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

A continuous/discontinuous Galerkin solution of the shallow water  equations with dynamic viscosity, high-order wetting and drying, and implicit  time integration