An iterative starting method to control parasitism for the Leapfrog method

The Leapfrog method is a time–symmetric multistep method, widely used to solve the Euler equations and other Hamiltonian systems, due to its low cost and geometric properties. A drawback with Leapfrog is that it suffers from parasitism. This paper describes an iterative starting method, which may be used to reduce to machine precision the size of the parasitic components in the numerical solution at the start of the computation. The severity of parasitic growth is also a function of the differential equation, the main method and the time–step. When the tendency to parasitic growth is relatively mild, computational results indicate that using this iterative starting method may significantly increase the time–scale over which parasitic effects remain acceptably small. Using an iterative starting method, Leapfrog is applied to the cubic Schrödinger equation. The computational results show that the Hamiltonian and soliton behaviour are well–preserved over long time-scales.