  In this work we present a framework for enforcing discrete maximum principles
in discontinuous Galerkin (DG) discretizations. The developed schemes are
applicable to scalar conservation laws as well as hyperbolic systems. Our
methodology for limiting volume terms is similar to recently proposed methods
for continuous Galerkin approximations, while DG flux terms require novel
stabilization techniques. Piecewise Bernstein polynomials are employed as shape
functions for the DG spaces, thus facilitating the use of very high order
spatial approximations. We discuss the design of a new, provably invariant
domain preserving DG scheme that is then extended by state-of-the-art subcell
flux limiters to obtain a high-order bound preserving approximation. The
limiting procedures can be formulated in the semi-discrete setting. Thus
convergence to steady state solutions is not inhibited by the algorithm. We
present numerical results for a variety of benchmark problems. Conservation
laws considered in this study are linear and nonlinear scalar problems, as well
as the Euler equations of gas dynamics and the shallow water system.
