Mixed collocation methods for y″=f(x,y)

The collocation methods introduced here are based on linear combinations of trigonometric functions and powers. The motivation is to provide better approximations for oscillatory solutions of initial-value problems for differential equations of the special form y″=f(x,y). The resulting methods, for two or more collocation points, are implicit Runge–Kutta–Nyström methods with coefficients which depend on both the fitted angular frequency and the steplength. Algebraic and trigonometric order conditions are considered and the stability properties of some methods are examined. Particular mixed collocation methods, and other methods for the same class of problems, are compared by applying them to a variety of test problems.