Analysis of trigonometric implicit Runge-Kutta methods

Authors:
Hoang Si Nguyen;Roger B. Sidje;Nguyen Huu Cong
Affiliations:
Faculty of Mathematics-Mechanics-Informatics, College of Sciences, Vietnam National University, Hanoi, Vietnam;Advanced Computational Modelling Centre, Department of Mathematics, University of Queensland, Brisbane, Australia;School of Graduate Studies, Vietnam National University, Hanoi, Vietnam
Venue:
Journal of Computational and Applied Mathematics
Year:
2007

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Abstract

Using generalized collocation techniques based on fitting functions that are trigonometric (rather than algebraic as in classical integrators), we develop a new class of multistage, one-step, variable stepsize, and variable coefficients implicit Runge-Kutta methods to solve oscillatory ODE problems. The coefficients of the methods are functions of the frequency and the stepsize. We refer to this class as trigonometric implicit Runge-Kutta (TIRK) methods. They integrate an equation exactly if its solution is a trigonometric polynomial with a known frequency. We characterize the order and A-stability of the methods and establish results similar to that of classical algebraic collocation RK methods.