SPURIOUS RESONANCE IN SEMIDISCRETE METHODS FOR THE KORTEWEG–DE VRIES EQUATION

A multiple scales analysis of semidiscrete methods for the Korteweg–de Vries equation is conducted. Methods that approximate the spatial derivatives by finite differences with arbitrary order accuracy and the limiting method, the Fourier pseudospectral method, are considered. The analysis reveals that a resonance effect can occur in the semidiscrete solution but not in the solution of the continuous equation. It is shown for the Fourier pseudospectral discretization that resonance can only be caused by aliased modes. The spurious semidiscrete solutions are investigated in numerical experiments and we suggest methods for avoiding spurious resonance.