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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 <= \alpha <= k+1) divided difference of the DG error in the L2-norm is of order k+(3-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 k+(3-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 (3k/2+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 L2-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.
|Number of pages||47|
|Early online date||8 Aug 2016|
|Publication status||Published - May 2017|
- Discontinuous Galerkin method
- Nonlinear hyperbolic conservation laws
- Negative-order norm error estimates
- Divided difference
- SIAC filtering
- 2 Finished
USEFUL - Useful Superconvergence Extraction for Filtering of Underresolved Solutions - Xiong Meng Fellowship
Ryan, J. & Meng, X.
1/09/14 → 31/08/16
Exploiting Superconvergence in Discontinuous Galerkin Methods for Improved Time-Stepping and Visualization
1/02/13 → 31/07/16
- 15 Citations (Scopus)
- 1 Article
Divided difference estimates and accuracy enhancement of discontinuous Galerkin methods for nonlinear symmetric systems of hyperbolic conservation lawsMeng, X. & Ryan, J. K., 25 Jan 2018, In: IMA Journal of Numerical Analysis. 38, 1, p. 125–155 31 p.
Research output: Contribution to journal › Article › peer-reviewOpen AccessFile7 Citations (Scopus)11 Downloads (Pure)