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Global approximations to solutions of initial value problems


Author: Luis Kramarz
Journal: Math. Comp. 32 (1978), 35-59
MSC: Primary 65L05
DOI: https://doi.org/10.1090/S0025-5718-1978-0461917-1
MathSciNet review: 0461917
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Abstract | References | Similar Articles | Additional Information

Abstract: A wide class of implicit one-step methods for the construction of global approximations to solutions of initial value problems is studied. Approximations more general than piecewise polynomials can be constructed to exploit certain characteristics of the differential equation. Error bounds are given for the general class of methods but emphasis is placed on methods based on Hermite interpolation, for which higher rates of convergence are obtained for special choices of interpolation points. Computational examples are presented.


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  • [1] J. H. AHLBERG, E. N. NILSON & J. L. WALSH, "Convergence properties of generalized splines," Proc. Nat. Acad. Sci. U.S.A., v. 54, 1965, pp. 344-350. MR 0223800 (36:6847)
  • [2] G. D. ANDRIA, G. D. BYRNE & D. R. HILL, "Natural block implicit methods," BIT, v. 13, 1973, pp. 131-144. MR 0323110 (48:1468)
  • [3] P. M. ANSELONE, Collectively Compact Operator Approximation Theory and Applications to Integral Equations, Prentice-Hall, Englewood Cliffs, N. J., 1971. MR 0443383 (56:1753)
  • [4] G. BIRKHOFF, M. H. SCHULTZ & R. S. VARGA, "Piecewise Hermite interpolation in one and two variables with applications to partial differential equations," Numer. Math., v. 11, 1968, pp. 232-256. MR 0226817 (37:2404)
  • [5] E. D. CALLENDER, "Single step methods and low order splines for solutions of ordinary differential equations," SIAM J. Numer. Anal., v. 8, 1971, pp. 61-66. MR 0315897 (47:4446)
  • [6] G. J. COOPER, "Interpolation and quadrature methods for ordinary differential equations," Math. Comp., v. 22, 1968, pp. 69-76. MR 0224289 (36:7333)
  • [7] C. R. DE BOOR & B. SWARTZ, "Collocation at Gaussian points," SIAM J. Numer. Anal., v. 10, 1973, pp. 582-606. MR 0373328 (51:9528)
  • [8] N. DUNFORD & J. T. SCHWARTZ, Linear Operators, Interscience, New York, 1957.
  • [9] W. GAUTSCHI, "Numerical integration of ordinary differential equations based on trigonometric polynomials," Numer. Math., v. 3, 1961, pp. 381-397. MR 0138200 (25:1647)
  • [10] H. L. HULME, "Discrete Galerkin and related one-step methods for ordinary differential equations," Math. Comp., v. 26, 1972, pp. 881-891. MR 0315899 (47:4448)
  • [11] H. L. HULME, "One-step piecewise polynomial Galerkin methods for initial value problems," Math. Comp., v. 26, 1972, pp. 415-426. MR 0321301 (47:9834)
  • [12] E. ISAACSON & H. B. KELLER, Analysis of Numerical Methods, Wiley, New York, 1966. MR 0201039 (34:924)
  • [13] L. GR. IXARU, "A new method for solving inhomogeneous second order differential equations," Rev. Roumaine Math. Pures Appl., v. 19, 1974, pp. 199-203. MR 0345414 (49:10150)
  • [14] D. JACKSON, The Theory of Approximation, Amer. Math. Soc. Colloq. Publ., vol. 12, Amer. Math. Soc., Providence, R. I., 1930.
  • [15] L. KRAMARZ, Global Approximations to Solutions of Initial Value Problems, Ph.D. Thesis, Georgia Inst, of Tech., Atlanta, 1977. MR 0461917 (57:1899)
  • [16] T. LYCHE & L. L. SCHUMAKER, Local Spline Approximation Methods, Tech. Summary Report 1417, Math. Res. Center, U. S. Army, Univ. of Wisconsin, Madison, 1974. MR 0397249 (53:1108)
  • [17] GH. MICULA, "The numerical solution of nonlinear differential equations by deficient spline functions," Z. Angew. Math. Mech., v. 55, 1975, pp. T.254-T.256. MR 0433891 (55:6862)
  • [18] S. A. PRUESS, "Solving linear boundary value problems by approximating the coefficients," Math. Comp., v. 27, 1973, pp. 551-561. MR 0371100 (51:7321)
  • [19] 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)
  • [20] I. J. SCHOENBERG, "On monosplines of least deviation and best quadrature formulae," SIAM J. Numer. Anal., v. 2, 1965, pp. 144-170. MR 0202309 (34:2182)
  • [21] M. H. SCHULTZ & R. S. VARGA, "L-splines," Numer. Math., v. 10, 1967, pp. 345-369. MR 0225068 (37:665)
  • [22] A. SPITZBART, "A generalization of Hermite's interpolation formula," Amer. Math. Monthly, v. 67, 1960, pp. 42-46. MR 0137945 (25:1393)
  • [23] D. D. STANCU & A. H. STROUD, "Quadrature formulas with simple Gaussian nodes and multiple fixed nodes," Math. Comp., v. 17, 1963, pp. 384-394. MR 0157485 (28:718)
  • [24] A. H. STROUD & D. D. STANCU, "Quadrature formulas with multiple Gaussian nodes," SIAM J. Numer. Anal., v. 2, 1965, pp. 129-143. MR 0179940 (31:4177)
  • [25] P. TURÁN, "On the theory of the mechanical quadrature," Acta Sci. Math. (Szeged), v. 12, 1950, pp. 30-37. MR 0036797 (12:164b)
  • [26] H. A. WATTS & L. F. SHAMPINE, "A-stable block implicit one-step methods," BIT, v. 12, 1972, pp. 252-266. MR 0307483 (46:6603)
  • [27] J. WILLIAMS & F. DE HOOG, "A class of A-stable advanced multistep methods," Math. Comp., v. 28, 1974, pp. 163-177. MR 0356519 (50:8989)
  • [28] K. A. 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)

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Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1978-0461917-1
Keywords: Initial value problems, ordinary differential equations, one-step implicit methods, global approximations, collocation, projection methods, piecewise polynomials, spline
Article copyright: © Copyright 1978 American Mathematical Society

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