Abstract
In a recent paper, Cash and Moore have given a fourth order formula for the approximate numerical integration of two-point boundary value problems in O.D.E.s. The formula presented was in effect a “one-off” formula in that it was obtained using a trial and error approach. The purpose of the present paper is to describe a unified approach to the derivation of high order formulae for the numerical integration of two-point boundary value problems. It is shown that the formula derived by Cash and Moore fits naturally into this framework and some new formulae of orders 4, 6 and 8 are derived using this approach. A numerical comparison with certain existing finite difference methods is made and this comparison indicates the efficiency of the high order methods for problems having a suitably smooth solution.
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References
U. Ascher, J. Christiansen and R. Russell,A collocation solver for mixed order systems of boundary value problems, Math. Comp., 33 (1979), 659–680.
I. Barrowdale and A. Young,Computational experience in solving linear operator equations using the Chebyshev norm, Numerical Approximation to Functions and Data, J. G. Hayes ed., Athlone Press (1970), 115–142.
G. Birkhoff and R. S. Varga,Discretization errors for well-set Cauchy problems, J. Math. and Phys., 44 (1965), 1–23.
J. R. Cash and D. R. Moore,A high order method for the numerical solution of two-point boundary value problems, BIT, 20 (1980), 44–52.
M. M. Chawla,A sixth order tridiagonal finite difference method for nonlinear two-point boundary value problems, BIT, 17 (1977), 128–133.
M. M. Chawla,An eighth order tridiagonal finite difference method for nonlinear two-point boundary value problems, BIT, 17 (1977), 281–285.
M. M. Chawla,A fourth order tridiagonal finite difference method for general nonlinear two-point boundary value problems with mixed boundary conditions, J. IMA, 21 (1978), 83–95.
S. D. Conte,The numerical solution of linear boundary value problems, SIAM Rev., 8 (1966), 309–321.
G. Dahlquist,A special stability problem for linear multistep methods, BIT 3 (1963), 27–43.
P. Hall and G. Seminara,Centrifugal instability of a Stokes layer, Proc. R. Soc. Lond. A, 350 (1976), 299–317.
H. B. Keller, inNumerical Solution of Boundary Value Problems for O.D.E., ed. A. K. Aziz, Academic Press (1975), 27–88.
H. B. Keller,Numerical Methods for Two-Point Boundary Value Problems, Ginn-Blaisdell, Waltham, Mass., 1968.
H. B. Keller,Accurate difference methods for nonlinear two-point boundary value problems, SIAM J. Numer. Anal., 11 (1974), 305–320.
M. R. Scott,On the conversion of boundary value problems into stable initial value problems via several invariant embedding algorithms, inNumerical Solution of Boundary Value Problems for O.D.E., ed. A. K. Aziz, Academic Press (1975), 89–146.
G. Seminara,Centrifugal instability of a Stokes layer, ZAMP 30 (1979), 615–625.
A. Singhal,Implicit Runge-Kutta formulae for the numerical integration of O.D.E.s, Ph.D. Thesis, University of London, (1980).
H. J. Stetter,Analysis of Discretization Methods for Ordinary Differential Equations, Springer-Verlag, 1973.
R. Weiss,The application of implicit Runge-Kutta and collocation methods to boundary value problems, Math. Comp., 28 (1974), 449–464.
A. B. White,Numerical Solution of Two-Point Boundary Value Problems, Ph.D. Thesis, California Inst. of Tech., Pasadena, 1974.
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Cash, J.R., Singhal, A. High order methods for the numerical solution of two-point boundary value problems. BIT 22, 183–199 (1982). https://doi.org/10.1007/BF01944476
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DOI: https://doi.org/10.1007/BF01944476