An Ultra-Weak Discontinuous Galerkin Method for Schrödinger Equation in One Dimension
This work provides a rigorous numerical analysis for a specific numerical method applied to the Schrödinger equation, but it is an incremental contribution within the field of numerical PDEs.
The paper develops an ultra-weak discontinuous Galerkin method for the one-dimensional nonlinear Schrödinger equation, deriving stability conditions and optimal a priori L^2 error estimates for a general class of numerical fluxes, with numerical examples confirming the theory.
In this paper, we develop an ultra-weak discontinuous Galerkin (DG) method to solve the one-dimensional nonlinear Schrödinger 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.