*(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 |