Iterated deferred correction for linear two-point boundary value problems
This paper deals with the numerical solution of two-point boundary value problems by the so-called iterated deferred correction. [For a precise statement of this procedure and a study of its convergence see: \textit{R. D. Skeel}, SIAM J. Numer. Anal. 19, 171-196 (1982; Zbl 0489.65051)]. Here the authors propose deferred corrections based on successive application of symmetric implicit Runge-Kutta methods which increase the order by two units in each iteration. In the first part of the paper a study of the linear stability of Gaussian and Lobatto IIIC formulas is presented. More precisely they require that the numerical solutions provided by the method for \(y'(x) = \lambda y (x)\), \(x \in [x_0, x_N]\) be bounded for \(y_0\) given and \(\text{Re }\lambda \leq 0\) and \(y_\nu\) given and \(\text{Re } \lambda \geq 0\). Then, they show that Lobatto IIC methods can be written as mono-implicit Runge-Kutta methods and by using the previous experience of one of the authors with this type of methods in the numerical solution of two-point boundary value problems, they have constructed a code that uses Lobatto formulas up to order eight. Finally, some numerical experiments comparing the behaviour of two standard codes: COLSYS and TWPBVP (available from NETLIB) and two new codes based on successive application of Gauss and Lobatto formulas, are presented. The main conclusions derived from these numerical experiments are that the new codes have a good stability. Further, the code based on Lobatto formulas has the advantage of a lower computational cost than the Gaussian code.
scientific article; zbMATH DE number 890678