Sparse Grid Discretizations based on a Discontinuous Galerkin Method

TL;DR

This paper extends Sparse Grid discretization methods for PDEs using a Discontinuous Galerkin approach, demonstrating superior efficiency and accuracy in high-dimensional scalar wave equations up to 7D.

Contribution

It introduces a Discontinuous Galerkin extension to Sparse Grids for PDEs and compares their performance in high-dimensional problems.

Findings

Sparse Grids significantly reduce computational cost in high dimensions.

The method achieves high accuracy with fewer grid points.

Open source code enables reproducibility of results.

Abstract

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.