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A Staggered Discontinuous Galerkin Method with Local TV Regularization for the Burgers Equation

Part of: Optics, electromagnetic theory - General

Published online by Cambridge University Press:  10 November 2015

Hiu Ning Chan  and
Eric T. Chung
Hiu Ning Chan
Affiliation:
Department of Mathematics, The Chinese University of Hong Kong, Sha Tin, Hong Kong
Eric T. Chung*
Affiliation:
Department of Mathematics, The Chinese University of Hong Kong, Sha Tin, Hong Kong
*
*Corresponding author. Email address: chnhester@yahoo.com.hk (H. N. Chan), tschung@math.cuhk.edu.hk (Eric T. Chung)
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Abstract

The staggered discontinuous Galerkin (SDG) method has been recently developed for the numerical approximation 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 be 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.

Keywords

discontinuous Galerkin methodTV regularizationenergy conservationmass conservationstaggered grid

MSC classification

Secondary: 78A48: Composite media; random media
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 8 , Issue 4 , November 2015 , pp. 451 - 474
Copyright
Copyright © Global-Science Press 2015 

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