  We introduce a general framework for enforcing local or global maximum
principles in high-order space-time discretizations of a scalar hyperbolic
conservation law. We begin with sufficient conditions for a space
discretization to be bound preserving (BP) and satisfy a semi-discrete maximum
principle. Next, we propose a global monolithic convex (GMC) flux limiter which
has the structure of a flux-corrected transport (FCT) algorithm but is
applicable to spatial semi-discretizations and ensures the BP property of the
fully discrete scheme for strong stability preserving (SSP) Runge-Kutta time
discretizations. To circumvent the order barrier for SSP time integrators, we
constrain the intermediate stages and/or the final stage of a general
high-order RK method using GMC-type limiters. In this work, our theoretical and
numerical studies are restricted to explicit schemes which are provably BP for
sufficiently small time steps. The new GMC limiting framework offers the
possibility of relaxing the bounds of inequality constraints to achieve higher
accuracy at the cost of more stringent time step restrictions. The ability of
the presented limiters to preserve global bounds and recognize well-resolved
smooth solutions is verified numerically for three representative RK methods
combined with weighted essentially nonoscillatory (WENO) finite volume space
discretizations of linear and nonlinear test problems in 1D.
