Convergence, error analysis and longtime behavior of the Scalar Auxiliary Variable method for the nonlinear Schr\"odinger equation

TL;DR

This paper analyzes the convergence, error bounds, and long-term behavior of the Scalar Auxiliary Variable (SAV) method for the nonlinear Schrödinger equation, highlighting its energy conservation advantages.

Contribution

It provides the first rigorous convergence analysis and long-time error estimates for the SAV method applied to the nonlinear Schrödinger equation.

Findings

The SAV method preserves a modified Hamiltonian at the discrete level.

It achieves second-order global error bounds.

The SAV method exhibits better energy conservation than classical schemes.

Abstract

We carry out the convergence analysis of the Scalar Auxiliary Variable (SAV) method applied to the nonlinear Schr\"odinger equation which preserves a modified Hamiltonian on the discrete level. We derive a weak and strong convergence result, establish second-order global error bounds and present long time error estimates on the modified Hamiltonian. In addition, we illustrate the favorable energy conservation of the SAV method compared to classical splitting schemes in certain applications.