Numerical long-time energy conservation for the nonlinear Schrödinger equation. (English) Zbl 1433.65236
Summary: Long-time near-conservation of energy by the widely used split-step Fourier method applied to the cubic nonlinear Schrödinger 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.
MSC:
65M70 | Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs |
35Q55 | NLS equations (nonlinear Schrödinger equations) |