A new perspective on flux and slope limiting in discontinuous Galerkin methods for hyperbolic conservation laws

Abstract

In this work, we discuss and develop multidimensional limiting techniques for discontinuous Galerkin (DG) discretizations of scalar hyperbolic problems. To ensure that each cell average satisfies a local discrete maximum principle (DMP), we impose inequality constraints on the local Lax-Friedrichs fluxes of a piecewise-linear (P 1 ) approximation. Since the piecewise-constant (P 0 ) version corresponds to a property-preserving low-order finite volume method, the validity of DMP conditions can always be enforced using slope and/or flux limiters. We show that the (currently rather uncommon) use of direct flux limiting makes it possible to construct more accurate DMP-satisfying approximations in which a weak form of slope limiting is used to prevent unbounded growth of solution gradients. Moreover, both fluxes and slopes can be limited in a manner which produces nonlinear problems with welldefined residuals even at steady state. We derive/present slope limiters based on different kinds of inequality constraints, discuss their properties and introduce new anisotropic limiters for problems that require different treatment of different space directions. At the flux limiting stage, the anisotropy of the problem at hand can be taken into account by using a customized definition of local bounds for the DMP constraints. At the slope limiting stage, we adjust the magnitude of individual directional derivatives using low-order reconstructions from cell averages to define the bounds. In this way, we avoid unnecessary limiting of well-resolved derivatives at smooth peaks and in internal/boundary layers. The properties of selected algorithms are explored in numerical studies for DG-P 1 discretizations of two-dimensional test problems. In the context of hp-adaptive DG methods, the new limiting procedures can be used in P 1 subcells of macroelements marked as 'troubled' by a smoothness indicator.