Volume 6, Issue 1
A Straightforward $hp$-Adaptivity Strategy for Shock-Capturing with High-Order Discontinuous Galerkin Methods

Hongqiang Lu & Qiang Sun

Adv. Appl. Math. Mech., 6 (2014), pp. 135-144.

Published online: 2014-06

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  • Abstract

In this paper, high-order Discontinuous Galerkin (DG) method is used to solve the two-dimensional Euler equations. A shock-capturing method based on the artificial viscosity technique is employed to handle physical discontinuities. Numerical tests show that the shocks can be captured within one element even on very coarse grids. The thickness of the shocks is dominated by the local mesh size and the local order of the basis functions. In order to obtain better shock resolution, a straightforward $hp$-adaptivity strategy is introduced, which is based on the high-order contribution calculated using hierarchical basis. Numerical results indicate that the $hp$-adaptivity method is easy to implement and better shock resolution can be obtained with smaller local mesh size and higher local order.

  • Keywords

$hp$-adaptivity, shock capturing, discontinuous Galerkin.

  • AMS Subject Headings

35L67, 65M60

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{AAMM-6-135, author = {}, title = {A Straightforward $hp$-Adaptivity Strategy for Shock-Capturing with High-Order Discontinuous Galerkin Methods}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2014}, volume = {6}, number = {1}, pages = {135--144}, abstract = {

In this paper, high-order Discontinuous Galerkin (DG) method is used to solve the two-dimensional Euler equations. A shock-capturing method based on the artificial viscosity technique is employed to handle physical discontinuities. Numerical tests show that the shocks can be captured within one element even on very coarse grids. The thickness of the shocks is dominated by the local mesh size and the local order of the basis functions. In order to obtain better shock resolution, a straightforward $hp$-adaptivity strategy is introduced, which is based on the high-order contribution calculated using hierarchical basis. Numerical results indicate that the $hp$-adaptivity method is easy to implement and better shock resolution can be obtained with smaller local mesh size and higher local order.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.2013.m-s1}, url = {http://global-sci.org/intro/article_detail/aamm/9.html} }
TY - JOUR T1 - A Straightforward $hp$-Adaptivity Strategy for Shock-Capturing with High-Order Discontinuous Galerkin Methods JO - Advances in Applied Mathematics and Mechanics VL - 1 SP - 135 EP - 144 PY - 2014 DA - 2014/06 SN - 6 DO - http://doi.org/10.4208/aamm.2013.m-s1 UR - https://global-sci.org/intro/article_detail/aamm/9.html KW - $hp$-adaptivity, shock capturing, discontinuous Galerkin. AB -

In this paper, high-order Discontinuous Galerkin (DG) method is used to solve the two-dimensional Euler equations. A shock-capturing method based on the artificial viscosity technique is employed to handle physical discontinuities. Numerical tests show that the shocks can be captured within one element even on very coarse grids. The thickness of the shocks is dominated by the local mesh size and the local order of the basis functions. In order to obtain better shock resolution, a straightforward $hp$-adaptivity strategy is introduced, which is based on the high-order contribution calculated using hierarchical basis. Numerical results indicate that the $hp$-adaptivity method is easy to implement and better shock resolution can be obtained with smaller local mesh size and higher local order.

Hongqiang Lu & Qiang Sun. (1970). A Straightforward $hp$-Adaptivity Strategy for Shock-Capturing with High-Order Discontinuous Galerkin Methods. Advances in Applied Mathematics and Mechanics. 6 (1). 135-144. doi:10.4208/aamm.2013.m-s1
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