  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 (P1) approximation. Since the piecewise-constant (P0) 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 well-defined
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-P1 discretizations of two-dimensional
test problems. In the context of hp-adaptive DG methods, the new limiting
procedures can be used in P1 subcells of macroelements marked as `troubled' by
a smoothness indicator.
