*(English)*Zbl 0847.65052

The authors discuss the concepts of a periodicity interval and $P$-stability in connection with linear multistep methods applied to initial value problems for second-order ordinary differential equations without the first derivative. Many of the known linear multistep methods are so called exponential-fitting methods because these methods are exact when the solution of the differential equation is a function belonging to a basis of functions which includes at least one exponential function with purely imaginary argument. The coefficients of such methods are functions of one or more fitted frequencies and the steplength.

The stability properties of several known exponential fitting methods are analysed and also a new $P$-stability criterion is proposed. An appendix investigates some two-step fourth-order exponential-fitting methods and stands out the particular cases of explicit methods with their periodicity intervals.

Exponential fitting methods may not perform well when they are applied to stiff oscillatory problems.

##### MSC:

65L20 | Stability and convergence of numerical methods for ODE |

65L06 | Multistep, Runge-Kutta, and extrapolation methods |

34A34 | Nonlinear ODE and systems, general |