Adaptive splitting methods for nonlinear Schr\"{o}dinger equations in the semiclassical regime

TL;DR

This paper analyzes the error behavior of exponential operator splitting methods for nonlinear Schrödinger equations in the semiclassical regime, providing insights into local errors and adaptive step size control.

Contribution

It determines the exact local error form for Lie and Strang splitting methods and introduces defect-based error estimators for adaptive time stepping.

Findings

Exact local error forms are derived for Lie and Strang splitting methods.

Dependence of errors on the semiclassical parameter is characterized.

Numerical examples validate the theoretical error estimates.

Abstract

The error behavior of exponential operator splitting methods for nonlinear Schr{\"o}dinger equations in the semiclassical regime is studied. For the Lie and Strang splitting methods, the exact form of the local error is determined and the dependence on the semiclassical parameter is identified. This is enabled within a defect-based framework which also suggests asymptotically correct a~posteriori local error estimators as the basis for adaptive time stepsize selection. Numerical examples substantiate and complement the theoretical investigations.