A class of two-step \(P\)-stable methods for the accurate integration of second order periodic initial value problems (Q1077137)

The author considers a two parameter family of two-step methods for the numerical integration of second order differential equations \(y''=f(t,y)\). The parameters are determined in such a way that the ''phase-lag'' of the method is minimal. This is equivalent to the fact that the local error is minimal for the linear problem \(y''+y=0\). It should be remarked that the proposed method is of order six for linear differential equations with constant coefficients, but it is only of order two for the general nonlinear problem. Numerical results are presented only for the linear test equation.