  This work is devoted to the design of interior penalty discontinuous Galerkin
(dG) schemes that preserve maximum principles at the discrete level for the
steady transport and convection-diffusion problems and the respective transient
problems with implicit time integration. Monotonic schemes that combine
explicit time stepping with dG space discretization are very common, but the
design of such schemes for implicit time stepping is rare, and it had only been
attained so far for 1D problems. The proposed scheme is based on an artificial
diffusion that linearly depends on a shock detector that identifies the
troublesome areas. In order to define the new shock detector, we have
introduced the concept of discrete local extrema. The diffusion operator is a
graph-Laplacian, instead of the more common finite element discretization of
the Laplacian operator, which is essential to keep monotonicity on general
meshes and in multi-dimension. The resulting nonlinear stabilization is
non-smooth and nonlinear solvers can fail to converge. As a result, we propose
a smoothed (twice differentiable) version of the nonlinear stabilization, which
allows us to use Newton with line search nonlinear solvers and dramatically
improve nonlinear convergence. A theoretical numerical analysis of the proposed
schemes show that they satisfy the desired monotonicity properties. Further,
the resulting operator is Lipschitz continuous and there exists at least one
solution of the discrete problem, even in the non-smooth version. We provide a
set of numerical results to support our findings.
