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 $L^2$ 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.

翻译：本文通过引入基于勒让德矩的重构算子，构建了一个降阶间断伽辽金（RDG）空间，该空间能在使用比标准间断伽辽金（DG）空间更少的自由度（DoFs）的同时，达到相同的逼近精度。这种基于“窄模板”的重构算子设计能够保持高阶DG方法的局部数据结构特性。利用RDG空间，我们对非定常非线性对流-扩散-反应方程采用隐式-显式时间推进的局部间断伽辽金（LDG）方法，其中自由度数目的减少使我们能够实现更高的效率。在理论分析方面，我们给出了重构算子的适定性和逼近性质，以及半离散LDG格式的L²误差估计。若干代表性数值试验展示了该方法在捕捉边界层时的高精度与优越性能。