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Numerical integration of non-linear two-point boundary-value problems using iterated deferred corrections—I

1986; Elsevier BV; Volume: 12; Issue: 10 Linguagem: Inglês

10.1016/0898-1221(86)90009-x

1873-7668

J. R. Cash,

Electromagnetic Simulation and Numerical Methods

In several recent papers, the idea has been proposed of using implicit Runge-Kutta formulae for the approximate integration of non-linear two-point boundary-value problems. These formulae often take a very special form so that computational efficiencies are obtained in certain well-defined circumstances. In this paper we examine the relative merits of these different classes of formulae and discuss their efficient implementation in a deferred correction framework. A theoretical comparison of these formulae is important, firstly because they call for different computational tasks to be performed, and secondly because they are not always applicable to exactly the same classes of problems. The most efficient of these 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 Runge-Kutta methods can be generally competitive with the other methods considered.