A collocation solver for mixed order systems of boundary value problems
HTML articles powered by AMS MathViewer
- 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, 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.
- 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, 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.
- 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, 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.
- 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, 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.
- 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, Numerical Solution of the Flow between Two-Counter-Rotating Infinite Plane Disks by Multiple Shooting, Rep. #7621, Technische Universität München, 1976.
- 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 Numerical Methods for Differential Systems, Academic Press, New York, 1976, pp. 197-227.
- 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
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