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Runge-Kutta methods adapted to the numerical integration of oscillatory problems. (English) Zbl 1057.65043
The material is devoted to the construction of Runge-Kutta methods specially adapted to the numerical integration of initial value problems with oscillatory solutions. The coefficients of these methods are frequency-dependent such that certain particular oscillatory solutions are computed exactly. The first section is given by an introduction concerning this type of Runge-Kutta methods. Section two presents a detailed exposition of the ideas that lead to the formulation of the adapted Runge-Kutta methods. Explicit Runge-Kutta methods for oscillatory ordinary differential equations (ODEs) and order conditions are exposed. This part contains a modification of the classical Runge-Kutta algorithms such that the oscillatory linear problem is solved without truncation errors. Necessary and sufficient order conditions for the adapted Runge-Kutta methods are derived by using the B-series theory and the rooted trees. These conditions reduce to the classical order conditions for the classical Runge-Kutta methods when the parameter $v= \omega h\to 0$, $\omega$ being the main frequency of the oscillatory solutions and $h$ the step-size. In section three one derives explicit adapted Runge-Kutta methods with order 3 and 4 as well as embedded pairs of adapted Runge-Kutta methods of orders 3 and 4. The fourth section presents some numerical experiments which support the efficiency of the adapted Runge-Kutta methods derived in the paper, when they are compared with other classical Runge-Kutta methods. Two types of comparisons with both fixed and variable step-size are exposed, each one for a linear model problem test, respectively a nonlinear model problem test. Thus the numerical results show the excellent behaviour of the new methods when they are compared with standard Runge-Kutta methods. Section five contains the conclusions and also some research perspective, like the extension of these methods to the case of semilinear ODE systems $(y'= Ay+ g(t, y))$, where the stiffness matrix $A$ contains implicitly the frequencies of the problem.

##### MSC:
 65L06 Multistep, Runge-Kutta, and extrapolation methods 65L05 Initial value problems for ODE (numerical methods) 34A34 Nonlinear ODE and systems, general 34C10 Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory 65L50 Mesh generation and refinement (ODE)
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