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

Tie Zhang & Jingna Liu

DOI: 10.4208/nmtma.2014.1216nm

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 = {Tie Zhang and Jingna Liu}, 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 AU - Tie Zhang & Jingna Liu 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 and Jingna Liu. (2014). 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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