Paper Info
Title
Error estimates and post-processing of local discontinuous Galerkin method for Schrödinger equations.
Abstract
In this paper, we present L2 and negative-order norm estimates for the local discontinuous Galerkin (LDG) method solving variable coefficient Schrödinger equations. For these special solutions the LDG method exhibits “hidden accuracy”, and we are able to extract it through the use of a convolution kernel that is composed of a linear combination of B-splines. This technical was initially introduced by Cockburn, Luskin, Shu, and Süli for linear hyperbolic equations and extended by Ryan et al. as a smoothness-increasing accuracy-conserving (SIAC) filter. We demonstrate that it is possible to extend the SIAC filter on Schrödinger equations. When polynomials of degree k are used, we can prove theoretically the LDG method solutions are of order k+1, whereas the post-processed solutions that convolution with the SIAC filter are of order at least 2k. Additionally, we present numerical results to confirm that the accuracy of LDG solutions can be improved from O(hk+1) to at least O(h2k+1) for Schrödinger equations by using alternating numerical fluxes.
Year
DOI
Venue
2019
10.1016/j.cam.2019.01.033
Journal of Computational and Applied Mathematics
Keywords
DocType
Volume
Local discontinuous Galerkin method,Error estimates,Negative-order norm,Schrödinger equations,SIAC filter,Post-processing
Journal
356
ISSN
Citations 
PageRank 
0377-0427
1
0.35
References 
Authors
17
2
Name
Order
Citations
PageRank
Tao Qi11512.43
Yinhua Xia29710.49