INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int. J. Numer. Meth. Engng 2010; 81:805–834 Published online 31 July 2009 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nme.2708 An essentially non-oscillatory semi-Lagrangian method for tidal flow simulations Mofdi El-Amrani 1, ∗, † and Mohammed Sea¨ ıd 2 1 Departamento de Matem´ atica Aplicada, Universidad Rey Juan Carlos, 28933 M´ ostoles-Madrid, Spain 2 School of Engineering, University of Durham, South Road, Durham DH1 3LE, U.K. SUMMARY We develop an essentially non-oscillatory semi-Lagrangian method for solving two-dimensional tidal flows. The governing equations are derived from the incompressible Navier–Stokes equations with assumptions of shallow water flows including bed frictions, eddy viscosity, wind shear stresses and Coriolis forces. The method employs the modified method of characteristics to discretize the convective term in a finite element framework. Limiters are incorporated in the method to reconstruct an essentially non-oscillatory algorithm at minor additional cost. The central idea consists in combining linear and quadratic interpolation procedures using nodes of the finite element where departure points are localized. The resulting semi- discretized system is then solved by an explicit Runge–Kutta Chebyshev scheme with extended stages. This scheme adds in a natural way a stabilizing stage to the conventional Runge–Kutta method using the Chebyshev polynomials. The proposed method is verified for the recirculation tidal flow in a channel with forward-facing step. We also apply the method for simulation of tidal flows in the Strait of Gibraltar. In both test problems, the proposed method demonstrates its ability to handle the interaction between water free-surface and bed frictions. Copyright 2009 John Wiley & Sons, Ltd. Received 12 November 2008; Revised 28 May 2009; Accepted 16 June 2009 KEY WORDS: tidal flows; shallow water equations; semi-Lagrangian method; finite elements; non- oscillatory interpolation; Runge–Kutta Chebyshev; Strait of Gibraltar 1. INTRODUCTION Mathematical modelling of tidal flows in water systems is based on the formulation and solution of the appropriate equations of continuity and motion of water. In general, tidal flows represent a three-dimensional turbulent Newtonian flow in complicated geometrical domains. The costs of ∗ Correspondence to: Mofdi El-Amrani, Departamento de Matem´ atica Aplicada, Universidad Rey Juan Carlos, 28933 M´ ostoles-Madrid, Spain. † E-mail: mofdi.elamrani@urjc.es Contract/grant sponsor: Agencia Espa˜ nola de Cooperaci´ on Internacional (AECI); contract/grant number: A/7346/06 Copyright 2009 John Wiley & Sons, Ltd.