Projects per year
Abstract
In this paper, an analysis of the accuracyenhancement for the discontinuous Galerkin (DG) method applied to onedimensional 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 alphath order (1 <= \alpha <= k+1) divided difference of the DG error in the L2norm is of order k+(3alpha)/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+(3alpha)/2 in the negativeorder norm, demonstrating that it is possible
to extend the SmoothnessIncreasing AccuracyConserving filter to nonlinear conservation laws to obtain at least (3k/2+1)th order superconvergence for postprocessed solutions. As a byproduct, for variable coefficient hyperbolic equations, we provide an explicit proof for optimal convergence results of order k+1 in the L2norm for the divided differences of DG errors and thus (2k+1)th order superconvergence in
negativeorder norm holds. Numerical experiments are given that confirm the theoretical results.
Original language  English 

Pages (fromto)  27–73 
Number of pages  47 
Journal  Numerische Mathematik 
Volume  136 
Issue number  1 
Early online date  8 Aug 2016 
DOIs  
Publication status  Published  May 2017 
Keywords
 Discontinuous Galerkin method
 Nonlinear hyperbolic conservation laws
 Negativeorder norm error estimates
 Postprocessing
 Divided difference
 SIAC filtering
Projects
 2 Finished

USEFUL  Useful Superconvergence Extraction for Filtering of Underresolved Solutions  Xiong Meng Fellowship
Ryan, J. & Meng, X.
1/09/14 → 31/08/16
Project: Research

Exploiting Superconvergence in Discontinuous Galerkin Methods for Improved TimeStepping and Visualization
Ryan, J.
1/02/13 → 31/07/16
Project: Research
Research output
 13 Citations (Scopus)
 1 Article

Divided difference estimates and accuracy enhancement of discontinuous Galerkin methods for nonlinear symmetric systems of hyperbolic conservation laws
Meng, X. & Ryan, J. K., 25 Jan 2018, In: IMA Journal of Numerical Analysis. 38, 1, p. 125–155Research output: Contribution to journal › Article › peerreview
Open AccessFile6 Citations (Scopus)9 Downloads (Pure)