On Generalized Gauss–Radau Projections and Optimal Error Estimates of Upwind-Biased DG Methods for the Linear Advection Equation on Special Simplex Meshes
On Generalized Gauss–Radau Projections and Optimal Error Estimates of Upwind-Biased DG Methods for the Linear Advection Equation on Special Simplex Meshes
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DOI：
10.1007/s10915-023-02166-w
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发表时间：
2023-03
影响因子：
2.5
通讯作者：
Zheng Sun;Y. Xing
中科院分区：
文献类型：
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作者：
Zheng Sun;Y. Xing
广义Gauss-Radau（GGR）投影是一种全局投影算子，被广泛用于广义数值通量间断Galerkin（DG）方法的误差分析。在以前的工作中，GGR投影构建笛卡尔网格，并通过代数方法进行分析。在本文中，我们首先提出了一种替代的能量方法来分析一维GGR投影，它不需要组装和显式求解一个全球系统在整个计算域的代数方法。然后，我们推广这个能量参数，以构建一个全球性的投影算子在特殊的单形网格在多维满足所谓的流动条件。利用该投影，证明了这些网格上线性平流方程的迎风偏置DG方法的最佳误差估计，这推广了Cockburn等人（SIAM J Numer Anal 46（3）：1250-1265，2008）在时间依赖设置中对纯迎风情况的误差分析。
Generalized Gauss–Radau (GGR) projections are global projection operators that are widely used for the error analysis of discontinuous Galerkin (DG) methods with generalized numerical fluxes. In previous work, GGR projections were constructed for Cartesian meshes and analyzed through an algebraic approach. In this paper, we first present an alternative energy approach for analyzing the one-dimensional GGR projection, which does not require assembling and explicitly solving a global system over the entire computational domain as that in the algebraic approach. We then generalize this energy argument to construct a global projection operator on special simplex meshes in multidimensions satisfying the so-called flow condition. With this projection, optimal error estimates are proved for upwind-biased DG methods for the linear advection equation on these meshes, which generalizes the error analysis for the purely upwind case by Cockburn et al. (SIAM J Numer Anal 46(3):1250–1265, 2008) in a time-dependent setting.