An improvement of classical slope limiters for high‐order discontinuous Galerkin method

Abstract

SUMMARYIn 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 convergingdiverging 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.

A discontinuous Galerkin method for solving the fluid and magnetohydrodynamic equations in astrophysical simulations

“…with slope limiting, flux limiting, shock capturing, or weighted essentially non-oscillatory (WENO) approaches) (Luo, Baum & Löhner 2008), which is also an issue for FV methods. Some DG formulations are found to be more sensitive to certain shock limiters than their FV counterparts, but techniques exist to prevent unphysical oscillatory solutions in high-order DG methods (Hoteit et al 2004;Ghostine et al 2009;Luo, Baum & Löhner 2008).Our DG implementation falls into the class of centroidal Taylor basis procedures developed by Luo, Baum & Löhner (2008). This formulation of DG is relatively new and is quite different from the more widespread approach that employs nodal basis value functions (which would not be generalizable to a moving Voronoi c 2013 RAS…”

“…with slope limiting, flux limiting, shock capturing, or weighted essentially non-oscillatory (WENO) approaches) (Luo, Baum & Löhner 2008), which is also an issue for FV methods. Some DG formulations are found to be more sensitive to certain shock limiters than their FV counterparts, but techniques exist to prevent unphysical oscillatory solutions in high-order DG methods (Hoteit et al 2004;Ghostine et al 2009;Luo, Baum & Löhner 2008).…”

An eigenvector‐based linear reconstruction approach for time stepping in discontinuous Galerkin scheme used to solve shallow water equations

“…The 2D shallow water equations are solved using a DG finite element method [11,16,17]. The domain is discretised into triangular computational cells represented in Figure 1.…”

Adaptive hierarchic transformations for dynamically $p$-enriched slope-limiting over discontinuous Galerkin systems of generalized equations

“…For example, in a hierarchical basis (as shown explicitly below) the degrees of freedom grow nonlinearly as a function of p and each degree of freedom ends up carrying information of potentially pathological (or undesirable) overshoots and undershoots which have developed over the native (or non-limited) solution space. It turns out that this complication introduces a substantial technical difficulty in practice, which many papers on numerical shock capturing [1,6,7,12,14,16,25,27,28] tend to avoid addressing directly. Most noteworthy is the observation that slopelimiters tend to limit the coefficients in their chosen basis independently of each other, in the sense that each component is adjusted based on information about the surrounding solution on a relatively local submanifold of the total domain.…”