Superconvergence of Discontinuous Galerkin and Local Discontinuous Galerkin Schemes for Linear Hyperbolic and Convection-Diffusion Equations in One Space Dimension

Abstract

In this paper, we study the superconvergence property for the discontinuous Galerkin (DG) and the local discontinuous Galerkin (LDG) methods, for solving one-dimensional time dependent linear conservation laws and convection-diffusion equations. We prove superconvergence towards a particular projection of the exact solution when the upwind flux is used for conservation laws and when the alternating flux is used for convection-diffusion equations. The order of superconvergence for both cases is proved to be k + 3 2 when piecewise P k polynomials with k ≥ 1 are used. The proof is valid for arbitrary non-uniform regular meshes and for piecewise P k polynomials with arbitrary k ≥ 1, improving upon the results in [8, 9] in which the proof based on Fourier analysis was given only for uniform meshes with periodic boundary condition and piecewise P 1 polynomials.

Superconvergent functional output for time-dependent problems using finite differences on summation-by-parts form

“…It was shown that when discretizing the equations in a dual consistent [9,17] way, the order of accuracy of the output functional was higher than the FD solution itself. This superconvergent behaviour was seen already in [3] for FEM and in [7] for DG, but it had not been previously proven for finite difference schemes.…”

“…In order to derive a stable and spatially dual consistent scheme, (45) has to be rewritten as a first order system in the same way as in the local discontinuous Galerkin (LDG) method [46]. It has been shown that the LDG method has interesting superconvergent features not only for functionals, but also for the solution itself [7,29,47]. We hence adapt the LDG formulation and rewrite (45) as…”

“…Since then, duality and adjoint equations have been vastly studied in the context of finite element methods (FEM) [2] and more recently using discontinuous Galerkin (DG) methods [7,8,9,10], finite volume methods (FVM) [11] and spectral difference methods [12].…”

On the long-time simulation accuracy of the discontinuous Galerkin method for 1D transport equation

“…Тогда в многомерном случае на неструктурированной сетке при сохранении качества сеточных элементов DG обладает аппроксимацион-ной ошибкой O(h p ) и позволяет найти решение с точностью O(h p+1/2 ) в норме L 2 [2]. В одномерном случае на неравномерной сетке порядок аппроксимации также равен p, но порядок точности равен p + 1 [3].…”

Convergence analysis of the LDG method for singularly perturbed two-point boundary value problems

“…Among these is the local discontinuous Galerkin (LDG) method, which was proposed in [3] and [9] by separating higher order operators into systems of first order equations so that classical DG methods can be extended to problems with second order operators, especially for convection-diffusion and hyperbolic equations. The state of the art of the development of these methods and their applications can be found in [1,2,5,7,10].In this work, we apply the LDG method to 1-D singularly perturbed problems with Dirichlet boundary conditions: …”