Damping and phase analysis for some methods for solving second‐order ordinary differential equations

We consider numerical methods for initial value problems for second-order systems of ordinary differential equations, analysing them by applying them to the test equation We discuss conditions which ensure an oscillatory numerical solution and the desirability of such a property. We also use a slightly more general test equation to derive conditions which ensure that the numerical forced oscillation is in phase with the true forced oscillation. To illustrate the theory, we consider the damping and phase properties of some frequently used methods.