Shock capturing by Bernstein polynomials for scalar conservation laws

Jan Glaubitz

Applied Mathematics and Computation, 2019, vol. 363, issue C, -

Abstract: A main disadvantage of many high-order methods for hyperbolic conservation laws lies in the famous Gibbs–Wilbraham phenomenon, once discontinuities appear in the solution. Due to the Gibbs–Wilbraham phenomenon, the numerical approximation will be polluted by spurious oscillations, which produce unphysical numerical solutions and might finally blow up the computation. In this work, we propose a new shock capturing procedure to stabilise high-order spectral element approximations. The procedure consists of going over from the original (polluted) approximation to a convex combination of the original approximation and its Bernstein reconstruction, yielding a stabilised approximation. The coefficient in the convex combination, and therefore the procedure, is steered by a discontinuity sensor and is only activated in troubled elements. Building up on classical Bernstein operators, we are thus able to prove that the resulting Bernstein procedure is total variation diminishing and preserves monotone (shock) profiles. Further, the procedure can be modified to not just preserve but also to enforce certain bounds for the solution, such as positivity. In contrast to other shock capturing methods, e. g. artificial viscosity methods, the new procedure does not reduce the time step or CFL condition and can be easily and efficiently implemented into any existing code. Numerical tests demonstrate that the proposed shock-capturing procedure is able to stabilise and enhance spectral element approximations in the presence of shocks.

Keywords: Hyperbolic conservation laws; Spectral/hp element methods; Sub-cell shock capturing; Bernstein polynomials; Gibbs phenomenon; Total variation diminishing (search for similar items in EconPapers)
Date: 2019
References: View complete reference list from CitEc
Citations: View citations in EconPapers (1)

Downloads: (external link)
http://www.sciencedirect.com/science/article/pii/S0096300319305855
Full text for ScienceDirect subscribers only

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:eee:apmaco:v:363:y:2019:i:c:41

DOI: 10.1016/j.amc.2019.124593

Access Statistics for this article

Applied Mathematics and Computation is currently edited by Theodore Simos

More articles in Applied Mathematics and Computation from Elsevier
Bibliographic data for series maintained by Catherine Liu ().