Optimal convergence and superconvergence of semi-Lagrangian discontinuous Galerkin methods for linear convection equations in one space dimension-猫眼课题宝

DOI：

10.1090/mcom/3527

发表时间：

2020-03

期刊：

Math. Comput.

影响因子：

通讯作者：

中科院分区：

其他

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作者：

Yang Yang-Yang;Xiaofeng Cai;Jing-Mei Qiu

本文将半拉格朗日间断Galerkin（SLDG）方法应用于一维线性双曲型方程，并在范数下分析了数值解与精确解之间的误差。在所有以前的工作中，SLDG方法的理论分析将建议一个次优的收敛速度，由于随着时间步长的误差积累。然而，数值实验表明，最佳的收敛速度，如果终端时间是大的，超收敛速度。在本文中，我们将证明最优收敛和最优超收敛速度。主要有三个困难：1。重叠网格的误差分析。由于半拉格朗日时间离散的性质，我们需要引入背景欧拉网格和移位网格。两种网格交错排列，不容易构造局部投影和处理时间演化过程中的误差积累。2.范数下含时项的超收敛性。数值解和精确解的误差可以分为两部分，投影误差和时间相关超收敛项。投影强烈依赖于超收敛速度。因此，我们需要构造一个投影序列，逐步提高超收敛速度。3.投影序列的停止准则。投影序列基本上具有相同的形式。我们需要证明投影存在到某个特定的阶，因为超收敛速度不可能是无穷大。因此，我们将寻求投影存在的一些“隐藏”条件。在本文中，我们将解决所有这三个困难，并构造几个局部投影，以证明最优收敛和超收敛速度。数值实验验证了理论结果。引用

In this paper, we apply semi-Lagrangian discontinuous Galerkin (SLDG) methods for linear hyperbolic equations in one space dimension and analyze the error between the numerical and exact solutions under the-norm. In all the previous works, the theoretical analysis of the SLDG method would suggest a suboptimal convergence rate due to the error accumulation over time steps. However, numerical experiments demonstrate an optimal convergence rate and, if the terminal time is large, a superconvergence rate. In this paper, we will prove optimal convergence and optimal superconvergence rates. There are three main difficulties: 1. The error analysis on overlapping meshes. Due to the nature of the semi-Lagrangian time discretization, we need to introduce the background Eulerian mesh and the shifted mesh. The two meshes are staggered, and it is not easy to construct local projections and to handle the error accumulation during time evolution. 2. The superconvergence of time-dependent terms under the-norm. The error of the numerical and exact solutions can be divided into two parts, the projection error and the time-dependent superconvergence term. The projection strongly depends on the superconvergence rates. Therefore, we need to construct a sequence of projections and gradually improve the superconvergence rates. 3. The stopping criterion of the sequence of projections. The sequence of projections are basically of the same form. We need to show that the projections exist up to some certain order since the superconvergence rate cannot be infinity. Hence, we will seek some “hidden” condition for the existence of the projections. In this paper, we will solve all the three difficulties and construct several local projections to prove the optimal convergence and superconvergence rates. Numerical experiments verify the theoretical findings. References

Optimal convergence and superconvergence of semi-Lagrangian discontinuous Galerkin methods for linear convection equations in one space dimension

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