Abstract
In Sect. V.6 we have seen a convergence result for one-leg methods (Theorem 6.10) applied to nonlinear problems satisfying a one-sided Lipschitz condition. An extension to linear multistep methods has been given in Theorem 6.11. A different and direct proof of this result will be the first goal of this section. Unfortunately, such a result is valid only for A -stable methods (whose order cannot exceed two). The subsequent parts of this section are then devoted to convergence results for nonlinear problems, where the assumptions on the method are relaxed (e.g., A(α) -stability), but the class of problems considered is restricted. We shall present two different theories: the multiplier technique of Nevanlinna & Odeh (1981) and Lubich’s perturbation approach via the discrete variation of constants formula (Lubich 1991).
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© 1996 Springer-Verlag Berlin Heidelberg
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Hairer, E., Wanner, G. (1996). Convergence for Nonlinear Problems. In: Solving Ordinary Differential Equations II. Springer Series in Computational Mathematics, vol 14. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-05221-7_23
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DOI: https://doi.org/10.1007/978-3-642-05221-7_23
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