Comparison of Split-Step and Hamiltonian Integration Methods for Simulation of the Nonlinear Schr\"odinger Equation

TL;DR

This paper compares the split-step and Hamiltonian integration methods for solving the nonlinear Schrödinger equation, highlighting that HIM offers smaller errors and allows larger stable time steps than SS2.

Contribution

It provides a systematic comparison showing that HIM conserves the Hamiltonian exactly and enables larger stable time steps compared to SS2.

Findings

HIM has systematically smaller numerical error than SS2.

HIM allows for significantly larger time steps while maintaining stability.

SS2's time step is limited by numerical stability thresholds.

Abstract

We provide a systematic comparison of two numerical methods to solve the widely used nonlinear Schr\"odinger equation. The first one is the standard second order split-step (SS2) method based on operator splitting approach. The second one is the Hamiltonian integration method (HIM). It allows the exact conservation of the Hamiltonian at the cost of requiring the implicit time stepping. We found that numerical error for HIM method is systematically smaller than the SS2 solution for the same time step. At the same time, one can take orders of magnitude larger time steps in HIM compared with SS2 still ensuring numerical stability. In contrast, SS2 time step is limited by the numerical stability threshold.