INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS Int. J. Numer. Meth. Fluids 2009; 59:423–442 Published online 17 June 2008 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/fld.1823 An improvement of classical slope limiters for high-order discontinuous Galerkin method R. Ghostine 1, ∗, † , G. Kesserwani 1 , R. Mos´ e 1 , J. Vazquez 1 and A. Ghenaim 2 1 U.P.R. Syst` emes Hydrauliques Urbains, Ecole Nationale du G´ enie de l’Eau et de l’Environnement de Strasbourg, 1 quai Koch BP 61039 F, 67070 Strasbourg Cedex, France 2 INSA, Institut National des Sciences Appliqu´ ees, 24 boulevard de la Victoire, 67084 Strasbourg Cedex, France SUMMARY In this paper, we describe some existing slope limiters (Cockburn and Shu’s slope limiter and Hoteit’s slope limiter) for the two-dimensional Runge–Kutta discontinuous Galerkin (RKDG) method on arbitrary unstructured triangular grids. We describe the strategies for detecting discontinuities and for limiting spurious oscillations near such discontinuities, when solving hyperbolic systems of conservation laws by high-order discontinuous Galerkin methods. The disadvantage of these slope limiters is that they depend on a positive constant, which is, for specific hydraulic problems, difficult to estimate in order to eliminate oscillations near discontinuities without decreasing the high-order accuracy of the scheme in the smooth regions. We introduce the idea of a simple modification of Cockburn and Shu’s slope limiter to avoid the use of this constant number. This modification consists in: slopes are limited so that the solution at the integration points is in the range spanned by the neighboring solution averages. Numerical results are presented for a nonlinear system: the shallow water equations. Four hydraulic problems of discontinuous solutions of two-dimensional shallow water are presented. The idealized dam break problem, the oblique hydraulic jump problem, flow in a channel with concave bed and the dam break problem in a converging– diverging channel are solved by using the different slope limiters. Numerical comparisons on unstructured meshes show a superior accuracy with the modified slope limiter. Moreover, it does not require the choice of any constant number for the limiter condition. Copyright 2008 John Wiley & Sons, Ltd. Received 23 November 2007; Revised 12 March 2008; Accepted 13 March 2008 KEY WORDS: discontinuous Galerkin method; two-dimensional shallow water equations; slope limiter; steady; transient; unstructured grids 1. INTRODUCTION The success of the discontinuous Galerkin methods in approximating various physical prob- lems, notably hyperbolic systems of conservative laws, has attracted the hydraulic engineering ∗ Correspondence to: R. Ghostine, U.P.R. Syst` emes Hydrauliques Urbains, Ecole Nationale du G´ enie de l’Eau et de l’Environnement de Strasbourg, 1 quai Koch BP 61039 F, 67070 Strasbourg Cedex, France. † E-mail: rabih.ghostine@engees.u-strasbg.fr Copyright 2008 John Wiley & Sons, Ltd.