×

zbMATH — the first resource for mathematics

Numerical integration of nonlinear two-point boundary-value problems using iterated deferred corrections. I: A survey and comparison of some one-step formulae. (English) Zbl 0618.65071
This is the first paper in the series of the titled theme announced by the author. Here, for the approximate integration of nonlinear two-point boundary value problems, the relative merits of using implicit Runge- Kutta formulae are examined with the discussion of their efficient implementation in a deferred correction framework. Nonlinear problems are assumed to be treated by the Newton iteration. The most efficient of these formulae, including fully-implicit, mono-implicit and diagonally- implicit Runge-Kutta formulae, are then compared with methods based on finite differences and with methods based on spline collocation for some simple, smooth test functions. On the basis of operation counts and numerical experimentation, it is concluded that RK methods can be generally competitive with the other methods considered.
Reviewer: T.Mitsui

MSC:
65L10 Numerical solution of boundary value problems involving ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
Software:
COLSYS
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Cash, J.R., A variable order deferred correction algorithm for the numerical solution of nonlinear two-point boundary value problems, Comput. math. applic., 9, 257-265, (1983) · Zbl 0521.65069
[2] Cash, J.R., Diagonally implicit Runge-Kutta formulae for the numerical integration of nonlinear two-point boundary value problems, Comput. math. applic., 10, 123-137, (1984) · Zbl 0565.34012
[3] Cash, J.R., Adaptive Runge-Kutta methods for nonlinear two-point boundary value problems with mild boundary layers, Comput. math. applic., 11, 605-620, (1985) · Zbl 0579.65079
[4] Cash, J.R.; Singhal, A., High order methods for the numerical solution of two-point boundary value problems, Bit, 22, 184-199, (1982) · Zbl 0494.65049
[5] Cash, J.R.; Singhal, A., Mono-implicit Runge-Kutta formulae for the numerical integration of stiff differential equations, IMA jl numer. analysis, 2, 211-227, (1982) · Zbl 0488.65031
[6] Gupta, S., An adaptive boundary value Runge-Kutta solver for first order boundary value problems, SIAM jl numer. analysis, 22, 114-126, (1985) · Zbl 0569.65060
[7] Weiss, R., The application of implicit Runge-Kutta and colloction methods to boundary value problems, Maths comput., 28, 449-464, (1974) · Zbl 0284.65067
[8] Keller, H.B., Accurate difference methods for nonlinear two-point boundary value problems, SIAM jl numer. analysis, 11, 305-320, (1974) · Zbl 0282.65065
[9] Ascher, U.; Christiansen, J.; Russell, R.D., A collocation solver for mixed order systems of boundary value problems, Maths comput., 33, 659-679, (1979) · Zbl 0407.65035
[10] Ascher, U.; Christiansen, J.; Russell, R.D., Collocation software for boundary value odes, ACM trans. math. software, 7, 209-222, (1981) · Zbl 0455.65067
[11] Ascher, U.; Christiansen, J.; Russell, R.D., COLSYS: a collocation code for boundary value problems, (), 164-185 · Zbl 0459.65061
[12] Wright, K., Some relationships between implicit runga-Kutta, collocation and Lanczos τ methods and their stability properties, Bit, 20, 217-227, (1970) · Zbl 0208.41602
[13] Ascher, U.; Pruess, S.; Russell, R.D., On spline basis selection for solving differential equations, SIAM jl. numer. analysis, 20, 121-142, (1983) · Zbl 0525.65060
[14] Ascher, U.; Weiss, R.; Ascher, U.; Weiss, R.; Ascher, U.; Weiss, R., Collocation for singular perturbation problems. part III, SIAM jl numer. analysis, Maths comput., SIAM jl scient. statist. comput., 5, 811-829, (1984) · Zbl 0558.65060
[15] Cash, J.R., A class of implicit Runge-Kutta methods for the numerical integration of stiff ordinary differential equations, J. ass. comput. Mach., 4, 504-511, (1975) · Zbl 0366.65029
[16] Van Bokhoven, W.M.G., Efficient higher order implicit one step methods for integration of stiff differential equations, Bit, 20, 34-43, (1980) · Zbl 0448.65047
[17] Butcher, J.C., Coefficients for the study of Runge-Kutta integration processes, J. aust. math. soc., 3, 185-201, (1963) · Zbl 0223.65031
[18] Butcher, J.C., On Runge-Kutta processes of high order, J. aust. math. soc., 4, 179-194, (1964) · Zbl 0244.65046
[19] Cash, J.R., On the design of high order exponentially fitted formulae for the numerical integration of stiff systems, Numer. math., 36, 253-266, (1981) · Zbl 0488.65028
[20] Cash, J.R., On improving the absolute stability of local extrapolation, Numer. math., 40, 329-337, (1982) · Zbl 0529.65045
[21] Muir, P., Runge-Kutta methods for two-point boundary value problems, () · Zbl 0594.65064
[22] Cash, J.R.; Moore, D.R., A high order method for the numerical solution of two-point boundary value problems, Bit, 20, 44-52, (1980) · Zbl 0448.65048
[23] Lentini, M.; Pereya, V., An adaptive finite difference solver for nonlinear two-point boundary value problems with mild boundary layers, SIAM jl. numer. analysis, 14, 91-111, (1977) · Zbl 0358.65069
[24] White, A.B., Numerical solution of two-point boundary value problems, ()
[25] Skeel, R.D., A theoretical framework for proving accuracy results for deferred corrections, SIAM jl numer. analysis, 19, 171-196, (1982) · Zbl 0489.65051
[26] Burrage, K.; Butcher, J.C., Non-linear stability of a general class of differential equation methods, Bit, 20, 185-203, (1980) · Zbl 0431.65051
[27] Varah, J.M., A comparison of some numerical results for two-point boundary value problems, Maths comput., 28, 743-755, (1974) · Zbl 0292.65045
[28] Russell, R.D., A comparison of collocation and finite difference methods for two-point boundary value problems, SIAM jl numer. analysis, 14, 19-39, (1977) · Zbl 0359.65075
[29] Keller, H.B., Numerical solution of boundary value problems for ordinary differential equations: survey and some recent results on difference methods, (), 27-88 · Zbl 0311.65050
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.