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

Full-text PDF Free Access

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.

**[1]**J. H. Ahlberg, E. N. Nilson, and J. L. Walsh,*Convergence properties of generalized splines*, Proc. Nat. Acad. Sci. U.S.A.**54**(1965), 344–350. MR**0223800****[2]**George D. Andria, George D. Byrne, and David R. Hill,*Natural spline block implicit methods*, Nordisk Tidskr. Informationsbehandling (BIT)**13**(1973), 131–144. MR**0323110****[3]**Philip M. Anselone,*Collectively compact operator approximation theory and applications to integral equations*, Prentice-Hall, Inc., Englewood Cliffs, N. J., 1971. With an appendix by Joel Davis; Prentice-Hall Series in Automatic Computation. MR**0443383****[4]**G. Birkhoff, M. H. Schultz, and R. S. Varga,*Piecewise Hermite interpolation in one and two variables with applications to partial differential equations*, Numer. Math.**11**(1968), 232–256. MR**0226817**, https://doi.org/10.1007/BF02161845**[5]**E. David Callender,*Single step methods and low order splines for solutions of ordinary differential equations*, SIAM J. Numer. Anal.**8**(1971), 61–66. MR**0315897**, https://doi.org/10.1137/0708008**[6]**G. J. Cooper,*Interpolation and quadrature methods for ordinary differential equations*, Math. Comp.**22**(1968), 69–76. MR**0224289**, https://doi.org/10.1090/S0025-5718-1968-0224289-7**[7]**Carl de Boor and Blâir Swartz,*Collocation at Gaussian points*, SIAM J. Numer. Anal.**10**(1973), 582–606. MR**0373328**, https://doi.org/10.1137/0710052**[8]**N. DUNFORD & J. T. SCHWARTZ,*Linear Operators*, Interscience, New York, 1957.**[9]**Walter Gautschi,*Numerical integration of ordinary differential equations based on trigonometric polynomials*, Numer. Math.**3**(1961), 381–397. MR**0138200**, https://doi.org/10.1007/BF01386037**[10]**Bernie L. Hulme,*Discrete Galerkin and related one-step methods for ordinary differential equations*, Math. Comp.**26**(1972), 881–891. MR**0315899**, https://doi.org/10.1090/S0025-5718-1972-0315899-8**[11]**Bernie L. Hulme,*One-step piecewise polynomial Galerkin methods for initial value problems*, Math. Comp.**26**(1972), 415–426. MR**0321301**, https://doi.org/10.1090/S0025-5718-1972-0321301-2**[12]**Eugene Isaacson and Herbert Bishop Keller,*Analysis of numerical methods*, John Wiley & Sons, Inc., New York-London-Sydney, 1966. MR**0201039****[13]**L. Gr. Ixaru,*A new method for solving inhomogeneous second order differential equations*, Rev. Roumaine Math. Pures Appl.**19**(1974), 199–203. MR**0345414****[14]**D. JACKSON,*The Theory of Approximation*, Amer. Math. Soc. Colloq. Publ., vol. 12, Amer. Math. Soc., Providence, R. I., 1930.**[15]**Luis Kramarz,*Global approximations to solutions of initial value problems*, Math. Comp.**32**(1978), no. 141, 35–59. MR**0461917**, https://doi.org/10.1090/S0025-5718-1978-0461917-1**[16]**Tom Lyche and Larry L. Schumaker,*Local spline approximation methods*, J. Approximation Theory**15**(1975), no. 4, 294–325. MR**0397249****[17]**Gh. Micula,*The numerical solution of nonlinear differential equations by deficient spline functions*, Z. Angew. Math. Mech.**55**(1975), no. 4, T254–T255. Vorträge der Wissenschaftlichen Jahrestagung der Gesellschaft für Angewandte Mathematik und Mechanik (Bochum, 1974). MR**0433891****[18]**Steven A. Pruess,*Solving linear boundary value problems by approximating the coefficients*, Math. Comp.**27**(1973), 551–561. MR**0371100**, https://doi.org/10.1090/S0025-5718-1973-0371100-1**[19]**R. D. Russell and L. F. Shampine,*A collocation method for boundary value problems*, Numer. Math.**19**(1972), 1–28. MR**0305607**, https://doi.org/10.1007/BF01395926**[20]**I. J. Schoenberg,*On monosplines of last deviation and best quadrature formulae*, J. Soc. Indust. Appl. Math. Ser. B Numer. Anal.**2**(1965), 144–170. MR**0202309****[21]**M. H. Schultz and R. S. Varga,*𝐿-splines*, Numer. Math.**10**(1967), 345–369. MR**0225068**, https://doi.org/10.1007/BF02162033**[22]**A. Spitzbart,*A generalization of Hermite’s interpolation formula*, Amer. Math. Monthly**67**(1960), 42–46. MR**0137945**, https://doi.org/10.2307/2308924**[23]**D. D. Stancu and A. H. Stroud,*Quadrature formulas with simple Gaussian nodes and multiple fixed nodes*, Math. Comp.**17**(1963), 384–394. MR**0157485**, https://doi.org/10.1090/S0025-5718-1963-0157485-3**[24]**A. H. Stroud and D. D. Stancu,*Quadrature formulas with multiple Gaussian nodes*, J. Soc. Indust. Appl. Math. Ser. B Numer. Anal.**2**(1965), 129–143. MR**0179940****[25]**P. Turán,*On the theory of the mechanical quadrature*, Acta Sci. Math. Szeged**12**(1950), no. Leopoldo Fejér et Frederico Riesz LXX annos natis dedicatus, Pars A, 30–37. MR**0036797****[26]**H. A. Watts and L. F. Shampine,*𝐴-stable block implicit one-step methods*, Nordisk Tidskr. Informationsbehandling (BIT)**12**(1972), 252–266. MR**0307483****[27]**Jack Williams and Frank de Hoog,*A class of 𝐴-stable advanced multistep methods*, Math. Comp.**28**(1974), 163–177. MR**0356519**, https://doi.org/10.1090/S0025-5718-1974-0356519-8**[28]**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**0400724**

Retrieve articles in *Mathematics of Computation*
with MSC:
65L05

Retrieve articles in all journals with MSC: 65L05

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