Volume 7, Issue 2
Accuracy Enhancement of Discontinuous Galerkin Method for Hyperbolic Systems

Tie Zhang & Jingna Liu

Numer. Math. Theor. Meth. Appl., 7 (2014), pp. 214-233.

Published online: 2014-07

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  • Abstract

We study the enhancement of accuracy, by means of the convolution post-processing technique, for discontinuous Galerkin(DG) approximations to hyperbolic problems. Previous investigations have focused on the superconvergence obtained by this technique for elliptic, time-dependent hyperbolic and convection-diffusion problems. In this paper, we demonstrate that it is possible to extend this post-processing technique to the hyperbolic problems written as the Friedrichs' systems by using an upwind-like DG method. We prove that the $L_2$-error of the DG solution is of order $k+1/2$, and further the post-processed DG solution is of order $2k+1$ if $Q_k$-polynomials are used. The key element of our analysis is to derive the $(2k+1)$-order negative norm error estimate. Numerical experiments are provided to illustrate the theoretical analysis.

  • Keywords

Discontinuous Galerkin method, hyperbolic problem, accuracy enhancement, post-processing, negative norm error estimate.

  • AMS Subject Headings

65N30, 65M60

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{NMTMA-7-214, author = {}, title = {Accuracy Enhancement of Discontinuous Galerkin Method for Hyperbolic Systems}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2014}, volume = {7}, number = {2}, pages = {214--233}, abstract = {

We study the enhancement of accuracy, by means of the convolution post-processing technique, for discontinuous Galerkin(DG) approximations to hyperbolic problems. Previous investigations have focused on the superconvergence obtained by this technique for elliptic, time-dependent hyperbolic and convection-diffusion problems. In this paper, we demonstrate that it is possible to extend this post-processing technique to the hyperbolic problems written as the Friedrichs' systems by using an upwind-like DG method. We prove that the $L_2$-error of the DG solution is of order $k+1/2$, and further the post-processed DG solution is of order $2k+1$ if $Q_k$-polynomials are used. The key element of our analysis is to derive the $(2k+1)$-order negative norm error estimate. Numerical experiments are provided to illustrate the theoretical analysis.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.2014.1216nm}, url = {http://global-sci.org/intro/article_detail/nmtma/5872.html} }
TY - JOUR T1 - Accuracy Enhancement of Discontinuous Galerkin Method for Hyperbolic Systems JO - Numerical Mathematics: Theory, Methods and Applications VL - 2 SP - 214 EP - 233 PY - 2014 DA - 2014/07 SN - 7 DO - http://doi.org/10.4208/nmtma.2014.1216nm UR - https://global-sci.org/intro/article_detail/nmtma/5872.html KW - Discontinuous Galerkin method, hyperbolic problem, accuracy enhancement, post-processing, negative norm error estimate. AB -

We study the enhancement of accuracy, by means of the convolution post-processing technique, for discontinuous Galerkin(DG) approximations to hyperbolic problems. Previous investigations have focused on the superconvergence obtained by this technique for elliptic, time-dependent hyperbolic and convection-diffusion problems. In this paper, we demonstrate that it is possible to extend this post-processing technique to the hyperbolic problems written as the Friedrichs' systems by using an upwind-like DG method. We prove that the $L_2$-error of the DG solution is of order $k+1/2$, and further the post-processed DG solution is of order $2k+1$ if $Q_k$-polynomials are used. The key element of our analysis is to derive the $(2k+1)$-order negative norm error estimate. Numerical experiments are provided to illustrate the theoretical analysis.

Tie Zhang & Jingna Liu. (2020). Accuracy Enhancement of Discontinuous Galerkin Method for Hyperbolic Systems. Numerical Mathematics: Theory, Methods and Applications. 7 (2). 214-233. doi:10.4208/nmtma.2014.1216nm
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