On Fourier Time-Splitting Methods for Nonlinear Schrödinger Equations in the Semiclassical Limit

We prove an error estimate for a Lie--Trotter splitting operator associated with the Schrödinger--Poisson equation in the semiclassical regime, when the WKB approximation is valid. In finite time, and so long as the solution to a compressible Euler--Poisson equation is smooth, the error between the numerical solution and the exact solution is controlled in Sobolev spaces, in a suitable phase/amplitude representation. As a corollary, we infer the numerical convergence of the quadratic observables with a time step independent of the Planck constant. A similar result is established for the nonlinear Schrödinger equation in the weakly nonlinear regime.