Sparse Grid Discretizations based on a Discontinuous Galerkin Method
Sparse Grid Discretizations based on a Discontinuous Galerkin Method
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基于间断伽辽金法的稀疏网格离散化
DOI：
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发表时间：
2017
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通讯作者：
E. Schnetter
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作者：
Alexander Atanasov;E. Schnetter
我们研究和扩展稀疏网格作为偏微分方程（PDE）的离散化方法。在$D $维中求解偏微分方程的成本随着常用方法的增加而增加，为$O（N ^D）$。即使对于中等的$D $（例如$D = 3 $），这对于增加问题大小$N $来说也很快变得昂贵得令人望而却步。这种效应被称为“幻觉诅咒”。稀疏网格提供了一种替代的离散化方法，其成本要小得多，为O（N log ^{D-1} N）$。在本文中，我们向读者介绍稀疏网格，并通过不连续伽辽金方法扩展该方法。然后，我们解决标量波动方程在高达$6 + 1 $维，比较完整和稀疏网格之间的成本和精度。稀疏网格的性能远远优于上级网格，即使在三维中也是如此。我们的代码作为开源代码免费提供，我们鼓励读者复制我们展示的结果。
We examine and extend Sparse Grids as a discretization method for partial differential equations (PDEs). Solving a PDE in $D$ dimensions has a cost that grows as $O(N^D)$ with commonly used methods. Even for moderate $D$ (e.g. $D=3$), this quickly becomes prohibitively expensive for increasing problem size $N$. This effect is known as the Curse of Dimensionality. Sparse Grids offer an alternative discretization method with a much smaller cost of $O(N log^{D-1}N)$. In this paper, we introduce the reader to Sparse Grids, and extend the method via a Discontinuous Galerkin approach. We then solve the scalar wave equation in up to $6+1$ dimensions, comparing cost and accuracy between full and sparse grids. Sparse Grids perform far superior, even in three dimensions. Our code is freely available as open source, and we encourage the reader to reproduce the results we show.