  The high-order numerical solution of the non-linear shallow water equations
(and of hyperbolic systems in general) is susceptible to unphysical Gibbs
oscillations that form in the proximity of strong gradients. The solution to
this problem is still an active field of research as no general cure has been
found yet. In this paper, we tackle this issue by presenting a dynamically
adaptive viscosity based on a residual-based sub-grid scale model that has the
following properties: $(i)$ it removes the spurious oscillations in the
proximity of strong wave fronts while preserving the overall accuracy and
sharpness of the solution. This is possible because of the residual-based
definition of the dynamic diffusion coefficient. $(ii)$ For coarse grids, it
prevents energy from building up at small wave-numbers. $(iii)$ The model has
no tunable parameter. Our interest in the shallow water equations is tied to
the simulation of coastal inundation, where a careful handling of the
transition between dry and wet surfaces is particularly challenging for
high-order Galerkin approximations. In this paper, we extend to a unified
continuous/discontinuous Galerkin (CG/DG) framework a very simple, yet
effective wetting and drying algorithm originally designed for DG [Xing, Zhang,
Shu (2010)]. We show its effectiveness for problems in one and two dimensions
on domains of increasing characteristic lengths varying from centimeters to
kilometers. Finally, to overcome the time-step restriction incurred by the
high-order Galerkin approximation, we advance the equations forward in time via
a three stage, second order explicit-first-stage, singly diagonally implicit
Runge-Kutta (ESDIRK) time integration scheme. Via ESDIRK, we are able to
preserve numerical stability for an advective CFL number 10 times larger than
its explicit counterpart.
