One-step piecewise polynomial multiple collocation methods for initial value problems

Hennart, J. P.

doi:10.1090/S0025-5718-1977-0431686-9

One-step piecewise polynomial multiple collocation methods for initial value problems
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by J. P. Hennart PDF

Math. Comp. 31 (1977), 24-36 Request permission

Abstract:

New methods are proposed for the numerical solution of systems of first-order differential equations. On each subinterval of a given mesh of size h, a polynomial of degree l is constructed, its parameters being determined by a multiple collocation technique. The resulting piecewise polynomial approximation is of order $O({h^{l + 1}})$ at the mesh points and between them. In addition, the jth derivatives of the approximation on each subinterval provide approximations of order $O({h^{l + 1 - j}})$, $j = 1, \ldots ,l$. Some of the methods proposed are shown to be A-stable or even strongly A-stable.

References

George D. Andria, George D. Byrne, and David R. Hill, Natural spline block implicit methods, Nordisk Tidskr. Informationsbehandling (BIT) 13 (1973), 131–144. MR 323110, DOI 10.1007/bf01933485
George D. Andria, George D. Byrne, and David R. Hill, Integration formulas and schemes based on $g$-splines, Math. Comp. 27 (1973). MR 339460, DOI 10.1090/S0025-5718-1973-0339460-5

J. Royal Aeronautical Soc.

Garrett Birkhoff and Richard S. Varga, Discretization errors for well-set Cauchy problems. I, J. Math. and Phys. 44 (1965), 1–23. MR 179952

The Mathematics of Finite Elements and Applications

George D. Byrne and Donald N. H. Chi, Linear multistep formulas based on $g$-splines, SIAM J. Numer. Anal. 9 (1972), 316–324. MR 311111, DOI 10.1137/0709031
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 315897, DOI 10.1137/0708008
Philip J. Davis, Interpolation and approximation, Blaisdell Publishing Co. [Ginn and Co.], New York-Toronto-London, 1963. MR 0157156
Germund G. Dahlquist, A special stability problem for linear multistep methods, Nordisk Tidskr. Informationsbehandling (BIT) 3 (1963), 27–43. MR 170477, DOI 10.1007/bf01963532
Jim Douglas Jr. and Todd Dupont, Galerkin methods for parabolic equations, SIAM J. Numer. Anal. 7 (1970), 575–626. MR 277126, DOI 10.1137/0707048
Byron L. Ehle, $A$-stable methods and Padé approximations to the exponential, SIAM J. Math. Anal. 4 (1973), 671–680. MR 331787, DOI 10.1137/0504057
G. Fix and N. Nassif, On finite element approximations to time-dependent problems, Numer. Math. 19 (1972), 127–135. MR 311122, DOI 10.1007/BF01402523

Trans. Amer. Nuclear Soc.

Peter Henrici, Discrete variable methods in ordinary differential equations, John Wiley & Sons, Inc., New York-London, 1962. MR 0135729
Bernie L. Hulme, Piecewise polynomial Taylor methods for initial value problems, Numer. Math. 17 (1971), 367–381. MR 298953, DOI 10.1007/BF01436086
Bernie L. Hulme, One-step piecewise polynomial Galerkin methods for initial value problems, Math. Comp. 26 (1972), 415–426. MR 321301, DOI 10.1090/S0025-5718-1972-0321301-2
Bernie L. Hulme, Discrete Galerkin and related one-step methods for ordinary differential equations, Math. Comp. 26 (1972), 881–891. MR 315899, DOI 10.1090/S0025-5718-1972-0315899-8
P. M. Hummel and C. L. Seebeck Jr., A generalization of Taylor’s expansion, Amer. Math. Monthly 56 (1949), 243–247. MR 28907, DOI 10.2307/2304764

Nuclear Sci. and Engrg.

Frank R. Loscalzo and Thomas D. Talbot, Spline function approximations for solutions of ordinary differential equations, SIAM J. Numer. Anal. 4 (1967), 433–445. MR 221756, DOI 10.1137/0704038
Frank R. Loscalzo, An introduction to the application of spline functions to initial value problems, Theory and Applications of Spline Functions (Proceedings of Seminar, Math. Research Center, Univ. of Wisconsin, Madison, Wis., 1968) Academic Press, New York, 1969, pp. 37–64. MR 0240989

Internat. J. Numer. Methods Engrg.

Harvey S. Price and Richard S. Varga, Error bounds for semidiscrete Galerkin approximations of parabolic problems with applications to petroleum reservoir mechanics, Numerical Solution of Field Problems in Continuum Physics (Proc. Sympos. Appl. Math., Durham, N.C., 1968) Amer. Math. Soc., Providence, R.I., 1970, pp. 74–94. MR 0266452
Blair Swartz and Burton Wendroff, Generalized finite-difference schemes, Math. Comp. 23 (1969), 37–49. MR 239768, DOI 10.1090/S0025-5718-1969-0239768-7
Richard S. Varga, On higher order stable implicit methods for solving parabolic partial differential equations, J. Math. and Phys. 40 (1961), 220–231. MR 140191

Comput. Methods Appl. Mech. Engrg.

Olof B. Widlund, A note on unconditionally stable linear multistep methods, Nordisk Tidskr. Informationsbehandling (BIT) 7 (1967), 65–70. MR 215533, DOI 10.1007/bf01934126
K. Wright, Some relationships between implicit Runge-Kutta, collocation Lanczos $\tau$ methods, and their stability properties, Nordisk Tidskr. Informationsbehandling (BIT) 10 (1970), 217–227. MR 266439, DOI 10.1007/bf01936868

Internat. J. Numer. Methods Engrg.

O. C. Zienkiewicz, The finite element method in engineering science, McGraw-Hill, London-New York-Düsseldorf, 1971. The second, expanded and revised, edition of The finite element method in structural and continuum mechanics. MR 0315970