Sediment transport models in Shallow Water equations and numerical approach by high order finite volume methods q M.J. Castro Dı ´az a , E.D. Ferna ´ndez-Nieto b, * , A.M. Ferreiro c a Dpto. Ana ´ lisis Matema ´ tico, Universidad de Ma ´ laga, Campus Teatinos s/n, Ma ´ laga, Spain b Departamento de Matema ´ tica Aplicada I, Universidad de Sevilla, 41012 Sevilla, Spain c Dpto. Ana ´ lisis Ecuaciones Diferenciales y Ana ´ lisis Nume ´rico, Universidad de Sevilla, C/ Tarfia S/N, Sevilla, Spain Received 5 May 2006; received in revised form 30 October 2006; accepted 10 July 2007 Available online 29 September 2007 Abstract This paper is concerned with the numerical approximation of bedload sediment transport due to water evolution. For the hydrody- namical component we consider Shallow Water equations. The morphodynamical component is defined by a continuity equation, which is defined in function of the solid transport discharge. We present several deterministic models, such as Meyer-Peter & Mu ¨ller, Van Rijn or Grass model. We also present an unified definition for the solid transport discharge, and we compare with Grass model. Both com- ponents define a coupled system of equations that can be rewrite as a non-conservative hyperbolic system. To discretize it, we consider finite volume methods with or without flux limiters and high order state reconstructions. Finally we present several tests, where we observe numerically the order of the numerical schemes. Comparisons with analytical solutions and experimental data are also presented. Ó 2007 Elsevier Ltd. All rights reserved. 1. Introduction The sediment can be defined as a fragmented material from rocks that has been formed by different physical and/or chemical process. The study of sediment transport processes includes movement of rocks in a mountain as material diffusion in water, among other processes. Trans- port is caused by gravity effects and by friction effects with the air or the fluid containing the sediment. Sediment transport is usually divided into three types: bedload, saltation and suspension (see Fig. 1). Bedload transport is defined as the type of transport where sediment grains roll or slide along the bed. Saltation transport is defined as the type of transport where single grains jump over the bed a length proportional to their diameter, losing for instants the contact with the soil. Sediment is suspended when the flux is intense enough such as the sediment grains reach height over the bed. In this paper we face the study of bedload sediment transport. To model bedload sediment transport process caused by the movement of a fluid in contact with the sed- iment layer, we consider a coupled model constituted by a hydrodynamical component and a morphodynamical component. The hydrodynamical component is modeled by Shallow Water equations, which is are used to study fluid move- ment in rivers, channel, coast areas, etc., while a sediment transport equation, depending on solid transport flux, is considered to model the morphodynamical component. In literature different equations to model the solid trans- port sediment flux could be found: (Grass equation [25], Meyer-Peter & Mu ¨ ller’s equation [38], Van Rijn’s equation [54–56], Nielsen’s equation [39], Kalinske [32,33], Einstein’s equation [18,19,31,60], etc., generally obtained by empirical methods. Among all these formulae, some are deterministic formulae and others are based on probabilistic terms. In this paper we only consider deterministic equations (see Section 2.3). 0045-7930/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.compfluid.2007.07.017 q This research was partially supported by Spanish Government Research Projects BFM2003-07530-C02-01 and BFM2003-07530-C02-02. * Corresponding author. E-mail addresses: castro@anamat.cie.uma.es (M.J. Castro Dı ´az), edofer@us.es (E.D. Ferna ´ndez-Nieto), anafefe@us.es (A.M. Ferreiro). www.elsevier.com/locate/compfluid Available online at www.sciencedirect.com Computers & Fluids 37 (2008) 299–316