Numerical long-time energy conservation for the nonlinear Schrodinger equation

摘要

Long-time near-conservation of energy by the widely used split-step Fourier method applied to the cubic nonlinear Schrodinger on a torus is investigated. For initial values that are small in the Sobolev space H-1, it is shown that the energy is nearly preserved on time intervals that scale polynomially in the inverse of the size of the initial values. This result holds under a nonresonance condition on the time step size of the method. It is shown with the help of a completely resonant modulated Fourier expansion.