SIAM Journal on Scientific Computing, Volume 42, Issue 5, Page A2943-A2973, January 2020.
In this paper, we develop an adaptive multiresolution discontinuous Galerkin (DG) scheme for scalar hyperbolic conservation laws in multidimensions. Compared with previous work for linear hyperbolic equations [W. Guo and Y. Cheng, SIAM J. Sci. Comput., 38 (2016), pp. A3381--A3409, W. Guo and Y. Cheng, SIAM J. Sci. Comput., 39 (2017), pp. A2962--A2992], a class of interpolatory multiwavelets are applied to efficiently compute the nonlinear integrals over elements and edges in DG schemes. The resulting algorithm, therefore, can achieve similar computational complexity as the sparse grid DG method for smooth solutions. Theoretical and numerical studies are performed taking into consideration the accuracy and stability with regard to the choice of the interpolatory multiwavelets. Artificial viscosity is added to capture the shock and only acts on the leaf elements taking advantage of the multiresolution representation. Adaptivity is realized by auto error thresholding based on hierarchical surplus. Accuracy and robustness are demonstrated by several numerical tests.

中文翻译：

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多维标量双曲守恒律的人工黏性自适应多分辨率不连续Galerkin方法

SIAM科学计算杂志，第42卷，第5期，第A2943-A2973页，2020年1月。
在本文中，我们针对多维标量双曲守恒律，开发了一种自适应多分辨率不连续伽勒金（DG）方案。与先前关于线性双曲方程的研究比较[W．郭和郑成，暹罗科学。计算（38）（2016），A3381-A3409，郭伟，郑元,、 SIAM J. [计算]（39（2017），第A2962--A2992页）中，应用一类插值多小波来有效地计算DG方案中元素和边上的非线性积分。因此，对于平滑解决方案，所得算法可以实现与稀疏网格DG方法相似的计算复杂性。进行理论和数值研究时要考虑到插值多小波的选择的准确性和稳定性。添加了人工粘度以捕获冲击，并利用多分辨率表示仅作用于叶元素。自适应是通过基于分层剩余的自动错误阈值实现的。几个数值测试证明了准确性和鲁棒性。