Linearly implicit energy-preserving integrating factor methods for the 2D nonlinear Schr\"odinger equation with wave operator and convergence analysis

TL;DR

This paper introduces a new class of linear, energy-preserving integrating factor methods for the 2D nonlinear Schrödinger equation with wave operator, achieving high-order accuracy and rigorous convergence analysis without grid restrictions.

Contribution

The paper develops a novel energy-preserving integrating factor method combining scalar auxiliary variables, with proven energy conservation and optimal convergence for the 2D NLSW, extending to arbitrary high order.

Findings

The second-order scheme is rigorously energy-preserving.

The methods achieve optimal convergence in the $H^1$ norm.

Numerical experiments confirm theoretical results and advantages.

Abstract

In this paper, we develop a novel class of linear energy-preserving integrating factor methods for the 2D nonlinear Schr\"odinger equation with wave operator (NLSW), combining the scalar auxiliary variable approach and the integrating factor methods. A second-order scheme is first proposed, which is rigorously proved to be energy-preserving. By using the energy methods, we analyze its optimal convergence in the $H^1$ norm without any restrictions on the grid ratio, where a novel technique and an improved induction argument are proposed to overcome the difficulty posed by the unavailability of a priori $L^\infty$ estimates of numerical solutions. Based on the integrating factor Runge-Kutta methods, we extend the proposed scheme to arbitrarily high order, which is also linear and conservative. Numerical experiments are presented to confirm the theoretical analysis and demonstrate the advantages of the proposed methods.