An Ultra-Weak Discontinuous Galerkin Method for Schr\"odinger Equation in One Dimension

TL;DR

This paper introduces an ultra-weak discontinuous Galerkin method for solving the one-dimensional nonlinear Schrödinger equation, providing stability analysis, error estimates, and numerical verification of the scheme's effectiveness.

Contribution

The paper develops a novel ultra-weak DG scheme for 1D Schrödinger equations with comprehensive stability and error analysis, including cases with element-wise and global projections.

Findings

Optimal $L^2$ error estimates achieved for many parameter choices

Numerical examples confirm theoretical stability and accuracy

Projection analysis depends on flux parameters and matrix structures

Abstract

In this paper, we develop an ultra-weak discontinuous Galerkin (DG) method to solve the one-dimensional nonlinear Schr\"odinger equation. Stability conditions and error estimates are derived for the scheme with a general class of numerical fluxes. The error estimates are based on detailed analysis of the projection operator associated with each individual flux choice. Depending on the parameters, we find out that in some cases, the projection can be defined element-wise, facilitating analysis. In most cases, the projection is global, and its analysis depends on the resulting $2\times2$ block-circulant matrix structures. For a large class of parameter choices, optimal $\textit{a priori}$ $L^2$ error estimates can be obtained. Numerical examples are provided verifying theoretical results.