On Optimal Convergence Rates for the Nonlinear Schr\"{o}dinger Equation with a Wave Operator via Localized Orthogonal Decomposition

TL;DR

This paper introduces a Localized Orthogonal Decomposition method for the 2D nonlinear Schrödinger equation with a wave operator, achieving unconditional superconvergent error estimates and preserving conservation laws.

Contribution

The paper develops a novel LOD method for the nonlinear Schrödinger equation with wave operator, providing theoretical error bounds and conservation law preservation.

Findings

Unconditional optimal-order superconvergent error estimates in L^p norm.

Method preserves conservation laws.

Numerical simulations verify theoretical results.

Abstract

In this paper, we develop a Localized Orthogonal Decomposition (LOD) method for the two-dimensional time-dependent nonlinear Schr\"{o}dinger equation with a wave operator. We prove that our method preserves conservation laws and admits a unique numerical solution; furthermore, we obtain unconditional (i.e., time-step restriction-free) optimal-order superconvergent \(L^p\) error estimates. To complement the theoretical analysis, we present a series of numerical simulations that verify the analytical results and further illustrate structural aspects of the problem.