SIAM Journal on Numerical Analysis, Volume 59, Issue 3, Page 1299-1324, January 2021.
In this paper, we propose a novel discontinuous Galerkin (DG) method to control the spurious oscillations when solving the scalar hyperbolic conservation laws. Usually, the high order linear numerical schemes would generate spurious oscillations when the solution of the hyperbolic conservation laws contains discontinuities. The spurious oscillations may be harmful to the numerical simulation, as it not only generates some artificial structures not belonging to the problems, but also causes many overshoots and undershoots that make the numerical scheme less robust. To overcome this difficulty, in this paper we introduce a numerical damping term to control spurious oscillations based on the classic DG formulation. In comparison to the classic DG method, the proposed DG method still maintains many good properties, such as the extremely local data structure, conservation, $L^2$-boundedness, optimal error estimates, and superconvergence. We also provide some numerical examples to show the good performance of the proposed DG scheme and verify our theoretical results.

中文翻译：

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标量双曲守恒律的无振荡非连续Galerkin方法

SIAM数值分析学报，第59卷，第3期，第1299-1324页，2021年1月。
在本文中，我们提出了一种新颖的不连续伽勒金（DG）方法，以在求解标量双曲守恒定律时控制杂散振荡。通常，当双曲守恒律的解包含不连续性时，高阶线性数值方案会产生伪振荡。寄生振荡可能对数值模拟有害，因为它不仅会生成一些不属于问题的人为结构，而且还会导致许多过冲和下冲，从而使数值方案的鲁棒性降低。为了克服这个困难，在本文中，我们引入了一个基于经典DG公式的数值阻尼项来控制杂散振荡。与传统的DG方法相比，所提出的DG方法仍保持许多良好的性能，例如极其局部的数据结构，保守性，$ L ^ 2 $有界，最优误差估计和超收敛。我们还提供了一些数值示例，以显示所提出的DG方案的良好性能并验证我们的理论结果。