An efficient Eulerian finite element method for the shallow water equations Emmanuel Hanert a,b, * , Daniel Y. Le Roux c , Vincent Legat b , Eric Deleersnijder a a Institut d’Astronomie et de Ge ´ ophysique G. Lemaı ˆtre, Universite ´ Catholique de Louvain, 2 Chemin du Cyclotron, B-1348 Louvain-la-Neuve, Belgium b Centre for Systems Engineering and Applied Mechanics, Universite ´ Catholique de Louvain, 4 Avenue Georges Lemaı ˆtre, B-1348 Louvain-la-Neuve, Belgium c De ´partement de Mathe ´matiques et de Statistique, Universite ´ Laval, Que ´bec, QC, Canada G1K 7P4 Received 14 November 2003; received in revised form 2 March 2004; accepted 17 June 2004 Available online 3 November 2004 Abstract The accuracy and efficiency of an Eulerian method is assessed by solving the non-linear shallow water equations and compared with the performances of an existing semi-Lagrangian method. Both methods use a linear non-conforming finite element discretization for velocity and a linear conforming finite element discretization for surface elevation. This finite element pair is known to be computationally efficient and free of pressure modes. The model equations are carefully derived and a comparison is performed by sim- ulating the propagation of slow Rossby waves in the Gulf of Mexico. Simulations show that the Eulerian model performs well and gives results comparable to high order semi-Lagrangian schemes using kriging interpolators. Ó 2004 Elsevier Ltd. All rights reserved. Keywords: Finite elements; Euleurian; Semi-Lagrangian; Shallow water equations; Rossby waves; Non-conforming linear interpolation; Kriging 1463-5003/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.ocemod.2004.06.006 * Corresponding author. Address: Institut dÕAstronomie et de Ge ´ophysique G. Lemaı ˆtre, Universite ´ Catholique de Louvain, 2 Chemin du Cyclotron, B-1348 Louvain-la-Neuve, Belgium. Tel.: +32 10 472354. E-mail address: hanert@astr.ucl.ac.be (E. Hanert). www.elsevier.com/locate/ocemod Ocean Modelling 10 (2005) 115–136