Sparse Grid Central Discontinuous Galerkin Method for Linear Hyperbolic Systems in High Dimensions

TL;DR

This paper introduces a sparse grid central discontinuous Galerkin method for high-dimensional linear hyperbolic systems, effectively reducing computational complexity while maintaining stability and accuracy.

Contribution

It combines the CDG framework with sparse grid techniques and introduces a hierarchical polynomial representation for non-periodic problems.

Findings

The method achieves $L^2$ stability and error estimates for scalar problems.

Numerical results demonstrate efficiency for acoustic and elastic wave simulations.

CFL conditions are analyzed and compared across methods.

Abstract

In this paper, we develop sparse grid central discontinuous Galerkin (CDG) scheme for linear hyperbolic systems with variable coefficients in high dimensions. The scheme combines the CDG framework with the sparse grid approach, with the aim of breaking the curse of dimensionality. A new hierarchical representation of piecewise polynomials on the dual mesh is introduced and analyzed, resulting in a sparse finite element space that can be used for non-periodic problems. Theoretical results, such as $L^2$ stability and error estimates are obtained for scalar problems. CFL conditions are studied numerically comparing discontinuous Galerkin (DG), CDG, sparse grid DG and sparse grid CDG methods. Numerical results including scalar linear equations, acoustic and elastic waves are provided.