(k, l)-Algebraic Stability of Runge-Kutta Methods
- 1 July 1988
- journal article
- research article
- Published by Oxford University Press (OUP) in IMA Journal of Numerical Analysis
- Vol. 8 (3) , 385-400
- https://doi.org/10.1093/imanum/8.3.385
Abstract
For ordinary differential equations satisfying a one-sided Lipschitz condition with Lipschitz constant v, the solutions satisfy with l=hv, so that, in the case of Runge-Kutta methods, estimates of the form ‖yn‖2≤k(l)‖yn−1‖2 are desirable. Burrage (1986) has investigated the behaviour of the error-bounding function k for positive l for the family of s-stage Gauss methods of order 2s, and has shown that k(l)=exp 2l+O(l3) (l↓0) for s≥3. In this paper, we extend the analysis of k to any irreducible algebraically stable Runge-Kutta method, and obtain results about the maximum order of k as an approximation to exp 2l. As a particular example, we investigate the function k for all algebraically stable methods of order 2s−1.Keywords
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