New two‐dimensional slope limiters for discontinuous Galerkin methods on arbitrary meshes

Abstract

SUMMARYIn this paper, we introduce an extension of Van Leer's slope limiter for two-dimensional discontinuous Galerkin (DG) methods on arbitrary unstructured quadrangular or triangular grids. The aim is to construct a non-oscillatory shock capturing DG method for the approximation of hyperbolic conservative laws without adding excessive numerical dispersion. Unlike some splitting techniques that are limited to linear approximations on rectangular grids, in this work, the solution is approximated by means of piecewise quadratic functions. The main idea of this new reconstructing and limiting technique follows a well-known approach where local maximum principle regions are defined by enforcing some constraints on the reconstruction of the solution. Numerical comparisons with some existing slope limiters on structured as well as on unstructured meshes show a superior accuracy of our proposed slope limiters.

Computational extended magneto-hydrodynamical study of shock structure generated by flows past an obstacle

“…There are small oscillations near the shock region, which are expected since positivity-preserving limiter (PP-limiter) is the only limiter used in this scheme, which does not smooth away oscillations as a total variation diminishing limiter (TVD) would. The use of a slope limiter 13 can remove all the oscillations. No noticeable difference is found using a finer mesh, which implies the scheme is converged with a resolution as low as 400 cells in each dimension.…”

Application of a Higher Order Discontinuous Galerkin

“…Oscillations in high order DG schemes can be cured by using slope limiters in the scheme and in recent years a number of authors have contributed to the issue of limiters for DG schemes (e.g. see [8,14,22,24,31,48]). One promising direction of research is design of DG limiters based on weighted essentially non-oscillatory (WENO) finite volume methodology.…”

Modélisation du transport de soluté en milieux poreux par la méthode d’éléments finis mixtes hybrides – développement d’un limiteur de flux

“…Dans la littérature, les auteurs recourent à la technique d'opérateurs de spliting. Les EFMH sont appliqués pour la résolution du terme dispersif de l'équation de transport, alors que la méthode des éléments finis discontinue est appliquée pour le terme convectif; avec un limiteur de pente qui permet de réduire les oscillations HOTEIT et al, 2002;HOTEIT et al, 2004;MAZZIA et al, 2002;OLTEAN et BUÈS, 2001;SIEGEL et al, 1997). D'après CARRAYROU et al (2005), chaque méthode introduit une erreur intrinsèque dans la solution.…”