arXivDaily arXiv每日学术速递 周一至周五更新
arXiv 2008.13170math.NAcs.NA
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非连续Galerkin方法的通用精度增强滤波器设计

How to Design A Generic Accuracy-Enhancing Filter for Discontinuous Galerkin Methods

Xiaozhou Li

摘要:

高阶精度($k+1$在$L^2$范数中的阶)是间断Galerkin(DG)方法的众所周知的有益性质之一。此外,许多研究已经证明了半离散DG方法的超收敛性($2k+1$在负范数下的阶)。人们可以利用这种超收敛性能的后处理技术,以提高DG解决方案的准确性。在$L^2$范数中将收敛速率从阶$k+1$提高到阶$2k+1$的一类流行的后处理技术是平滑度增加精度保持(SIAC)滤波。除了提高精度之外,SIAC滤波还增加了DG解决方案的元件间平滑度。SIAC滤波是Cockburn等人于2003年在求解线性双曲型方程的DG方法中引入的。从那时起,已经提出了SIAC滤波的许多推广。然而,SIAC滤波的发展从未超出使用样条函数(主要是B样条)来构造滤波函数的框架。在本文中,我们首先调查的一般基函数(超越样条函数),可用于构建SIAC滤波器。一般的基函数的研究放松SIAC滤波器的结构,并提供更具体的属性,如额外的光滑度等。其次,我们研究了基函数的分布,并提出了一种新的SIAC滤波器称为紧凑SIAC滤波器,显着降低了原来的SIAC滤波器的支持大小,同时保留(甚至提高)其能力,以提高DG解决方案的准确性。我们证明了新的SIAC滤波器提取超收敛的能力,并提供了数值结果来证实理论结果,并显示新发现的良好的数值性能。

英文摘要:

Higher-order accuracy (order of $k+1$ in the $L^2$ norm) is one of the well known beneficial properties of the discontinuous Galerkin (DG) method. Furthermore, many studies have demonstrated the superconvergence property (order of $2k+1$ in the negative norm) of the semi-discrete DG method. One can take advantage of this superconvergence property by post-processing techniques to enhance the accuracy of the DG solution. A popular class of post-processing techniques to raise the convergence rate from order $k+1$ to order $2k+1$ in the $L^2$ norm is the Smoothness-Increasing Accuracy-Conserving (SIAC) filtering. In addition to enhancing the accuracy, the SIAC filtering also increases the inter-element smoothness of the DG solution. The SIAC filtering was introduced for the DG method of the linear hyperbolic equation by Cockburn et al. in 2003. Since then, there are many generalizations of the SIAC filtering have been proposed. However, the development of SIAC filtering has never gone beyond the framework of using spline functions (mostly B-splines) to construct the filter function. In this paper, we first investigate the general basis function (beyond the spline functions) that can be used to construct the SIAC filter. The studies of the general basis function relax the SIAC filter structure and provide more specific properties, such as extra smoothness, etc. Secondly, we study the basis functions' distribution and propose a new SIAC filter called compact SIAC filter that significantly reduces the original SIAC filter's support size while preserving (or even improving) its ability to enhance the accuracy of the DG solution. We show that the proofs of the new SIAC filters' ability to extract the superconvergence and provide numerical results to confirm the theoretical results and demonstrate the new finding's good numerical performance.

机构、总结、中文摘要均由AI分析而来,可能存在错误,请以论文正文为准。

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