Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

A collocation solver for mixed order systems of boundary value problems


Authors: U. Ascher, J. Christiansen and R. D. Russell
Journal: Math. Comp. 33 (1979), 659-679
MSC: Primary 65L10
DOI: https://doi.org/10.1090/S0025-5718-1979-0521281-7
MathSciNet review: 521281
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [1] U. ASCHER, "Discrete least squares approximations for ordinary differential equations," SIAM J. Numer. Anal., v. 15, 1978, pp. 478-496. MR 491701 (81e:65043)
  • [2] U. ASCHER, J. CHRISTIANSEN & R. D. RUSSELL, A Collocation Solver for Mixed Order Systems of Boundary Value Problems, Comp. Sci. Tech. Rep. 77-13, Univ. of British Columbia, 1977.
  • [3] 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.
  • [4] C. DE BOOR, "On calculating with B-splines," J. Approximation Theory, v. 6, 1972, pp. 50-62. MR 0338617 (49:3381)
  • [5] C. DE BOOR, Good Approximation by Splines with Variable Knots. II, Lecture Notes in Math., vol. 363, Springer-Verlag, Berlin and New York, 1973. MR 0431606 (55:4603)
  • [6] C. DE BOOR, "Package for calculating with B-splines," SIAM J. Numer. Anal., v. 14, 1977, pp. 441-472. MR 0428691 (55:1711)
  • [7] C. DE BOOR & B. SWARTZ, "Collocation at Gaussian points," SIAM J. Numer. Anal., v. 10, 1973, pp. 582-606. MR 0373328 (51:9528)
  • [8] C. DE BOOR & R. WEISS, 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.
  • [9] C. BROYDEN, "Recent developments in solving nonlinear, algebraic systems," in Numerical Methods for Nonlinear Algebraic Equations, P. Rabinowitz (ed.), Gordon & Breach, New York, 1977. MR 0343586 (49:8327)
  • [10] R. BULIRSCH, J. STOER & P. DEUFLHARD, Numerical Solution of Nonlinear Two-Point Boundary Value Problems. I, Numer. Math. Handbook Series Approximation, 1976.
  • [11] J. CERUTTI, Collocation for Systems of Ordinary Differential Equations, Comp. Sci. Tech. Rep. 230, Univ. Wisconsin-Madison, 1974.
  • [12] J. CHRISTIANSEN & R. D. RUSSELL, "Error analysis for spline collocation methods with application to knot selection," Math. Comp., v. 32, 1978, pp. 415-419. MR 0494963 (58:13736)
  • [13] P. CIARLET, M. SCHULTZ & R. VARGA, "Numerical methods of high-order accuracy for nonlinear boundary value problems. I. One dimensional problem," Numer. Math., v. 9, 1967, pp. 394-430. MR 0221761 (36:4813)
  • [14] J. DANIEL & A. MARTIN, "Implementing deferred corrections for Numerov's difference method for second-order two-point boundary-value problems," SIAM J. Numer. Anal., v. 14, 1977, pp. 1033-1050. MR 0464599 (57:4526)
  • [15] J. DENNIS & J. MORÉ, "Quasi-Newton methods, motivation and theory," SIAM Rev., v. 19, 1977, pp. 46-89. MR 0445812 (56:4146)
  • [16] P. DEUFLHARD, "A relaxation strategy for the modified Newton method," Opfimization and Optimal Control, Bulirsch, Oettli and Stoer (eds.), Lecture Notes in Math., vol. 477, Springer, Berlin and New York, 1975, pp. 59-73. MR 0501896 (58:19128)
  • [17] H. J. DIEKOFF, P. LORY, H. J. OBERLE, H. J. PESCH, P. RENTROP & R. SEYDEL, "Comparing routines for the numerical solution of initial value problems of ordinary differential equations in multiple shooting," Numer. Math., v. 27, 1977, pp. 449-469. MR 0445845 (56:4179)
  • [18] D. DODSON, Optimal Order Approximation by Polynomial Spline Functions, Ph. D. thesis, Purdue Univ., 1972.
  • [19] R. ENGLAND, N. NICHOLS & J. REID, Subroutine D003AD, Harwell subroutine library, Harwell, England, 1973.
  • [20] 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.
  • [21] J. E. FLAHERTY & R. E. O'MALLEY, JR., "The numerical solution of boundary value problems for stiff differential equations," Math. Comp., v. 31, 1977, pp. 66-93. MR 0657396 (58:31859)
  • [22] P. HEMKER, A Numerical Study of Stiff Two-Point Boundary Problems, Math. Centrum, Amsterdam, 1977. MR 0488784 (58:8294)
  • [23] E. HOUSTIS, "A collocation method for systems of nonlinear ordinary differential equations," J. Math. Anal. Appl., v. 62, 1978, pp. 24-37. MR 0488785 (58:8295)
  • [24] M. LENTINI & V. PEREYRA, "A variable order finite difference method for nonlinear multipoint boundary value problems," Math. Comp., v. 28, 1974, pp. 981-1004. MR 0386281 (52:7139)
  • [25] M. LENTINI & V. PEREYRA, "An adaptive finite difference solver for nonlinear two point boundary problems with mild boundary layers," SIAM J. Numer. Anal., v. 14, 1977, pp. 91-111. MR 0455420 (56:13658)
  • [26] J. B. McLEOD & S. V. PARTER, "On the ttow between two counter rotating infinite plane disks," Arch. Rational Mech. Anal., v. 54, 1974, pp. 301-327. MR 0349122 (50:1616)
  • [27] H. J. PESCH & P. RENTROP, Numerical Solution of the Flow between Two-Counter-Rotating Infinite Plane Disks by Multiple Shooting, Rep. #7621, Technische Universität München, 1976.
  • [28] P. RENTROP, "Numerical solution of the singular Ginzburg-Landau equations by multiple shooting," Computing, v. 16, 1976, pp. 61-67. MR 0408257 (53:12022)
  • [29] R. D. RUSSELL, "Collocation for systems of boundary value problems," Numer. Math., v. 23, 1974, pp. 119-133. MR 0416074 (54:4150)
  • [30] R. D. RUSSELL, "Efficiencies of B-splines methods for solving differential equations," Proc. Fifth Conference on Numerical Mathematics, Utilitas Math., Winnipeg, Manitoba, 1975, pp. 599-617. MR 0405871 (53:9663)
  • [31] R. D. RUSSELL, "A comparison of collocation and finite differences for two-point boundary value problems," SIAM J. Numer. Anal., v. 14, 1977, pp. 19-39. MR 0451745 (56:10027)
  • [32] R. D. RUSSELL & J. CHRISTIANSEN, "Adaptive mesh selection strategies for solving boundary value problems," SIAM J. Numer. Anal., v. 15, 1978, pp. 59-80. MR 0471336 (57:11071)
  • [33] R. D. RUSSELL & L. F. SHAMPINE, "A collocation method for boundary value problems," Numer. Math., v. 19, 1972, pp. 1-28. MR 0305607 (46:4737)
  • [34] R. D. RUSSELL & L. F. SHAMPINE, "Numerical methods for singular boundary value problems," SIAM J. Numer. Anal., v. 12, 1975, pp. 13-36. MR 0400723 (53:4553)
  • [35] M. L. SCOTT & H. A. WATTS, "Computational solutions of linear two-point boundary value problems via orthonormalization," SIAM J. Numer. Anal., v. 14, 1977, pp. 40-70. MR 0455425 (56:13663)
  • [36] M. L. SCOTT & H. A. WATTS, "A systematized collection of codes for solving twopoint boundary-value problems," in Numerical Methods for Differential Systems, Academic Press, New York, 1976, pp. 197-227.
  • [37] J. M. VARAH, "Alternate row and column elimination for solving certain linear systems," SIAM J. Numer. Anal., v. 13, 1976, pp. 71-75. MR 0411199 (53:14937)
  • [38] K. WITTENBRINK, "High order projection methods of moment and collocation type for nonlinear boundary value problems," Computing, v. 11, 1973, pp. 255-274. MR 0400724 (53:4554)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65L10

Retrieve articles in all journals with MSC: 65L10


Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1979-0521281-7
Article copyright: © Copyright 1979 American Mathematical Society

American Mathematical Society