Continuous, discontinuous and coupled discontinuous–continuous Galerkin finite element methods for the shallow water equations

We consider the approximation of the depth‐averaged two‐dimensional shallow water equations by both a traditional continuous Galerkin (CG) finite element method as well as two discontinuous Galerkin (DG) approaches. The DG method is locally conservative, flux‐continuous on each element edge, and is suitable for both smooth and highly advective flows. A novel technique of coupling a DG method for continuity with a CG method for momentum is developed. This formulation is described in detail and validation via numerical testing is presented. Comparisons between a widely used CG approach, a conventional DG method, and the novel coupled discontinuous–continuous Galerkin method illustrates advantages and disadvantages in accuracy and efficiency.