Section: New Results

Uniformly accurate time-splitting schemes for NLS in the semiclassical limit

In [42] , we construct new numerical methods for the nonlinear Schrödinger equation in the semiclassical limit. We introduce time-splitting schemes for a phase-amplitude reformulation of the equation where the dimensionless Planck constant is not a singular parameter anymore. Our methods have an accuracy which is spectral in space, of second or fourth-order in time, and independent of the Planck constant before the formation of caustics. The scheme of second-order preserves exactly the $L^2$ norm of the solution, as the flow of the nonlinear Schrödinger equation does. In passing, we introduce a new time-splitting method for the eikonal equation, whose precision is spectral in space and of second or fourth-order in time.