M. E. Vázquez-Cendón

Unstructured finite volume discretisation of bed friction and convective flux in solute transport models linked to the shallow water equations

The finite volume discretisation of the shallow water equations has been the subject of many previous studies, most of which deal with a well-balanced conservative discretisation of the convective flux and bathymetry. However, the bed friction discretisation has not been so profusely analysed in previous works, while it may play a leading role in certain applications of shallow water models. In this paper we analyse the numerical discretisation of the bed friction term in the two-dimensional shallow water equations, and we propose a new unstructured upwind finite volume discretisation for this term. The new discretisation proposed improves the accuracy of the model in problems in which the bed friction is a relevant force in the momentum equation, and it guarantees a perfect balance between gravity and bed friction under uniform flow conditions. The relation between the numerical scheme used to solve the hydrodynamic equations and the scheme used to solve a scalar transport model linked to the shallow water equations, is also analysed in the paper. It is shown that the scheme used in the scalar transport model must take into consideration the scheme used to solve the hydrodynamic equations. The most important implication is that a well-balanced and conservative scheme for the scalar transport equation cannot be formulated just from the water depth and velocity fields, but has to consider also the way in which the hydrodynamic equations have been solved.

A projection hybrid finite volume/element method for low-Mach number flows

The purpose of this article is to introduce a projection hybrid finite volume/element method for low-Mach number flows of viscous or inviscid fluids. Starting with a 3D tetrahedral finite element mesh of the computational domain, the equation of the transport-diffusion stage is discretized by a finite volume method associated with a dual mesh where the nodes of the volumes are the barycenters of the faces of the initial tetrahedra. The transport-diffusion stage is explicit. Upwinding of convective terms is done by classical Riemann solvers as the Q-scheme of van Leer or the Rusanov scheme. Concerning the projection stage, the pressure correction is computed by a piecewise linear finite element method associated with the initial tetrahedral mesh. Passing the information from one stage to the other is carefully made in order to get a stable global scheme. Numerical results for several test examples aiming at evaluating the convergence properties of the method are shown.

Design and analysis of ADER-type schemes for model advection–diffusion–reaction equations

Abstract We construct, analyze and assess various schemes of second order of accuracy in space and time for model advection–diffusion–reaction differential equations. The constructed schemes are meant to be of practical use in solving industrial problems and are derived following two related approaches, namely ADER and MUSCL-Hancock. Detailed analysis of linear stability and local truncation error are carried out. In addition, the schemes are implemented and assessed for various test problems. Empirical convergence rate studies confirm the theoretically expected accuracy in both space and time.

A projection hybrid high order finite volume/finite element method for incompressible turbulent flows

Abstract In this paper the projection hybrid FV/FE method presented in [1] is extended to account for species transport equations. Furthermore, turbulent regimes are also considered thanks to the k–e model. Regarding the transport diffusion stage new schemes of high order of accuracy are developed. The CVC Kolgan-type scheme and ADER methodology are extended to 3D. The latter is modified in order to profit from the dual mesh employed by the projection algorithm and the derivatives involved in the diffusion term are discretized using a Galerkin approach. The accuracy and stability analysis of the new method are carried out for the advection–diffusion–reaction equation. Within the projection stage the pressure correction is computed by a piecewise linear finite element method. Numerical results are presented, aimed at verifying the formal order of accuracy of the scheme and to assess the performance of the method on several realistic test problems.

A Projection Hybrid Finite Volume-ADER/Finite Element Method for Turbulent Navier-Stokes

We present a second order finite volume/finite element projection method for low-Mach number flows. Moreover, transport of species law is also considered and turbulent regime is solved using a k −ɛ standard model. Starting with a 3D tetrahedral finite element mesh of the computational domain, the momentum equation is discretized by a finite volume method associated with a dual finite volume mesh where the nodes of the volumes are the barycenter of the faces of the initial tetrahedra. The resolution of Navier-Stokes equations coupled with a k −ɛ turbulence model requires the use of a high order scheme. The ADER methodology is extended to compute the flux terms with second order accuracy in time and space. Finally, the order of convergence is analysed by means of academic problems and some numerical results are presented.