On Fourier time-splitting methods for nonlinear Schrodinger equations in the semi-classical limit

TL;DR

This paper provides error estimates for Fourier time-splitting methods applied to nonlinear Schrödinger equations in the semi-classical limit, demonstrating convergence and stability under certain smoothness conditions.

Contribution

It establishes rigorous error bounds for Lie-Trotter splitting operators in the semi-classical regime, including convergence of quadratic observables independent of the Planck constant.

Findings

Error estimates for splitting methods in semi-classical limit

Convergence of quadratic observables with fixed time step

Results extend to weakly nonlinear Schrödinger equations

Abstract

We prove an error estimate for a Lie-Trotter splitting operator associated to the Schrodinger-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 Schrodinger equation in the weakly nonlinear regime.