Abstract
In this paper, an analysis of the accuracy-enhancement for the discontinuous Galerkin (DG) method applied to one-dimensional scalar nonlinear hyperbolic conservation laws is carried out. This requires analyzing the divided difference of the errors for the DG solution. We therefore first prove that the alpha -th order (1 le alpha le {k+1}) divided difference of the DG error in the L^2 norm is of order {k + frac{3}{2} - frac{alpha }{2}} when upwind fluxes are used, under the condition that |f'(u)| possesses a uniform positive lower bound. By the duality argument, we then derive superconvergence results of order {2k + frac{3}{2} - frac{alpha }{2}} in the negative-order norm, demonstrating that it is possible to extend the Smoothness-Increasing Accuracy-Conserving filter to nonlinear conservation laws to obtain at least ({frac{3}{2}k+1})th order superconvergence for post-processed solutions. As a by-product, for variable coefficient hyperbolic equations, we provide an explicit proof for optimal convergence results of order {k+1} in the L^2 norm for the divided differences of DG errors and thus ({2k+1})th order superconvergence in negative-order norm holds. Numerical experiments are given that confirm the theoretical results.
Highlights
In this paper, we study the accuracy-enhancement of semi-discrete discontinuous Galerkin (DG) methods for solving one-dimensional scalar conservation laws ut + f (u)x = 0, (x, t) ∈ (a, b) × (0, T ], u(x, 0) = u0(x), x ∈ = (a, b),(1.1a) (1.1b) where u0(x) is a given smooth function
As a by-product, for variable coefficient hyperbolic equations, we provide an explicit proof for optimal convergence results of order k + 1 in the L2 norm for the divided differences of DG errors and (2k + 1)th order superconvergence in negative-order norm holds
We study the accuracy-enhancement of semi-discrete discontinuous Galerkin (DG) methods for solving one-dimensional scalar conservation laws ut + f (u)x = 0, (x, t) ∈ (a, b) × (0, T ], u(x, 0) = u0(x), x ∈ = (a, b), (1.1a) (1.1b) where u0(x) is a given smooth function
Summary
The above framework is no longer valid for variable coefficient or nonlinear equations In this case, in order to derive superconvergent estimates about the post-processed solution, both the L2 norm and negative-order norm error estimates of divided differences should be established. Of this paper is to show L2 norm error estimates for divided differences, which are helpful for us to obtain a higher order of accuracy in the negative-order norm and the superconvergence of the post-processed solutions We remark that it requires | f (u)| having a uniform positive lower bound due to the technicality of the proof. The generalization from purely linear problems [11,15] to nonlinear hyperbolic equations in this paper involves several technical difficulties One of these is how to establish important relations between the spatial derivatives and time derivatives of a particular projection of divided differences of DG errors. In the appendix we provide the proofs for some of the more technical lemmas
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