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 0618.65071This 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 |

##### Keywords:

computational costs; numerical examples; implicit Runge-Kutta formulae; efficient implementation; deferred correction; Newton iteration; fully- implicit; mono-implicit; diagonally-implicit; finite differences; spline collocation; smooth test functions##### Software:

COLSYS
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\textit{J. R. Cash}, Comput. Math. Appl., Part A 12, 1029--1048 (1986; Zbl 0618.65071)

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##### References:

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