A review of the Verlet and Gautschi methods for the Wave Equation

The Verlet time-stepping method is much favoured for the solution of nonlinear wave equations, due to its efficiency and proven energy-preserving properties. Here, we investigate its properties close to the linear stability limit for the time-step, where spurious, but ultimately stable, humps can develop in the numerical solution. We consider a somewhat non-standard approach to linear stability to explain this phenomenon, and introduce a filter, motivated by related work for the unconditionally linearly-stable Gautschi method.