## A collocation solver for mixed order systems of boundary value problems

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- by U. Ascher, J. Christiansen and R. D. Russell PDF
- Math. Comp.
**33**(1979), 659-679 Request permission

## Abstract:

Implementation of a spline collocation method for solving boundary value problems for mixed order systems of ordinary differential equations is discussed. The aspects of this method considered include error estimation, adaptive mesh selection,*B*-spline basis function evaluation, linear system solution and nonlinear problem solution. The resulting general purpose code, COLSYS, is tested on a number of examples to demonstrate its stability, efficiency and flexibility.

## References

- Uri Ascher,
*Discrete least squares approximations for ordinary differential equations*, SIAM J. Numer. Anal.**15**(1978), no. 3, 478–496. MR**491701**, DOI 10.1137/0715031
U. ASCHER, J. CHRISTIANSEN & R. D. RUSSELL, - Carl de Boor,
*On calculating with $B$-splines*, J. Approximation Theory**6**(1972), 50–62. MR**338617**, DOI 10.1016/0021-9045(72)90080-9 - Carl de Boor,
*Good approximation by splines with variable knots. II*, Conference on the Numerical Solution of Differential Equations (Univ. Dundee, Dundee, 1973) Lecture Notes in Math., Vol. 363, Springer, Berlin, 1974, pp. 12–20. MR**0431606** - Carl de Boor,
*Package for calculating with $B$-splines*, SIAM J. Numer. Anal.**14**(1977), no. 3, 441–472. MR**428691**, DOI 10.1137/0714026 - Carl de Boor and Blâir Swartz,
*Collocation at Gaussian points*, SIAM J. Numer. Anal.**10**(1973), 582–606. MR**373328**, DOI 10.1137/0710052
C. DE BOOR & R. WEISS, - C. G. Broyden,
*Recent developments in solving nonlinear algebraic systems*, Numerical methods for nonlinear algebraic equations (Proc. Conf., Univ. Essex, Colchester, 1969) Gordon and Breach, London, 1970, pp. 61–73. MR**0343586**, DOI 10.1080/00031305.1970.10477185
R. BULIRSCH, J. STOER & P. DEUFLHARD, - J. Christiansen and R. D. Russell,
*Error analysis for spline collocation methods with application to knot selection*, Math. Comp.**32**(1978), no. 142, 415–419. MR**494963**, DOI 10.1090/S0025-5718-1978-0494963-2 - P. G. Ciarlet, M. H. Schultz, and R. S. Varga,
*Numerical methods of high-order accuracy for nonlinear boundary value problems. I. One dimensional problem*, Numer. Math.**9**(1966/67), 394–430. MR**221761**, DOI 10.1007/BF02162155 - James W. Daniel and Andrew J. Martin,
*Numerov’s method with deferred corrections for two-point boundary-value problems*, SIAM J. Numer. Anal.**14**(1977), no. 6, 1033–1050. MR**464599**, DOI 10.1137/0714071 - J. E. Dennis Jr. and Jorge J. Moré,
*Quasi-Newton methods, motivation and theory*, SIAM Rev.**19**(1977), no. 1, 46–89. MR**445812**, DOI 10.1137/1019005 - Peter Deuflhard,
*A relaxation strategy for the modified Newton method*, Optimization and optimal control (Proc. Conf., Oberwolfach, 1974) Lecture Notes in Math., Vol. 477, Springer, Berlin, 1975, pp. 59–73. MR**0501896** - H.-J. Diekhoff, P. Lory, H.-J. Oberle, H.-J. Pesch, P. Rentrop, and R. Seydel,
*Comparing routines for the numerical solution of initial value problems of ordinary differential equations in multiple shooting*, Numer. Math.**27**(1976/77), no. 4, 449–469. MR**445845**, DOI 10.1007/BF01399607
D. DODSON, - Joseph E. Flaherty and R. E. O’Malley Jr.,
*The numerical solution of boundary value problems for stiff differential equations*, Math. Comput.**31**(1977), no. 137, 66–93. MR**0657396**, DOI 10.1090/S0025-5718-1977-0657396-0 - P. W. Hemker,
*A numerical study of stiff two-point boundary problems*, Mathematical Centre Tracts, No. 80, Mathematisch Centrum, Amsterdam, 1977. MR**0488784** - Elias Houstis,
*A collocation method for systems of nonlinear ordinary differential equations*, J. Math. Anal. Appl.**62**(1978), no. 1, 24–37. MR**488785**, DOI 10.1016/0022-247X(78)90215-9 - M. Lentini and V. Pereyra,
*A variable order finite difference method for nonlinear multipoint boundary value problems*, Math. Comp.**28**(1974), 981–1003. MR**386281**, DOI 10.1090/S0025-5718-1974-0386281-4 - M. Lentini and V. Pereyra,
*An adaptive finite difference solver for nonlinear two-point boundary problems with mild boundary layers*, SIAM J. Numer. Anal.**14**(1977), no. 1, 94–111. MR**455420**, DOI 10.1137/0714006 - J. B. McLeod and Seymour V. Parter,
*On the flow between two counter-rotating infinte plane disks*, Arch. Rational Mech. Anal.**54**(1974), 301–327. MR**349122**, DOI 10.1007/BF00249193
H. J. PESCH & P. RENTROP, - P. Rentrop,
*Numerical solution of the singular Ginzburg-Landau equations by multiple shooting*, Computing**16**(1976), no. 1-2, 61–67 (English, with German summary). MR**408257**, DOI 10.1007/BF02241980 - Robert D. Russell,
*Collocation for systems of boundary value problems*, Numer. Math.**23**(1974), 119–133. MR**416074**, DOI 10.1007/BF01459946 - R. Russell,
*Efficiencies of $B$-spline methods for solving differential equations*, Proceedings of the Fifth Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1975) Congressus Numerantium, No. XVI, Utilitas Math. Publ., Winnipeg, Man., 1976, pp. 599–617. MR**0405871** - Robert D. Russell,
*A comparison of collocation and finite differences for two-point boundary value problems*, SIAM J. Numer. Anal.**14**(1977), no. 1, 19–39. MR**451745**, DOI 10.1137/0714003 - R. D. Russell and J. Christiansen,
*Adaptive mesh selection strategies for solving boundary value problems*, SIAM J. Numer. Anal.**15**(1978), no. 1, 59–80. MR**471336**, DOI 10.1137/0715004 - R. D. Russell and L. F. Shampine,
*A collocation method for boundary value problems*, Numer. Math.**19**(1972), 1–28. MR**305607**, DOI 10.1007/BF01395926 - R. D. Russell and L. F. Shampine,
*Numerical methods for singular boundary value problems*, SIAM J. Numer. Anal.**12**(1975), 13–36. MR**400723**, DOI 10.1137/0712002 - Melvin R. Scott and Herman A. Watts,
*Computational solution of linear two-point boundary value problems via orthonormalization*, SIAM J. Numer. Anal.**14**(1977), no. 1, 40–70. MR**455425**, DOI 10.1137/0714004
M. L. SCOTT & H. A. WATTS, "A systematized collection of codes for solving twopoint boundary-value problems," in - J. M. Varah,
*Alternate row and column elimination for solving certain linear systems*, SIAM J. Numer. Anal.**13**(1976), no. 1, 71–75. MR**411199**, DOI 10.1137/0713008 - K. A. Wittenbrink,
*High order projection methods of moment- and collocation-type for nonlinear boundary value problems*, Computing (Arch. Elektron. Rechnen)**11**(1973), no. 3, 255–274 (English, with German summary). MR**400724**, DOI 10.1007/bf02252915

*A Collocation Solver for Mixed Order Systems of Boundary Value Problems*, Comp. Sci. Tech. Rep. 77-13, Univ. of British Columbia, 1977. U. ASCHER & R. D. RUSSELL,

*Evaluation of B-Splines for Solving Systems of Boundary Value Problems*, Comp. Sci. Tech. Rep. 77-14, Univ. of British Columbia, 1977.

*Solveblok*:

*A Package for Solving Almost Block Diagonal Linear Systems, with Applications to Spline Approximation and the Numerical Solution of Ordinary Differential Equations*, MRC TSR #1625, Madison, Wisconsin, 1976.

*Numerical Solution of Nonlinear Two-Point Boundary Value Problems*. I, Numer. Math. Handbook Series Approximation, 1976. J. CERUTTI,

*Collocation for Systems of Ordinary Differential Equations*, Comp. Sci. Tech. Rep. 230, Univ. Wisconsin-Madison, 1974.

*Optimal Order Approximation by Polynomial Spline Functions*, Ph. D. thesis, Purdue Univ., 1972. R. ENGLAND, N. NICHOLS & J. REID,

*Subroutine*D003AD, Harwell subroutine library, Harwell, England, 1973. S. C. EISENSTADT, R. S. SCHREIBER & M. H. SCHULTZ,

*Finite Element Methods for Spherically Symmetric Elliptic Equations*, Res. Rept. #109, Comp. Sci., Yale Univ., 1977.

*Numerical Solution of the Flow between Two-Counter-Rotating Infinite Plane Disks by Multiple Shooting*, Rep. #7621, Technische Universität München, 1976.

*Numerical Methods for Differential Systems*, Academic Press, New York, 1976, pp. 197-227.

## Additional Information

- © Copyright 1979 American Mathematical Society
- Journal: Math. Comp.
**33**(1979), 659-679 - MSC: Primary 65L10
- DOI: https://doi.org/10.1090/S0025-5718-1979-0521281-7
- MathSciNet review: 521281