and Affiliations

  • Winfried Auzinger
    • 1
  • Othmar Koch
    • 1
  • Ewa Weinmüller
    • 1
  1. 1.Department of Applied Mathematics and Numerical AnalysisVienna University of TechnologyViennaAustria

Personalised recommendations