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. ?? 2006 Elsevier B.V. All rights
reserved.
