Discontinuous Galerkin Method Based on the Reduced Space for the Nonlinear Convection–Diffusion–Reaction Equation

In this paper, by introducing a reconstruction operator based on the Legendre moments, we construct a reduced discontinuous Galerkin (RDG) space that could achieve the same approximation accuracy but using fewer degrees of freedom (DoFs) than the standard discontinuous Galerkin (DG) space. The design of the “narrow-stencil-based” reconstruction operator can preserve the local data structure property of the high-order DG methods. With the RDG space, we apply the local discontinuous Galerkin (LDG) method with the implicit-explicit time marching for the nonlinear unsteady convection–diffusion–reaction equation, where the reduction of the number of DoFs allows us to achieve higher efficiency. In terms of theoretical analysis, we give the well-posedness and approximation properties for the reconstruction operator and the L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2$$\end{document} error estimate for the semi-discrete LDG scheme. Several representative numerical tests demonstrate the accuracy and the performance of the proposed method in capturing the layers.