Chan, Hiu Ning; Chung, Eric T. A staggered discontinuous Galerkin method with local TV regularization for the Burgers equation. (English) Zbl 1349.65442 Numer. Math., Theory Methods Appl. 8, No. 4, 451-474 (2015). Summary: 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. Cited in 5 Documents MSC: 65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs 35Q53 KdV equations (Korteweg-de Vries equations) 65M50 Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs Keywords:discontinuous Galerkin method; energy conservation; mass conservation; staggered grid; inviscid Burgers equation; total variation regularization; numerical results PDFBibTeX XMLCite \textit{H. N. Chan} and \textit{E. T. Chung}, Numer. Math., Theory Methods Appl. 8, No. 4, 451--474 (2015; Zbl 1349.65442) Full Text: DOI