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For tidal-stream energy to become a competitive renewable energy source, clustering multiple turbines into arrays is paramount. Array optimisation is thus critical for achieving maximum power performance and reducing cost of energy. However, ascertaining an optimal array layout is a complex problem, subject to specific site hydrodynamics and multiple inter-disciplinary constraints. In this work, we present a novel optimisation approach that combines an analytical-based wake model, FLORIS, with an ocean model, Thetis. The approach is demonstrated through applications of increasing complexity. By utilising the method of analytical wake superposition, the addition or alteration of turbine position does not require re-calculation of the entire flow field, thus allowing the use of simple heuristic techniques to perform optimisation at a fraction of the computational cost of more sophisticated methods. Using a custom condition-based placement algorithm, this methodology is applied to the Pentland Firth for arrays with turbines of $$3.05\,\hbox {m}/\hbox {s}$$ 3.05 m / s rated speed, demonstrating practical implications whilst considering the temporal variability of the tide. For a 24-turbine array case, micro-siting using this technique delivered an array 15.8% more productive on average than a staggered layout, despite flow speeds regularly exceeding the rated value. Performance was evaluated through assessment of the optimised layout within the ocean model that treats turbines through a discrete turbine representation. Used iteratively, this methodology could deliver improved array configurations in a manner that accounts for local hydrodynamic effects.

The present work is devoted to the derivation of a fully well-balanced and positivepreserving numerical scheme for the shallow water equations with Coriolis force. The first main issue consists in preserving all the steady states, including the geostrophic equilibrium. Our strategy relies on a Godunov-type scheme with suitable source term and steady state discretisations. The second challenge lies in improving the order of the scheme while preserving the fully well-balanced property. A modification of the classical methods is required since no conservative reconstruction can preserve all the steady states in the case of rotating shallow water equations. A steady state detector is used to overcome this matter. Some numerical experiments are presented to show the relevance and the accuracy of both first-order and second-order schemes.

Within the finite element community, discontinuous Galerkin (DG) and mixed finite element methods have become increasingly popular in simulating geophysical flows. However, robust and efficient solvers for the resulting saddle-point and elliptic systems arising from these discretizations continue to be an on-going challenge. One possible approach for addressing this issue is to employ a method known as hybridization, where the discrete equations are transformed such that classic static condensation and local post-processing methods can be employed. However, it is challenging to implement hybridization as performant parallel code within complex models, whilst maintaining separation of concerns between applications scientists and software experts. In this paper, we introduce a domain-specific abstraction within the Firedrake finite element library that permits the rapid execution of these hybridization techniques within a code-generating framework. The resulting framework composes naturally with Firedrake's solver environment, allowing for the implementation of hybridization and static condensation as runtime-configurable preconditioners via the Python interface to PETSc, petsc4py. We provide examples derived from second order elliptic problems and geophysical fluid dynamics. In addition, we demonstrate that hybridization shows great promise for improving the performance of solvers for mixed finite element discretizations of equations related to large-scale geophysical flows., Comment: Revisions for submission to GMD

We present the Lie symmetry analysis for a hyperbolic partial differential system known as the one-dimensional Saint-Venant-Exner model. The model describes shallow-water systems with bed evolution given by the Exner terms. The sediment flux is considered to be a power-law function of the velocity of the fluid. The admitted Lie symmetries are classified according to the power index of the sediment flux. Furthermore, the one-dimensional optimal system is determined in all cases. From the Lie symmetries we derive similarity transformations which are applied to reduce the hyperbolic system into a set of ordinary differential equations. Closed-form exact solutions, which have not been presented before in the literature, are presented. Finally, the initial value problem for the similarity solutions is discussed.

International audience; In this communication, we consider a numerical scheme for the shallow-water system. The scheme under consideration has been proven to preserve the positivity of the water height and to be fully well-balanced, i.e. to exactly preserve the smooth moving steady state solutions of the shallow-water equations with the topography source term. The goal of this work is to prove a discrete entropy inequality satisfied by this scheme.; Dans cette communication, nous nous intéressons à un schéma numérique pour le système de Saint-Venant. Nous avons prouvé que ce schéma préservait la positivité de la hauteur de l'eau et qu'il était complètement équilibre, c'est-à-dire qu'il préservait exactement les solutions stationnaires régulières à vitesse non nulle des équations de Saint-Venant avec terme source de topographie. Le but de ce travail est de prouver une inégalité d'entropie discrète satisfaite par ce schéma.

Sparse spectral methods for solving partial differential equations have been derived in recent years using hierarchies of classical orthogonal polynomials on intervals, disks, and triangles. In this work we extend this methodology to a hierarchy of non-classical orthogonal polynomials on disk slices (e.g. a half-disk) and trapeziums. This builds on the observation that sparsity is guaranteed due to the boundary being defined by an algebraic curve, and that the entries of partial differential operators can be determined using formulae in terms of (non-classical) univariate orthogonal polynomials. We apply the framework to solving the Poisson, variable coefficient Helmholtz, and Biharmonic equations., 38 pages, 7 figures

The spectral/hp element method combines the geometric flexibility of the classical h-type finite element technique with the desirable numerical properties of spectral methods, employing high-degree piecewise polynomial basis functions on coarse finite element-type meshes. The spatial approximation is based upon orthogonal polynomials, such as Legendre or Chebychev polynomials, modified to accommodate a C 0 - continuous expansion. Computationally and theoretically, by increasing the polynomial order p, high-precision solutions and fast convergence can be obtained and, in particular, under certain regularity assumptions an exponential reduction in approximation error between numerical and exact solutions can be achieved. This method has now been applied in many simulation studies of both fundamental and practical engineering flows. This paper briefly describes the formulation of the spectral/hp element method and provides an overview of its application to computational fluid dynamics. In particular, it focuses on the use of the spectral/hp element method in transitional flows and ocean engineering. Finally, some of the major challenges to be overcome in order to use the spectral/hp element method in more complex science and engineering applications are discussed.

Network flows and specifically water flow in open canals can be modeled by systems of balance laws defined on graphs. The shallow water or Saint-Venant system of balance laws is one of the most used model and present two phases: fluvial or sub-critical and torrential or super-critical. Phase transitions may occur within the same canal but transitions related to networks are less investigated. In this paper we provide a complete characterization of possible phase transitions for a case study of a simple scenario with two canals and one junction. However, our analysis allows the study of more complicate networks. Moreover, we provide some numerical simulations to show the theory at work.

The problem of a travelling wave in a semi-infinite channel over an arbitrary quasi-flat bathymetry is studied in the shallow-water formulation. It is shown how the streamfunction can be cast, in the vicinity of an elliptic equilibrium for the fluid flow, in the form of a nearly-integrable non-autonomous Hamiltonian with a perturbation which is aperiodic in time. The obtained model provides a natural example in the context of the aperiodically time-dependent Hamiltonian systems studied in Fortunati and Wiggins (2016) and the local stability of the perturbed streamlines is obtained as a consequence of this theory. The proofs use the tools of Perturbation Theory in the real-analytic setting.

Friction decoupling, i.e. the computation of friction vector components making separate use of the corresponding velocity components, is common in staggered grid models of the SWE simplifications (Zero-Inertia and Local Inertia Approximation), due to the programming simplicity and to the consequent calculations speed-up. In the present paper, the effect of friction decoupling has been studied from the theoretical and numerical point of view. First, it has been found that friction vector decoupling causes the reduction of the computed friction force and the rotation of the friction force vector. Second, it has been demonstrated that decoupled-friction models lack rotational invariance, i.e. model results depend on the alignment of the reference framework. These theoretical results have been confirmed by employing numerical experiments on flat and smoothly variable bed. Since the speed-up obtained with decoupled-friction models is modest, the coupled-friction models, which are more general, should be preferred in every case.

International audience; A theoretical framework and numerical techniques to solve optimal control problems with a spatial trace term in the terminal cost and governed by regularized nonlinear hyperbolic conservation laws are provided. Depending on the spatial dimension, the set at which the optimum of the trace term is reached under the action of the control function can be a point, a curve or a hypersurface. The set is determined by geometric parameters. Theoretically the lack of a convenient functional framework in the context of optimal control for hyperbolic systems leads us to consider a parabolic regularization for the state equation, in order to derive optimality conditions. For deriving these conditions, we use a change of variables encoding the sensitivity with respect to the geometric parameters. As illustration, we consider the shallow-water equations with the objective of maximizing the height of the wave at the final time, a wave whose location and shape are optimized via the geometric parameters. Numerical results are obtained in 1D and 2D, using finite difference schemes, combined with an immersed boundary method for iterating the geometric parameters.

In this work, we take a modern high-resolution finite-volume scheme for solving the rotational shallow-water equations and extend it with features required to run real-world ocean simulations. Our contributions include a spatially varying north vector and Coriolis term required for large scale domains, moving wet-dry fronts, a static land mask, bottom shear stress, wind forcing, boundary conditions for nesting in a global model, and an efficient model reformulation that makes it well-suited for massively parallel implementations. Our model order is verified using a grid convergence test, and we show numerical experiments using three different sections along the coast of Norway based on data originating from operational forecasts run at the Norwegian Meteorological Institute. Our simulation framework shows perfect weak scaling on a modern P100 GPU, and is capable of providing tidal wave forecasts that are very close to the operational model at a fraction of the cost. All source code and data used in this work are publicly available under open licenses. This research has mainly been funded by the Research Council of Norway under grant number 250935 (GPU Ocean), and partly by grant number 310515 (Havvarsel). The GPU Ocean project has received support in form of compute time on UNINETT Sigma2 - the National Infrastructure for High Performance Computing and Data Storage in Norway under project number nn9550k.

Tsunami modeling commonly accepts the shallow water system as governing equations where the major difficulty is the correct treatment of the nonconservative term due to bathymetry variations. The finite volume method for solving the shallow water equations with such source terms has received great attention in the two last decades. The built-in conservation property, the capacity to correctly treat discontinuities, and the ability to handle complex bathymetry configurations preserving some steady state configurations (well-balanced scheme) make the method very efficient. Nevertheless, it is still a challenge to build an efficient numerical scheme, with very few numerical artifacts (e.g., small numerical diffusion, correct propagation of the discontinuities, accuracy, and robustness), to be used in an operational environment, and that is able to better capture the dynamics of the wet-dry interface and the physical phenomena that occur in the inundation area. In the first part of this paper, we present a new second-order finite volume code. The code is developed for the shallow water equations with a nonconservative term based on the hydrostatic reconstruction technology to achieve a well-balanced scheme and an adequate dry/wet interface treatment. A detailed presentation of the numerical method is given. In the second part of the paper, we highlight the advantages of the new numerical technique. We benchmark the numerical code against analytical, experimental, and field results to assess the robustness and the accuracy of the numerical code. Finally, we use the 28 February 1969 North East Atlantic tsunami to check the performance of the code with real data., Historical data for Cascais and Lagos (1969 Lisbon Tsunami) are available at http://www.dgterritorio.pt/cartografia_e_geodesia/geodesia/redes_geodesicas/rede_maregrafica/. The tagus estuary data (typewriter document) are available at the Dom Luiz Institute library http://idl.ul.pt/node/33. This work is funded by the Portugal-France research agreement, through the research project GEONUM FCT-ANR/MAT-NAN/0122/2012. This research was financed by Portuguese Funds through FCT-Fundacao para a Ciencia e a Tecnologia, within the Project UID/MAT/00013/2013.

A two-dimensional non-hydrostatic shallow-water model for weakly dispersive waves is developed using the least-squares finite-element method. The model is based on the depth-averaged, nonlinear and non-hydrostatic shallow-water equations. The non-hydrostatic shallow-water equations are solved with the semi-implicit (predictor-corrector) method and least-squares finite-element method. In the predictor step, hydrostatic pressure at the previous step is used as an initial guess and an intermediate velocity field is calculated. In the corrector step, a Poisson equation for the non-hydrostatic pressure is solved and the final velocity and free-surface elevation is corrected for the new time step. The non-hydrostatic shallow-water model is verified and applied to both wave and flow driven fluid flows, including solitary wave propagation in a channel, progressive sinusoidal waves propagation over a submerged bar, von Karmann vortex street, and ocean circulations of Dongsha Atolls. It is found hydrostatic shallow-water model is efficient and accurate for shallow water flows. Non-hydrostatic shallow-water model requires 1.5 to 3.0 more cpu time than hydrostatic shallow-water model for the same simulation. Model simulations reveal that non-hydrostatic pressure gradients could affect the velocity field and free-surface significantly in case where nonlinearity and dispersion are important during the course of wave propagation.

We study mixed finite element methods for the rotating shallow water equations with linearized momentum terms but nonlinear drag. By means of an equivalent second-order formulation, we prove long-time stability of the system without energy accumulation. We also give rates of damping in unforced systems and various continuous dependence results on initial conditions and forcing terms. A priori error estimates for the momentum and free surface elevation are given in $$L^2$$ as well as for the time derivative and divergence of the momentum. Numerical results confirm the theoretical results regarding both energy damping and convergence rates.

International audience; This work is devoted to the derivation of a fully well-balanced numerical scheme for the well-known shallow-water model. During the last two decades, several well-balanced strategies have been introduced with a special attention to the exact capture of the stationary states associated with the so-called lake at rest. By fully well-balanced, we mean here that the proposed Godunov-type method is also able to preserve stationary states with non zero velocity. The numerical procedure is shown to preserve the positiveness of the water height and satisfies a discrete entropy inequality.

The general solution of the dam-break problem with partial uplift of the sluice-gate is presented in the framework of the one-dimensional Shallow-Water Equations, under the hypothesis that the classic Energy–Momentum (E–M) formulation is used to evaluate the flow characteristics at the gate. Due to the nonlinearity of the problem, there are ranges of the initial conditions for which the problem admits multiple solutions, or no solution. In the ranges in which there are multiple solutions, the maximization of the discharge under the gate is used as a disambiguation criterion. The exact solutions are used as a benchmark in order to evaluate the results of a simple Finite-Volume scheme, where the discharge under the gate and the forces exerted on the flow by the gate itself are calculated using the E–M formulation.

This article presents a non-intrusive reduced order model (NIROM) for general, dynamic partial differential equations. Based upon proper orthogonal decomposition (POD) and Smolyak sparse grid collocation, the method first projects the unknowns with full space and time coordinates onto a reduced POD basis. Then we introduce a new least squares fitting procedure to approximate the dynamical transition of the POD coefficients between subsequent time steps taking only a set of full model solution snapshots as the training data during the construction. Thus, neither the physical details nor further numerical simulations of the original PDE model is required by this methodology and the level of non-intrusiveness is improved compared to existing ROMs. Furthermore, we take adaptive measures to address the instability issue arising from reduced order iterations of the POD coefficients. This model can be applied to a wide range of physical and engineering scenarios and we test it on a couple problems in fluid dynamics. It is demonstrated that this reduced order approach captures the dominant features of the high fidelity models with reasonable accuracy while the computation complexity is reduced by several orders of magnitude.

In this article, we describe a non-intrusive reduction method for porous media multiphase flows using Smolyak sparse grids. This is the first attempt at applying such an non-intrusive reduced-order modelling (NIROM) based on Smolyak sparse grids to porous media multiphase flows. The advantage of this NIROM for porous media multiphase flows resides in that its non-intrusiveness, which means it does not require modifications to the source code of full model. Another novelty is that it uses Smolyak sparse grids to construct a set of hypersurfaces representing the reduced-porous media multiphase problem. This NIROM is implemented under the framework of an unstructured mesh control volume finite element multiphase model. Numerical examples show that the NIROM accuracy relative to the high-fidelity model is maintained, whilst the computational cost is reduced by several orders of magnitude.

Free-surface flows are usually modelled by means of the Shallow-water Equations: this system of hyperbolic equations exhibits a source term which is proportional to the product of the water depth by the bed slope, and which takes into account the effect of gravity onto fluid mass. Recently, much attention has been paid to the case in which bottom discontinuities are present in the physical domain to be represented: in this case, it is difficult to define the non-conservative product in the distributional sense. Here, the discontinuous-bottom Shallow-water Equations with hydrostatic pressure distribution at the bed step (Bernetti et al., 2006) are discussed in the context of the theory of Dal Maso et al. (1995) [9] ; finally, a first-order numerical scheme is presented, which is consistent for regular solutions, and which is able to capture contact discontinuities at bottom steps. Numerous tests are presented to show the feasibility of the scheme and its ability to converge to the exact solution in the cases of smooth as well as discontinuous bed profiles.

Limited-area models (LAMs) use higher resolutions and more advanced parameterizations of physical processes than global numerical weather prediction models, but suffer from one additional source of error—the lateral boundary condition (LBC). The large-scale model passes the information on its fields to the LAM only over the narrow coupling zone at discrete times separated by a coupling interval of several hours. The LBC temporal resolution can be lower than the time necessary for a particular meteorological feature to cross the boundary. A LAM user who depends on LBC data acquired from an independent prior analysis or parent model run can find that usual schemes for temporal interpolation of large-scale data provide LBC data of inadequate quality. The problem of a quickly moving depression that is not recognized by the operationally used gridpoint coupling scheme is examined using a simple one-dimensional model. A spectral method for nesting a LAM in a larger-scale model is implemented and tested. Results for a traditional flow-relaxation scheme combined with temporal interpolation in spectral space are also presented.

A non-intrusive model reduction computational method using hypersurfaces representation has been developed for reservoir simulation and further applied to 3D fluvial channel problems in this work. This is achieved by a combination of a radial basis function (RBF) interpolation and proper orthogonal decomposition (POD) method. The advantage of the method is that it is generic and non-intrusive, that is, it does not require modifications to the original complex source code, for example, a 3D unstructured mesh control volume finite element (CVFEM) reservoir model used here. The capability of this non-intrusive reduced order model (NIROM) based on hypersurfaces representation has been numerically illustrated in a horizontally layered porous media case, and then further applied to a 3D complex fluvial channel case. By comparing the results of the NIROM against the solutions obtained from the high fidelity full model, it is shown that this NIROM results in a large reduction in the CPU computation cost while much of the details are captured. [Received: May 19, 2017; Accepted: December 1, 2017]

The goal of this paper is to construct efficient parallel solvers for 2D hyperbolic systems of conservation laws with source terms and nonconservative products. The method of lines is applied: at every intercell a projected Riemann problem along the normal direction is considered which is discretized by means of well-balanced Roe methods. The resulting 2D numerical scheme is explicit and first-order accurate. In [M.J. Castro, J.A. García, J.M. González, C. Pares, A parallel 2D Finite Volume scheme for solving systems of balance laws with nonconservative products: Application to shallow flows, Comput. Methods Appl. Mech. Engrg. 196 (2006) 2788–2815] a domain decomposition method was used to parallelize the resulting numerical scheme, which was implemented in a PC cluster by means of MPI techniques.In this paper, in order to optimize the computations, a new parallelization of SIMD type is performed at each MPI thread, by means of SSE (“Streaming SIMD Extensions”), which are present in common processors. More specifically, as the most costly part of the calculations performed at each processor consists of a huge number of small matrix and vector computations, we use the Intel© Integrated Performance Primitives small matrix library. To make easy the use of this library, which is implemented using assembler and SSE instructions, we have developed a C++ wrapper of this library in an efficient way. Some numerical tests were carried out to validate the performance of the C++ small matrix wrapper. The specific application of the scheme to one-layer Shallow-Water systems has been implemented on a PC’s cluster. The correct behavior of the one-layer model is assessed using laboratory data.

International audience; This paper reports on the numerical modelling of flash flood propagation in urban areas after an excessive rain event or dam/dyke break wave. A two-dimensional (2-D) depth-averaged shallow-water model is used, with a refined grid of quadrilaterals and triangles for representing the urban area topography. The 2-D shallow-water equations are solved using the explicit second-order scheme that is adapted from MUSCL approach. Four applications are reported here to demonstrate the potential benefits and limits of 2-D modelling: (i) laboratory experimental dam-break wave in presence of an isolated building; (ii) flash flood over a physical model of the urbanized Toce river valley in Italy; (iii) extreme flood in October 1988 in the dense city of Nîmes (France); and (iv) dam-break flood in October 1982 in the town of Sumacárcel (Spain). Computed flow depths and velocities compare favourably with recorded data although for the experimental study on dam-break wave some discrepancies are observed around buildings, where the flow is strongly 3-D in character. The numerical simulations show that the flow depths and flood wave celerity are significantly affected by the presence of buildings in comparison with the original floodplain. Further, this study confirms the importance of topography and roughness coefficient for flood propagation simulation.; Cet article relate la modélisation numérique de la propagation de crues éclair en zone urbaine suite à de fortes pluies ou une onde de rupture de barrage. Les équations de Saint Venant bidimensionnelles sont résolues sur un maillage de quadrilatères et triangles représentant la topographie de la zone urbaine. Le schéma numérique est du second ordre du type MUSCL. Quatre applications sont décrites : une expérience en laboratoire d'onde de rupture de barrage autour d'un bâtiment isolé ; crue éclair d'un modèle physique de la vallée du Toce en Italie ; crue éclair d'octobre 1988 à Nîmes (France) ; onde de rupture de barrage dans la ville de Sumacarcel (Espagne) en octobre 1982. Les hauteurs d'eau et vitesses calculées sont en accord avec les données relevées bien que l'expérience de l'onde de rupture de barrage montre des différences autour des bâtiments où l'écoulement est fortement 3D. La modélisation numérique montre que les hauteurs d'eau et vitesses sont fortement modifiées par la présence de bâtiments. En outre, cette étude confirme l'importance de la topographie du terrain naturel et des coefficients de frottement.

© 2015 Mathematical Sciences Publishers. We present a high-order finite-volume approach for solving the shallow-water equations on the sphere, using multiblock grids on the cubed sphere. This approach combines a Runge-Kutta time discretization with a fourth-order-accurate spatial discretization and includes adaptive mesh refinement and refinement in time. Results of tests show fourth-order convergence for the shallow-water equations as well as for advection in a highly deformational flow. Hierarchical adaptive mesh refinement allows solution error to be achieved that is comparable to that obtained with uniform resolution of the most refined level of the hierarchy but with many fewer operations.

The Hamiltonian particle-mesh (HPM) method is generalized to the spherical shallow-water equations, utilizing constrained particle dynamics on the sphere and Merilees pseudospectral method (complexity (J2logJ) in the latitudinal gridsize) to approximate the inverse modified Helmholtz regularization operator. The time step for the explicit, symplectic integrator depends only on a uniform physical smoothing length. Copyright © 2004 Royal Meteorological Society

The article of record as published may be located at http://dx.doi.org/10.1002/qj.2474 We test the behaviour of a unified continuous/discontinuous Galerkin (CG/DG) shallowwater model in spherical geometry with curved elements on three different grids of ubiquitous use in atmospheric modelling: (i) the cubed-sphere, (ii) the reduced latitude–longitude, and (iii) the icosahedral grid. Both conforming and non-conforming grids are adopted including static and dynamically adaptive grids for a total of twelve mesh configurations. The behaviour of CG and DG on the different grids are compared for a nonlinear midlatitude perturbed jet and for a linear case that admits an analytic solution. Because the inviscid solution on certain grids shows a high sensitivity to the resolution, the viscous counterpart of the governing equations is also solved and the results compared. The logically unstructured element-based CG/DG model described in this article is flexible with respect to arbitrary grids. However, we were unable to define a best grid configuration that could possiblyminimize the error regardless of the characteristic geometry of the flow. This is especially true if the governing equations are not regularized by the addition of a sufficiently large, fully artificial, diffusion mechanism, as will be shown. The main novelty of this study lies in the unified implementation of two element-based Galerkin methods that share the same numericalmachinery and do not rely on any specific grid configuration to solve the shallow-water equation on the sphere. The authors gratefully acknowledge the support of the Office of Naval Research through program element PE-0602435N, the National Science Foundation (Division ofMathematical Sciences) through program element 121670, and the Air Force Office of Scientific Research through the Computational Mathematics program.

International audience; In this paper we present a method of optimal control developed in order to calculate the current corresponding to the observed sea level in a fluid domain $\Omega$ and during a time T. The control is the external stress $\f\ $. The cost function measures the distance between the observed and computed sea levels. The equations satisfied by the depth and the depth averaged velocity are of nonlinear shallow-water type. The existence and uniqueness of a solution for the direct problem are studied in the case of Dirichlet nonhomogeneous boundary conditions. We prove, by means of minimizing sequences, the existence of an optimal control $(\f\ ,\U\ )$ in the case of the small data and a very viscous fluid. To characterize it we build a sequence of problems corresponding to a linearization of the direct problem. We obtain the necessary conditions of optimality. The set of equations and the inequality characterizing the optimal control $(\f\ ,\U\ )$ is obtained as the limit of the penalization.

A parallel pseudo-spectral method for the simulation in distributed memory computers of the shallow-water equations in primitive form was developed and used on the study of turbulent shallow-waters LES models for orographic subgrid-scale perturbations. The main characteristics of the code are: momentum equations integrated in time using an accurate pseudo-spectral technique; Eulerian treatment of advective terms; and parallelization of the code based on a domain decomposition technique. The parallel pseudo-spectral code is efficient on various architectures. It gives high performance onvector computers and good speedup on distributed memory systems. The code is being used for the study of the interaction mechanisms in shallow-water ows with regular as well as random orography with a prescribed spectrum of elevations. Simulations show the evolution of small scale vortical motions from the interaction of the large scale flow and the small-scale orographic perturbations. These interactions transfer energy from the large-scale motions to the small (usually unresolved) scales. The possibility of including the parametrization of this effects in turbulent LES subgrid-stress models for the shallow-water equations is addressed.

The aim of this work is to develop a well-balanced finite-volume method for the accurate numerical solution of the equations governing suspended sediment and bed load transport in two-dimensional shallow-water flows. The modelling system consists of three coupled model components: (i) the shallow-water equations for the hydrodynamical model; (ii) a transport equation for the dispersion of suspended sediments; and (iii) an Exner equation for the morphodynamics. These coupled models form a hyperbolic system of conservation laws with source terms. The proposed finite-volume method consists of a predictor stage for the discretization of gradient terms and a corrector stage for the treatment of source terms. The gradient fluxes are discretized using a modified Roe's scheme using the sign of the Jacobian matrix in the coupled system. A well-balanced discretization is used for the treatment of source terms. In this paper, we also employ an adaptive procedure in the finite-volume method by monitoring the concentration of suspended sediments in the computational domain during its transport process. The method uses unstructured meshes and incorporates upwinded numerical fluxes and slope limiters to provide sharp resolution of steep sediment concentrations and bed load gradients that may form in the approximate solutions. Details are given on the implementation of the method, and numerical results are presented for two idealized test cases, which demonstrate the accuracy and robustness of the method and its applicability in predicting dam-break flows over erodible sediment beds. The method is also applied to a sediment transport problem in the Nador lagoon.

In this thesis a z?layer unstructured C-grid finite volume hydrostatic model is presented. An efficient and highly scalable implicit technique for the solution of the free surface equation is combined with an Eulerian approach for the advection of momentum. A consistent velocity reconstruc- tion procedure which not only satisfies the continuity law but also guarantees the discrete kinetic energy conservation is presented. It is shown that an ac- curate velocity reconstruction procedure is of crucial importance not only for discretization of the Coriolis term, but also for the correct advection of mo- mentum, especially in the multilayer case. Unlike other z?layer models the method presented here ensures that the staircase representation of bathymetry and free surface has no influence on the vertical structure of the flow. The method is therefore guaranteed to be strictly momentum conservative, also in the layers containing the free surface and bed. A number of test cases are presented to show that the model is able to accu- rately simulate Coriolis dominated flows and flooding and drying processes both in the depth-averaged case and in the presence of multiple z?layers. A simulation of the 2004 Indian Ocean tsunami is used to evaluate the ability of the method to simulate fast propagating tsunami waves and detailed inundation processes. Results obtained using two different rupture models are compared to the tide gauge arrival times, satellite altimetry data and the inundation ob- servations in the Banda Aceh area. The comparison is used not only to assess the quality of the underlying rupture models but also to determine the value of the available data sources for such an assessment. Preliminary results of the unstructured grid fine resolution tidal model the southern North Sea including the Rhine-Scheldt Delta region are presented. The model is able to correctly reproduce the essential characteristics of the M2 tide, as well as of the most important nonlinear shallow water overtides M4 or M6. The simulated velocity field was used to evaluate the Simpson- Hunter stratification parameter, the variability of frontal positions due to tidal advection and spring-neap adjustment.

This paper shows that in the linearized shallow-water equations, the numerical schemes can become weakly unstable for the 2Δx wave in the C grid when the Courant number is 1 in the forward–backward scheme and 0.5 in the leapfrog scheme because of the repeated eigenvalues in the matrices. The instability can be amplified and spread to other waves and smaller Courant number if the diffusion term is included. However, Shuman smoothing can control the instability.