Optimal Error Estimates of Conservative Local Discontinuous Galerkin Method for Nonlinear Schr\"odinger Equation

TL;DR

This paper introduces a conservative local discontinuous Galerkin method for the 1D nonlinear Schrödinger equation, achieving optimal convergence rates and preserving charge conservation, validated by numerical experiments.

Contribution

The paper develops a novel conservative LDG method with optimal convergence and charge conservation for nonlinear Schrödinger equations.

Findings

Achieves optimal convergence rate of O(h^{k+1})

Preserves charge conservation law

Numerical experiments confirm theoretical results

Abstract

In this paper, we propose a conservative local discontinuous Galerkin method for one-dimensional nonlinear Schr\"odinger equation. By using special upwind-biased numerical fluxes, we establish the optimal rate of convergence $\mathcal O(h^{k+1})$, with polynomial of degree $k$ and grid size $h$. Meanwhile, we show that this method preserves the charge conservation law and thus we call it a conservative local discontinuous Galerkin method. Numerical experiments verify our theoretical result.