INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS Int. J. Numer. Meth. Fluids 2008; 1:1 Prepared using fldauth.cls [Version: 2002/09/18 v1.01] Practical evaluation of five partly-discontinuous finite element pairs for the non-conservative shallow water equations Richard Comblen 1, ∗ , Jonathan Lambrechts 1 , Jean-Fran¸ cois Remacle 1 , Vincent Legat 1 1 Centre for Systems Engineering and Applied Mechanics, Universit´ e catholique de Louvain, 4 Avenue Georges Lemaˆ ıtre, B-1348 Louvain-la-Neuve, Belgium SUMMARY This paper provides a comparison of five finite element pairs for the shallow water equations. We consider continuous, discontinuous and partially discontinuous finite element formulations that are supposed to provide second order spatial accuracy. All of them rely on the same weak formulation, using Riemann solver to evaluate interface integrals. We define several asymptotic limit cases of the shallow water equations within their space of parameters. The idea is to develop a comparison of these numerical schemes in several relevant regimes of the subcritical shallow water flow. Finally, a new pair, using non-conforming linear elements for both velocities and elevation (P NC 1 -P NC 1 ), is presented, giving optimal rates of convergence in all test cases. P NC 1 -P1 and P DG 1 -P1 mixed formulations lacks of convergence for inviscid flows. P DG 1 -P2 pair is more expensive but provides accurate results for all benchmarks. P DG 1 -P DG 1 provides an efficient option, except for inviscid Coriolis-dominated flows, where a small lack of convergence is observed. Copyright c  2008 John Wiley & Sons, Ltd. key words: finite element, shallow water equations, discontinuous Galerkin, non-conforming element, Riemann solver, convergence 1. Introduction The shallow water equations are a classical model used in a wide area of physics and engineering. They govern flows in estuaries, enable modeling of dam-breaks, floods and tides, and are a key building block for ocean modeling as well as atmosphere modeling. Different numerical methods have been designed for the shallow water equations. Finite volumes are very popular for small scale applications as well as atmosphere modeling, whereas ocean models are mainly based on finite difference methods [1], as described for instance in the book [2]. In the finite element framework, major contributions have been developed with both discontinuous and continuous elements. The Discontinuous Galerkin (DG) method focuses growing interest since the late nineties, and gives accurate results for hyperbolic conservation laws. Basically, it consists in a volume * Correspondence to: richard.comblen@uclouvain.be – Tel.: +32 10 47 23 57 – fax: +32 10 47 21 80. Copyright c  2008 John Wiley & Sons, Ltd.