Strang splitting for a semilinear Schrödinger equation with damping and forcing

Abstract

We propose and analyze a Strang splitting method for a cubic semilinear Schrödinger equation with forcing and damping terms and subject to periodic boundary conditions. The nonlinear part is solved analytically, whereas the linear part – space derivatives, damping and forcing – is approximated by the exponential trapezoidal rule. The necessary operator exponentials and ϕ-functions can be computed efficiently by fast Fourier transforms if space is discretized by spectral collocation. Under natural regularity assumptions, we first show global existence of the problem in H4(T) and establish global bounds depending on properties of the forcing. The main result of our error analysis is first-order convergence in H1(T) and second-order convergence in L2(T) on bounded time-intervals.