论文标题
如何为不连续的Galerkin方法设计通用精度增强过滤器
How to Design A Generic Accuracy-Enhancing Filter for Discontinuous Galerkin Methods
论文作者
论文摘要
高阶精度（$ l^2 $ norm中的$ k+1 $的订单）是不连续的盖尔金（DG）方法的众所周知的有益属性之一。此外，许多研究证明了半分化DG方法的SuperConvergence属性（负标准为$ 2K+1 $）。可以通过后处理技术来利用这种超级融合属性，以增强DG解决方案的准确性。一类流行的后处理技术，可将收敛速度从$ k+1 $提高到$ l^2 $ norm中的订单$ 2K+1 $的融合率是平滑度的准确性（SIAC）过滤。除了提高精度外，SIAC滤波还增加了DG溶液的元素间平滑度。 Cockburn等人引入了SIAC滤波为线性双曲方程的DG方法。在2003年。从那时起，已经提出了许多SIAC滤波的概括。但是，SIAC滤波的开发从未超出使用样条函数（主要是B型）来构建滤波器函数的框架。在本文中，我们首先研究可用于构建SIAC滤波器的一般基础函数（超出样条函数）。一般基础函数的研究松弛SIAC滤波器结构并提供更具体的特性，例如额外的平滑度等。其次，我们研究了基础函数的分布，并提出了一种称为紧凑型SIAC滤波器的新型SIAC滤波器，可显着降低原始SIAC滤波器的支撑尺寸（甚至可以提高）增强DG溶液准确性的能力。我们表明，新的SIAC过滤器提取超授权并提供数值结果以确认理论结果并证明新发现的良好数值性能的证据。
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.
