The derivation of special purpose numerical methods for differential systems, i.e. adapted to accurately solve problems whose qualitative behaviour is supposed to be known a-priori, is usually carried out by means of non-polynomial fitting techniques. Exponential fitting (compare [2] and references therein) is certainly one of the most spread out techniques to obtain special purpose formulae in many fields of numerical analysis. In the context of numerical methods for ordinary differential equations, exponentially fitted Runge-Kutta formulae have been considered by many authors (see [3] for an updated state-of-art on the topic). The issue we want to revisit in this talk is the way of deriving the coefficients of such methods: we decide, indeed, to take into account the effect of the error inherited from the computation of the internal stages. Such contribution has always been neglected in previous version of exponentially fitted Runge-Kutta methods: on the contrary, we aim to make the propagation of the error along the stages visible. The revised technique is illustrated for hybrid methods and Runge-Kutta methods [1], for which we obtain new expressions of the coefficients, explicitly depending on the form of the system to be solved. The version obtained in this way is then compared for accuracy and stability with that achieved by means of the standard exponential fitting technique. Acknowledgments The authors express their gratitude to prof. Liviu Gr. Ixaru for the profitable discussions we had on the topic. References [1] R. D’Ambrosio, L. Gr. Ixaru, B. Paternoster, Construction of the EF-based Runge-Kutta methods revisited, Comp. Phys. Commun. 182, 322-329 (2011). [2] L. Gr. Ixaru and G.Vanden Berghe, Exponential Fitting, Kluwer Academic Publishers, Dordrecht (2004). [3] B. Paternoster, Present state-of-the-art in exponential fitting. A contribution dedicated to Liviu Ixaru on his 70-th anniversary, submitted.
Exponentially fitted numerical methods for differential systems with equation dependent coefficients
D'AMBROSIO, RAFFAELE;PATERNOSTER, Beatrice
2012-01-01
Abstract
The derivation of special purpose numerical methods for differential systems, i.e. adapted to accurately solve problems whose qualitative behaviour is supposed to be known a-priori, is usually carried out by means of non-polynomial fitting techniques. Exponential fitting (compare [2] and references therein) is certainly one of the most spread out techniques to obtain special purpose formulae in many fields of numerical analysis. In the context of numerical methods for ordinary differential equations, exponentially fitted Runge-Kutta formulae have been considered by many authors (see [3] for an updated state-of-art on the topic). The issue we want to revisit in this talk is the way of deriving the coefficients of such methods: we decide, indeed, to take into account the effect of the error inherited from the computation of the internal stages. Such contribution has always been neglected in previous version of exponentially fitted Runge-Kutta methods: on the contrary, we aim to make the propagation of the error along the stages visible. The revised technique is illustrated for hybrid methods and Runge-Kutta methods [1], for which we obtain new expressions of the coefficients, explicitly depending on the form of the system to be solved. The version obtained in this way is then compared for accuracy and stability with that achieved by means of the standard exponential fitting technique. Acknowledgments The authors express their gratitude to prof. Liviu Gr. Ixaru for the profitable discussions we had on the topic. References [1] R. D’Ambrosio, L. Gr. Ixaru, B. Paternoster, Construction of the EF-based Runge-Kutta methods revisited, Comp. Phys. Commun. 182, 322-329 (2011). [2] L. Gr. Ixaru and G.Vanden Berghe, Exponential Fitting, Kluwer Academic Publishers, Dordrecht (2004). [3] B. Paternoster, Present state-of-the-art in exponential fitting. A contribution dedicated to Liviu Ixaru on his 70-th anniversary, submitted.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
