Projects per year
Abstract
In this paper, we investigate the accuracyenhancement for the discontinuous Galerkin (DG) method for solving onedimensional nonlinear symmetric systems of hyperbolic conservation laws. For nonlinear equations, the divided difference estimate is an important tool that allows for superconvergence of the postprocessed solutions in the local L2norm. Therefore, we first prove that the L2norm of the αth order (1≤ α≤ k+1) divided difference of the DG error with upwind fluxes is of order k+(3α)/2, provided that the flux Jacobian matrix, f'(u), is symmetric positive definite. Furthermore, using the duality argument, we are able to derive superconvergence estimates of order 2k+(3α)/2 for the negativeorder norm, indicating that some particular compact kernels can be used to extract at least (3k/2+1)th order superconvergence for nonlinear systems of conservation laws.
Numerical experiments are shown to demonstrate the theoretical results.
Numerical experiments are shown to demonstrate the theoretical results.
Original language  English 

Pages (fromto)  125–155 
Journal  IMA Journal of Numerical Analysis 
Volume  38 
Issue number  1 
Early online date  20 Feb 2017 
DOIs  
Publication status  Published  25 Jan 2018 
Keywords
 discontinuous Galerkin
 nonlinear symmetric systems of hyperbolic conservation
 negativeorder norm estimates
 postprocessing
 Divided difference
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

Discontinuous Galerkin methods for nonlinear scalar hyperbolic conservation laws: divided difference estimates and accuracy enhancement
Meng, X. & Ryan, J., May 2017, In : Numerische Mathematik. 136, 1, p. 27–73 47 p.Research output: Contribution to journal › Article
Open AccessFile9 Citations (Scopus)10 Downloads (Pure) 
Exploiting Superconvergence Through SmoothnessIncreasing AccuracyConserving (SIAC) Filtering
Ryan, J., 2015, Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2014. Springer, Vol. 105. p. 87102 ( Lecture Notes in Computational Science and Engineering).Research output: Chapter in Book/Report/Conference proceeding › Conference contribution
Open AccessFile 
Superconvergent error estimates for positiondependent smoothnessincreasing accuracyconserving (SIAC) postprocessing of discontinuous Galerkin solutions
Ji, L., van Slingerland, P., Ryan, J. & Vuik, K., 2014, Mathematics of Computation, 83, p. 2239 2262 p.Research output: Contribution to specialist publication › Article