We investigate two efficient time discretizations for the post-processing technique of discontinuous Galerkin (DG) methods to solve hyperbolic conservation laws. The post-processing technique, which is applied at the final time of the DG method, can enhance the accuracy of the original DG solution (spatial superconvergence). One main difficulty of the post-processing technique is that the spatial superconvergence after post-processing needs to be matched with proper temporary accuracy. If the semi-discretized system (ODE system after spatial discretization) is under-resolved in time, then the space superconvergence will be concealed. In this paper, we focus our investigation on the recently designed SDG method and derive its explicit scheme from a correction process based on the DG weak formulation. We also introduce another similar technique, namely the spectral deferred correction (SDC) method. A comparison is made among both proposed time discretization techniques with the standard third-order Runge-Kutta method through several numerical examples, to conclude that both the SDG and SDC methods are efficient time discretization techniques for exploiting the spatial superconvergence of the DG methods.

翻译：本文研究两种高效时间离散方法，用于求解双曲守恒律的不连续Galerkin (DG) 方法的后处理技术。该后处理技术在DG方法的最终时刻应用，可提升原始DG解的精度（空间超收敛）。后处理技术的一个主要难点在于，后处理后的空间超收敛需与适当的时间精度相匹配。若半离散系统（空间离散后的常微分方程组）在时间上分辨率不足，则空间超收敛将被掩盖。本文重点研究近年来发展的SDG方法，并基于DG弱形式推导了其显式格式的校正过程。同时，我们引入另一类类似技术——谱延迟校正 (SDC) 方法。通过多个数值算例，将两种提出的时间离散技术与标准三阶龙格-库塔方法进行对比，结果表明SDG和SDC方法均是有效利用DG方法空间超收敛的高效时间离散技术。