Crank-Nicolson Galerkin approximations to nonlinear Schrödinger equations with rough potentials
Provides rigorous numerical analysis for Schrödinger equations with rough potentials, relevant for quantum physics simulations with disorder.
The paper proves convergence with rates in L∞(L2) for Galerkin finite element discretizations of nonlinear Schrödinger equations with rough, discontinuous potentials, achieving optimal rates for smooth potentials without CFL-type conditions.
This paper analyses the numerical solution of a class of non-linear Schrödinger equations by Galerkin finite elements in space and a mass- and energy conserving variant of the Crank-Nicolson method due to Sanz-Serna in time. The novel aspects of the analysis are the incorporation of rough, discontinuous potentials in the context of weak and strong disorder, the consideration of some general class of non-linearities, and the proof of convergence with rates in $L^{\infty}(L^2)$ under moderate regularity assumptions that are compatible with discontinuous potentials. For sufficiently smooth potentials, the rates are optimal without any coupling condition between the time step size and the spatial mesh width.