  The discontinuous Galerkin (DG) finite element method when applied to
hyperbolic conservation laws requires the use of shock-capturing limiters in
order to suppress unphysical oscillations near large solution gradients. In
this work we develop a novel shock-capturing limiter that combines key ideas
from the limiter of Barth and Jespersen [AIAA-89-0366 (1989)] and the maximum
principle preserving (MPP) framework of Zhang and Shu [Proc. R. Soc. A, 467
(2011), pp. 2752--2776]. The limiting strategy is based on traversing the mesh
element-by-element in order to (1) find local upper and lower bounds on
user-defined variables by sampling these variables on neighboring elements, and
(2) to then enforce these local bounds by minimally damping the high-order
corrections. The main advantages of this limiting strategy is that it is simple
to implement, effective at shock capturing, and retains high-order accuracy of
the solution in smooth regimes. The resulting numerical scheme is applied to
several standard numerical tests in both one and two-dimensions and on both
Cartesian and unstructured grids. These tests are used as benchmarks to verify
and assess the accuracy and robustness of the method.
