2 research outputs found
Sub-optimal convergence of discontinuous Galerkin methods with central fluxes for linear hyperbolic equations with even degree polynomial approximations
In this paper, we theoretically and numerically verify that the discontinuous
Galerkin (DG) methods with central fluxes for linear hyperbolic equations on
non-uniform meshes have sub-optimal convergence properties when measured in the
-norm for even degree polynomial approximations. On uniform meshes, the
optimal error estimates are provided for arbitrary number of cells in one and
multi-dimensions, improving previous results. The theoretical findings are
found to be sharp and consistent with numerical results.Comment: 27 pages, 1 figur
Sparse Grid Central Discontinuous Galerkin Method for Linear Hyperbolic Systems in High Dimensions
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 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