Large amplitude instability in finite difference approximations to the Klein–Gordon equation Original Research Article

In a paper by Jiminez and Vazquez (1990), four finite difference schemes for approximating the nonlinear Klein–Gordon equation were discussed. They observed undesirable characteristics in some of the numerical schemes, in particular a loss of spatial symmetry and the onset of instability for large values of a parameter in the initial condition of the equation. The proposition, put forward by Jiminez and Vazquez, that the observed instability is protected against by using energy conserving schemes is questioned. An analysis of the schemes as applied to a linear problem is carried out and these indicate that the instability arises from the use of explicit finite difference schemes rather than any failure of energy conservation. This conjecture is further supported by an analysis of two further schemes. Some general results applicable to the nonlinear finite difference schemes are also presented.