Sparse Grid Central Discontinuous Galerkin Method for Linear Hyperbolic Systems in High Dimensions
This work addresses the curse of dimensionality in solving high-dimensional hyperbolic systems, providing a new method with theoretical guarantees.
The paper develops a sparse grid central discontinuous Galerkin method for linear hyperbolic systems in high dimensions, achieving $L^2$ stability and error estimates for scalar problems, and demonstrating numerical results for acoustic and elastic waves.
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.