A Comparison of Continuous and Discontinuous Galerkin Algorithms for Shallow Water Transport

ADCIRC (ADvanced CIRCulation), developed over the last 20 years, is a hydrodynamic model capable of simulating water surface elevation and velocity fields in lakes, bays, estuaries, and oceans. The model is based on the full non-linear St. Venant (shallow water) equations, using the traditional hydrostatic pressure and Boussinesq approximations; the equations are discretized in space using linear finite elements, while time is discretized using an efficient split-step Crank-Nicolson algorithm. Model features include the following: both 2D depth integrated and 3D modes; linear or fully nonlinear equations; Cartesian or spherical coordinates; consistent or lumped mass matrices; Dirichlet, flux, or radiation boundary conditions; surface stress, flux, and tidal potential forcings; least squares analysis of harmonic constituents; wetting/drying of near-shore elements; fully-implicit time marching and, a preprocessor to optimize the code for various computer architectures. Over the years, the ADCIRC model has been supported by a number of federal and state agencies, with applications ranging from coastal dredging to larval transport to Naval fleet operations to hurricane storm surges. Recently, a transport algorithm has been added so that it is capable of prognostic baro-clinic simulations. Herein, we compare the performance of continuous and discontinuous Galerkin ("CG" and "DG," respectively) transport codes within the framework of the continuous hydrodynamic model; more specifically, the new codes are verified against known analytical solutions, examined for stability and convergence rates, and validated against experimental data. Results from verification studies show that both codes work well for low Peclet numbers (second order convergence), but only the DG code can capture the shock at high Peclet numbers. Results from the validation study show that both codes are able to capture the overall behavior of the "lock-exchange" experiment, although the DG is slightly more accurate. Parameter studies in the context of this lock-exchange experiment show that simulation results (both codes) are most sensitive to the value of the horizontal eddy viscosity. Global mass balance errors for both codes are much less than 1%.