Volume 8, Issue 4
A Staggered Discontinuous Galerkin Method with Local TV Regularization for the Burgers Equation

Hiu Ning Chan & Eric T. Chung

Numer. Math. Theor. Meth. Appl., 8 (2015), pp. 451-474.

Published online: 2015-08

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  • Abstract

The staggered discontinuous Galerkin (SDG) method has been recently developed for the numerical approxmation of partial differential equations. An important advantage of such methodology is that the numerical solution automatically satisfies some conservation properties which are also satisfied by the exact solution. In this paper, we will consider the numerical approximation of the inviscid Burgers equation by the SDG method. For smooth solutions, we prove that our SDG method has the properties of mass and energy conservation. It is well-known that extra care has to taken at locations of shocks and discontinuities. In this respect, we propose a local total variation (TV) regularization technique to suppress the oscillations in the numerical solution. This TV regularization is only performed locally where oscillation is detected, and is thus very efficient. Therefore, the resulting scheme will preserve the mass and energy away from the shocks and the numerical solution is regularized locally near shocks. Detailed description of the method and numerical results are presented.

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@Article{NMTMA-8-451, author = {}, title = {A Staggered Discontinuous Galerkin Method with Local TV Regularization for the Burgers Equation}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2015}, volume = {8}, number = {4}, pages = {451--474}, abstract = {

The staggered discontinuous Galerkin (SDG) method has been recently developed for the numerical approxmation of partial differential equations. An important advantage of such methodology is that the numerical solution automatically satisfies some conservation properties which are also satisfied by the exact solution. In this paper, we will consider the numerical approximation of the inviscid Burgers equation by the SDG method. For smooth solutions, we prove that our SDG method has the properties of mass and energy conservation. It is well-known that extra care has to taken at locations of shocks and discontinuities. In this respect, we propose a local total variation (TV) regularization technique to suppress the oscillations in the numerical solution. This TV regularization is only performed locally where oscillation is detected, and is thus very efficient. Therefore, the resulting scheme will preserve the mass and energy away from the shocks and the numerical solution is regularized locally near shocks. Detailed description of the method and numerical results are presented.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.2015.m1238}, url = {http://global-sci.org/intro/article_detail/nmtma/12418.html} }
TY - JOUR T1 - A Staggered Discontinuous Galerkin Method with Local TV Regularization for the Burgers Equation JO - Numerical Mathematics: Theory, Methods and Applications VL - 4 SP - 451 EP - 474 PY - 2015 DA - 2015/08 SN - 8 DO - http://doi.org/10.4208/nmtma.2015.m1238 UR - https://global-sci.org/intro/article_detail/nmtma/12418.html KW - AB -

The staggered discontinuous Galerkin (SDG) method has been recently developed for the numerical approxmation of partial differential equations. An important advantage of such methodology is that the numerical solution automatically satisfies some conservation properties which are also satisfied by the exact solution. In this paper, we will consider the numerical approximation of the inviscid Burgers equation by the SDG method. For smooth solutions, we prove that our SDG method has the properties of mass and energy conservation. It is well-known that extra care has to taken at locations of shocks and discontinuities. In this respect, we propose a local total variation (TV) regularization technique to suppress the oscillations in the numerical solution. This TV regularization is only performed locally where oscillation is detected, and is thus very efficient. Therefore, the resulting scheme will preserve the mass and energy away from the shocks and the numerical solution is regularized locally near shocks. Detailed description of the method and numerical results are presented.

Hiu Ning Chan & Eric T. Chung. (2020). A Staggered Discontinuous Galerkin Method with Local TV Regularization for the Burgers Equation. Numerical Mathematics: Theory, Methods and Applications. 8 (4). 451-474. doi:10.4208/nmtma.2015.m1238
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