Sparse Grid Discretizations based on a Discontinuous Galerkin Method

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.