Abstract
We develop optimal quadratic and cubic spline collocation methods for solving linear second-order two-point boundary value problems on non-uniform partitions. To develop optimal nonuniform partition methods, we use a mapping function from uniform to nonuniform partitions and develop expansions of the error at the nonuniform collocation points of some appropriately defined spline interpolants. The existence and uniqueness of the spline collocation approximations are shown, under some conditions. Optimal global and local orders of convergence of the spline collocation approximations and derivatives are derived, similar to those of the respective methods for uniform partitions. Numerical results on a variety of problems, including a boundary-layer problem, and a nonlinear problem, verify the optimal convergence of the methods, even under more relaxed conditions than those assumed by theory.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
D. Archer (1977) ArticleTitleAn O(h 4) cubic spline collocation method for quasilinear parabolic equations SIAM J. Numer. Anal 14 IssueID4 620–637 Occurrence Handle10.1137/0714042 Occurrence Handle0366.65054 Occurrence Handle461934
U. Ascher J. Christiansen R. D. Russell (1979) ArticleTitleA collocation solver for mixed order systems of boundary value problems Math. Comp. 33 IssueID146 659–679 Occurrence Handle521281 Occurrence Handle0407.65035
G. Bader U. Ascher (1987) ArticleTitleA new basis implementation for a mixed order boundary value ODE solver SIAM J. Sci. Stat. Comp. 8 IssueID4 483–500 Occurrence Handle10.1137/0908047 Occurrence Handle892301 Occurrence Handle0633.65084
Celia, M. A., Gray, W. G.: Numerical methods for differential equations. Prentice Hall 1992.
C. C. Christara (1994) ArticleTitleQuadratic spline collocation methods for elliptic partial differential equations BIT 34 33–61 Occurrence Handle10.1007/BF01935015 Occurrence Handle0815.65118 Occurrence Handle1429687
Christara, C., Ng, K. S.: Adaptive techniques for spline collocation. Computing 76, 259–277 (2006) (this issue).
J. W. Daniel B. K. Swartz (1975) ArticleTitleExtrapolated collocation for two-point boundary-value problems using cubic splines J. Inst. Maths Appl. 16 161–174 Occurrence Handle391519 Occurrence Handle10.1093/imamat/16.2.161 Occurrence Handle0402.65051
C. Boor Particlede B. Swartz (1973) ArticleTitleCollocation at Gaussian points SIAM J. Numer. Anal. 10 IssueID4 582–606 Occurrence Handle373328 Occurrence Handle0232.65065
D. J. Fyfe (1969) ArticleTitleThe use of cubic splines in the solution of two-point boundary value problems Comput. J. 12 188–192 Occurrence Handle10.1093/comjnl/12.2.188 Occurrence Handle0185.41404 Occurrence Handle243744
E. N. Houstis C. C. Christara J. R. Rice (1988) ArticleTitleQuadratic-spline collocation methods for two-point boundary value problems Int. J. Numer. Meth. Eng. 26 935–952 Occurrence Handle10.1002/nme.1620260412 Occurrence Handle933694 Occurrence Handle0654.65057
E. N. Houstis W. F. Mitchell J. R. Rice (1985) ArticleTitleCollocation software for second-order elliptic partial differential equations ACM Trans. Math. Soft. 11 IssueID4 379–412 Occurrence Handle828564 Occurrence Handle0584.65075
E. N. Houstis E. A. Vavalis J. R. Rice (1988) ArticleTitleConvergence of an O(h 4) cubic spline collocation method for elliptic partial differential equations SIAM J. Numer. Anal. 25 IssueID1 54–74 Occurrence Handle10.1137/0725006 Occurrence Handle923926 Occurrence Handle0637.65113
M. Irodotou-Ellina E. N. Houstis (1988) ArticleTitleAnO(h 6) quintic spline collocation method for fourth order two-point boundary value problems BIT 28 288–301 Occurrence Handle10.1007/BF01934092 Occurrence Handle938393 Occurrence Handle0651.65063
Ng, K. S.: Spline Collocation on adaptive grids and non-rectangular regions. PhD thesis, Department of Computer Science, University of Toronto, Toronto, Ontario, Canada, 2005 http://www.cs.toronto.edu/pub/reports/na/ccc/ngkit-05-phd.ps.gz
Prenter, P. M.: Splines and variational methods. New York: Wiley 1975.
R. D. Russell L. F. Shampine (1971) ArticleTitleA collocation method for boundary value problems Numer. Math. 19 1–28 Occurrence Handle305607
Zhu, Y.: Optimal quartic spline collocation methods for fourth order two-point boundary value problems. M.Sc. thesis, Department of Computer Science, University of Toronto, Toronto, Ontario, Canada. http://www.cs.toronto.edu/pub/reports/na/ccc/yz-01-msc.ps.gz, April 2001.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Christara, C.C., Ng, K.S. Optimal Quadratic and Cubic Spline Collocation on Nonuniform Partitions. Computing 76, 227–257 (2006). https://doi.org/10.1007/s00607-005-0140-4
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00607-005-0140-4