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Phase-fitted and amplification-fitted two-step hybrid methods for $y^{\prime\prime }=f(x,y)$. (English) Zbl 1141.65061
The author considers phase-fitted and amplitude-fitted two-step hybrid methods for the numerical integration of initial value problems for nonlinear ordinary differential equations featuring highly oscillatory solution behavior. The numerical approximation by standard methods often introduces a distorted frequency referred to as phase-lag. The present paper provides a general theoretical framework for new phase-fitting techniques for two-step hybrid methods which are designed to overcome this problem as well as reduce the amplification error by means of introducing two new parameters. Convergence of the new methods is analyzed by a standard approach investigating consistency and stability. As an example, a sixth-order method is constructed and the theoretical findings are validated by numerical comparisons with other well established methods.

##### MSC:
 65L05 Initial value problems for ODE (numerical methods) 34A34 Nonlinear ODE and systems, general 65L20 Stability and convergence of numerical methods for ODE 65L06 Multistep, Runge-Kutta, and extrapolation methods
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##### References:
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