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Analysis of trigonometric implicit Runge-Kutta methods. (English) Zbl 1106.65063
The idea of this paper is to present Runge-Kutta (RK) methods that exploit the known oscillatory behaviour of the solution of a differential equation to improve the quality of the numerical approximation. The method is based on the use of trigonometric fitting functions to develop a class of one-stage RK methods. The coefficients of the trigonometric implicit Runge-Kutta (TIRK) methods are functions of frequency and stepsize. The paper reviews alternative approaches to the problem, describes the development of the TIRK methods, analyses their stability and presents numerical examples.

##### MSC:
 65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations 65L05 Numerical methods for initial value problems 34A34 Nonlinear ordinary differential equations and systems, general theory 65L20 Stability and convergence of numerical methods for ordinary differential equations 65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
##### Software:
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