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

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.