Convergence analysis of high-order time-splitting pseudospectral methods for nonlinear schr?dinger equations

摘要

In this work, the issue of favorable numerical methods for the space and time discretization of low-dimensional nonlinear Schr?dinger equations is addressed. The objective is to provide a stability and error analysis of high-accuracy discretizations that rely on spectral and splitting methods. As a model problem, the time-dependent Gross-Pitaevskii equation arising in the description of Bose-Einstein condensates is considered. For the space discretization pseudospectral methods collocated at the associated quadrature nodes are analyzed. For the time integration highorder exponential operator splitting methods are studied, where the decomposition of the function defining the partial differential equation is chosen in accordance with the underlying spectral method. The convergence analysis relies on a general framework of abstract nonlinear evolution equations and fractional power spaces defined by the principal linear part. Essential tools in the derivation of a temporal global error estimate are further the formal calculus of Lie-derivatives and bounds for iterated Lie-commutators. Numerical examples for higher-order time-splitting pseudospectral methods applied to time-dependent Gross-Pitaevskii equations illustrate the theoretical result.