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：我们研究了两种应用于双曲守恒律间断Galerkin（DG）方法后处理技术的高效时间离散格式。该后处理技术应用于DG方法的最终时刻，可提升原始DG解的精度（空间超收敛）。该后处理技术的主要难点在于，后处理后的空间超收敛需与恰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）在时间方向上分辨率不足，空间超收敛将被掩盖。本文重点研究近期提出的SDG方法，并基于DG弱形式从校正过程中推导其显式格式。同时，我们引入了另一类相似技术——谱延迟校正（SDC）方法。通过多个数值算例，将上述两种时间离散技术与标准三阶Runge-Kutta方法进行对比，结果表明SDG与SDC方法均为利用DG方法空间超收敛的高效时间离散技术。