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. Copyright © 2006 John Wiley & Sons, Ltd.