L²stability analysis of the central discontinuous Galerkin method and a comparison between the central and regular discontinuous Galerkin methods

Abstract

Abstract.We prove stability and derive error estimates for the recently introduced central discontinuous Galerkin method, in the context of linear hyperbolic equations with possibly discontinuous solutions. A comparison between the central discontinuous Galerkin method and the regular discontinuous Galerkin method in this context is also made. Numerical experiments are provided to validate the quantitative conclusions from the analysis.Mathematics Subject Classification. 65M60.

A Central Discontinuous Galerkin Method for Hamilton-Jacobi Equations

“…2 when the Hamiltonian H in (1.1) is linear. The analysis is closely related to those in [6,31]. Though only the results for one dimensional cases are presented, similar results can be obtained for multi-dimensional cases.…”

“…Central DG methods were introduced in [29][30][31] for hyperbolic conservation laws, and they combine the central scheme [22,28,32] and the DG method. These methods evolve two copies of approximating solutions defined on overlapping meshes, and they avoid the use of Riemann solvers which can be complicated and costly for system of equations [26].…”

Central local discontinuous galerkin methods on overlapping cells for diffusion equations

“…In this paper we use overlapping cells and hence duplicative information, thereby avoiding numerical fluxes which is a distinct advantage of central schemes. This work is a continuation of our earlier work in [6,7] in which we presented and analyzed central DG schemes for hyperbolic PDEs. We would like to point out that the discussion on the difficulties related to numerical fluxes for DG methods solving diffusion equations has Keywords and phrases.…”

“…In Section 2.2 we provide an L 2 a priori error estimate for smooth solutions. In Section 2.3 we give a quantitative error estimate for this central LDG scheme for the polynomial degree up to 1 using Fourier analysis, similar to the technique used in [7,10,11]. As indicated before, we assume periodic boundary conditions.…”

“…The purely convective case was studied by us in [6,7]. The main focus of this paper is therefore to initiate the study of central LDG methods for the diffusive case, where we limit ourselves to a modest beginning with the linear heat equation…”