The numerical approximation of the shallow-water equations, which support geophysical wave propagation, namely the fast external Poincaré and Kelvin waves and the slow large-scale planetary Rossby waves, is a delicate and difficult problem. Indeed, the coupling between the momentum and the continuity equations may lead to the presence of erratic or spurious solutions for these waves, e.g. the spurious pressure and inertial modes. In addition, the presence of spurious branches and the emergence of spectral gaps in the dispersion relation at specific wavenumbers may lead to anomalous dissipation/dispersion in the representation of both fast and slow waves. The aim of the present study is to propose a class of possible discretization schemes, via the discontinuous Galerkin method, that is not affected by the above-mentioned problems. A Fourier/stability analysis of the 2D shallow-water model is performed by analysing the finite volume method and the linear discontinuous P1DG and non conforming P1NC Galerkin discretizations. These are stabilized by means of a family of numerical fluxes via the Polynomial Viscosity Matrix approach. A long time stability result is proven for all schemes and fluxes examined here. Further, a super-convergent result is demonstrated for the P1NC discrete frequencies compared to the P1DG ones, except for the slow mode in the Roe flux case. Indeed, we show that the Roe flux yields spurious frequencies and sub-optimal rates of convergence for the slow mode, for both the P1DG and P1NC methods. Finally, numerical solutions of linear and non-linear test problems confirm the theoretical results and reveal the computational cost efficiency of the P1NC approach.

This chapter presents an improved boundary feedback controller for the two- and three-dimensional Navier–Stokes equations, in a bounded domain Ω, for prescribed drag and lift coefficients. In order to determine the feedback control law, we consider an extended system coupling the equations governing the Navier–Stokes problem with an equation satisfied by the control on the bluff body, which is a part of the domain boundary. By using the Faedo–Galerkin method and a priori estimation techniques, a boundary control is built. This control law ensures the controllability of the discrete system. Then, a compactness result then allows us to pass to the limit in the non-linear system satisfied by the approximated solutions.

A Fourier/stability analysis of the third-order Korteweg–de Vries equation is presented subject to a class of local discontinuous Galerkin discretization using high-degree Lagrange polynomials. The selection of stability parameters involved in the method is made on the basis of the study of the higher frequency eigenmodes and the Fourier analysis. Explicit analytical dispersion relation and group velocity are obtained and the stability study of the discrete frequency is performed. The emergence of gaps in the imaginary part of the computed frequency is observed and studied for the first time to our knowledge. Further, a superconvergent result is demonstrated for the discrete frequency by obtaining an explicit analytical asymptotic formula for the latter.

We present a Fourier analysis of wave propagation problems subject to a class of continuous and discontinuous discretizations using high-degree Lagrange polynomials. This allows us to obtain explicit analytical formulas for the dispersion relation and group velocity and, for the first time to our knowledge, characterize analytically the emergence of gaps in the dispersion relation at specific wavenumbers, when they exist, and compute their specific locations. Wave packets with energy at these wavenumbers will fail to propagate correctly, leading to significant numerical dispersion. We also show that the Fourier analysis generates mathematical artifacts, and we explain how to remove them through a branch selection procedure conducted by analysis of eigenvectors and associated reconstructed solutions. The higher frequency eigenmodes, named erratic in this study, are also investigated analytically and numerically.

Compatible Galerkin methods (the Galerkin analogue of an Arakawa C-Grid) are growing in popularity for simulating geophysical fluid flows, due to their desirable characteristics, including but not limited to: energy conservation, higher-order accuracy, steady geostrophic modes and the absence of spurious stationary modes, such as pressure modes. However, these characteristics still do not guarantee good wave dispersion properties. In this work, we study the dispersion properties of two compatible Galerkin families for the 2D linear rotating shallow water equations on quadrilaterals: the Qn−Λk family from finite element exterior calculus and the newly developed MGDn family. These families are the extensions to quadrilaterals of the PnC−Pn−1DG and GDn−DGDn−1 pairs, respectively, studied for the 1D linear shallow water equations in [13]. A major finding from that paper was that the PnC−Pn−1DG pair has spectral gaps for n≥2 and the GDn−DGDn−1 does not. These spectral gaps are non-dimensional wavenumbers where the dispersion relationship is double-valued, and lead to anomalous dispersion and noise in numerical simulations. On quadrilaterals, previous work [24], [26] on the Qn−Λk family for inertia waves with n=2 and for gravity waves for arbitrary n has indicated the presence of spectral gaps, in the form of line discontinuities. The investigation of these gaps for the Qn−Λk family is extended in this paper to inertia-gravity waves for arbitrary n, including plots of the dispersion relationship for n=2 when using the lumping developed in [26], [35] that eliminates the spectral gaps. Additionally, the MGDn family is studied (including the use of reduced quadrature), which is found to be free of spectral gaps. For both families asymptotic convergence rates are established, effective resolutions determined and plots of the dispersion relationships for a range of n and Rossby radii are shown. Finally, a pair of numerical simulations are run to investigate the consequences of the spectral gaps and highlight the main differences between the two families.

In this work, we study the dispersion properties of two compatible Galerkin schemes for the 1D linearized shallow water equations: the PnC−Pn−1DG and the GDn−DGDn−1 element pairs. Pn is the order n Lagrange space, Pn−1DG is the order n−1 discontinuous Lagrange space, GDn is the order n Galerkin difference space, and DGDn−1 is the order n−1 discontinuous Galerkin difference space. Compatible Galerkin methods have many desirable properties, including energy conservation, steady geostrophic modes and the absence of spurious stationary modes, such as pressure modes. However, this does not guarantee good wave dispersion properties. Previous work on the P2C−P1DG pair has indeed indicated the presence of spectral gaps, and it is extended in this paper to the study of the PnC−Pn−1DG pair for arbitrary n. Additionally, an alternative element pair is introduced, the GDn−DGDn−1 pair, that is free of spectral gaps while benefiting from the desirable properties of compatible elements. Asymptotic convergence rates are established for both element pairs, including the use of inexact quadrature (which diagonalizes the velocity mass matrix) for the PnC−Pn−1DG pair and reduced quadrature for the GDn−DGDn−1 pair. Plots of the dispersion relationship and group velocities for a wide range of n and Rossby radii are shown. A brief investigation into the utility of mass lumping to remove the spectral gaps for the P3C−P2DG pair is performed. Finally, a pair of numerical simulations are run to investigate the consequences of the spectral gaps and highlight the main differences between the two elements.

In this work we study the exponential stabilization of the two and three-dimensional Navier-Stokes equations in a bounded domain $\Omega$, around a given steady-state flow, by means of a boundary control. In order to determine a feedback law, we consider an extended system coupling the Navier-Stokes equations with an equation satisfied by the control on the domain boundary. While most traditional approaches apply a feedback controller via an algebraic Riccati equation, the Stokes-Oseen operator or extension operators, a Galerkin method is proposed instead in this study. The Galerkin method permits to construct a stabilizing boundary control and by using energy a priori estimation technics, the exponential decay is obtained. A compactness result then allows us to pass to the limit in the system satisfied by the approximated solutions. The resulting feedback control is proven to be globally exponentially stabilizing the steady states of the two and three-dimensional Navier-Stokes equations.

For most of the discretization schemes, the numerical approximation of shallow-water models is a delicate problem. Indeed, the coupling between the momentum and the continuity equations usually leads to the appearance of spurious solutions and to anomalous dissipation/dispersion in the representation of the fast (Poincaré) and slow (Rossby) waves. In order to understand these difficulties and to select appropriate spatial discretization schemes, Fourier/dispersion analyses and the study of the null space of the associated discretized problems have proven beneficial. However, the cause of spurious oscillations and reduced convergence rates, that have been detected for most of mixed-order finite element shallow-water formulations, in simulating classical problems of geophysical fluid dynamics, is still an open question. The aim of the present study is to show that when spurious inertial solutions are present, they are mainly responsible for the aforementioned problems. Further, a criterion is found which determines the existence and the number of spurious inertial solutions. As it is delicate to cure spurious inertial modes, a class of possible discretization schemes is proposed, that is not affected by such spurious solutions.

The appearance of spurious pressure modes in early shallow-water (SW) models has resulted in two common strategies in the finite element (FE) community: using mixed primitive variable and generalized wave continuity equation (GWCE) formulations of the SW equations. One FE scheme in particular, the PNC 1 –P1 pair, combined with the primitive equations may be advantageously compared with the wave equation formulations and both schemes have similar data structures. Our focus here is on comparing these two approaches for a number of measures including stability, accuracy, efficiency, conservation properties, and consistency. The main part of the analysis centres on stability and accuracy results via Fourier-based dispersion analyses in the context of the linear SW equations. The numerical solutions of test problems are found to be in good agreement with the analytical results. Copyright 2011 John Wiley & Sons, Ltd.

SUMMARYThe appearance of spurious pressure modes in early shallow‐water (SW) models has resulted in two common strategies in the finite element (FE) community: using mixed primitive variable and generalized wave continuity equation (GWCE) formulations of the SW equations. One FE scheme in particular, the P − P1 pair, combined with the primitive equations may be advantageously compared with the wave equation formulations and both schemes have similar data structures. Our focus here is on comparing these two approaches for a number of measures including stability, accuracy, efficiency, conservation properties, and consistency. The main part of the analysis centres on stability and accuracy results via Fourier‐based dispersion analyses in the context of the linear SW equations. The numerical solutions of test problems are found to be in good agreement with the analytical results. Copyright © 2011 John Wiley & Sons, Ltd.

The finite-element spatial discretization of the linear shallow-water equations is examined in the context of several temporal discretization schemes. Three finite-element pairs are considered, namely, the $P^{}_{0}-P^{}_{1}$, $P^{NC}_{1}-P^{}_{1}$, and $RT^{}_{0}-P^{}_{0}$ schemes, and the backward and forward Euler, Crank–Nicolson, and second and third order Adams–Bashforth time stepping schemes are employed. A Fourier analysis is performed at the discrete level for the Poincaré waves, and it determines the stability limit of the schemes and the error in wave amplitude and phase that can be expected. Numerical solutions of test problems to simulate Poincaré waves illustrate the analytical results.

This Note deals with the regulation of water flow in open-channels employing the shallow-water model with variable bathymetry. By using energy it priori estimation technics and the compactness theory, we build a stabilizing boundary control. The control law is based on an arbitrary choice of the time dependent stabilization rate r. Say, the energy decreases like the exponential of - integral(t)(0) r(s) ds when the time t tends to infinity. To cite this article: A. Sene et al., C R. Acad. Sci. Paris, Ser. 1346 (2008). (C) 2008 Academie des sciences. Published by Elsevier Masson SAS. All rights reserved.

The goal of this study is to evaluate the effect of mass lumping oil the dispersion properties of four finite-element velocity/surface-elevation pairs that are used to approximate the linear shallow-water equations. For each pair, the dispersion relation, obtained using the mass lumping technique, is computed and analysed for both gravity and Rossby waves. The dispersion relations are compared with those obtained for the consistent schemes (without lumping) and the Continuous case. The P-0 - P-1, RT0 and P-1(NC) - P-1 pairs are shown to preserve good dispersive properties when the mass matrix is lumped. Test problems to simulate fast gravity and slow Rossby waves are in good agreement with the analytical results. Copyright (C) 2008 John Wiley & Sons, Ltd.

A Fourier analysis has been performed for a class of upwind finite volume schemes, including the study of phase speed, group velocity, damping and dispersion. In the first part, pure gravity waves are investigated. As expected, most upwind schemes lead to a significant damping, but they exhibit a better phase behavior than most centered schemes. In the second part, the Coriolis parameter is considered and the Rossby modes are studied. In this case, all selected upwind schemes lead to a severe damping. The numerical results are also compared with those obtained by using a slope limiter approach. It is concluded that most upwind schemes with or without slope limiters present poor results for an accurate calculation of the Rossby modes. Copyright © 2008 John Wiley & Sons, Ltd.

The behavior of planetary (Rossby) waves in finite-element numerical models is investigated by using a quasi-geostrophic approximation to the midlatitude, $\beta$-plane, two-dimensional linear shallow-water equations. A dispersion relation analysis is employed here to determine the possible occurrence of spurious modes and to ascertain the dispersive/dissipative nature of the finite-element Galerkin mixed formulation. Three finite-element pairs are considered by using a variety of mixed interpolation schemes. For each pair the frequency or dispersion relation is obtained and analyzed, and the dispersion properties are compared analytically and graphically with the continuous case. It is shown that certain choices of mixed interpolation schemes may lead to significant phase and group velocity errors and spurious solutions in the calculation of slow Rossby waves, due to the coupling between the momentum and continuity equations. Numerical solutions of two test problems to simulate slow Rossby waves are in good agreement with the analytical results.

In the late 1970's, the wave equation formulation of the shallow water equations appeared to be one of the few options for finite-element model development. Spurious pressure modes occurred in most primitive equation formulations because of the coupling between the gravity wave terms in the continuity and momentum equations. However, several new and rediscovered elements make a primitive equation approach viable and offer advantages over the wave equation methods. Some of the strengths of the wave equation formulation are the amplitude and phase accuracy for the explicit version, good efficiency, and absence of spurious pressure modes. Some major shortcomings are poor accuracy for the implicit version and poor stability when advection becomes important. A finite element that provides a close replacement for the wave equation formulation is the P1NC-P1 element which has linear non-conforming bases for velocity and linear conforming bases for sea level. The latter are the same as the linear bases used with the wave equation approach; hence, there is a close correspondence in data structure between the two approaches. Used in conjunction with the primitive equations, the approach with this element provides all the same strengths as the wave equation formulation but not the weaknesses. This approach has better amplitude and phase accuracy for both explicit and implicit methods, has good efficiency, has no spurious modes, and can be used with a wide variety of advection operators including ELM and semi-Lagrangian methods. In addition, investments in software infrastructure can be retained because of the similar data structure in the two approaches.

We present an existence theorem of a two-dimensional sedimentation model coupling a shallow water system with a sediment transport equation. The shallow water system includes Coriolis and friction terms. A Galerkin method is used to obtain a finite-dimensional problem which is solved using a Brouwer fixed point theorem. We prove that the limits of the resulting solution sequences satisfy the model equations.

A constructive linear algebra approach is developed to characterize the kernels of the discretized shallow-water equations. Three kernel relations are identified as necessary conditions for the discretized system to share the same stationary properties as the continuous system. This matrix kernel scheme is computed using MATLAB and applied to investigate the presence, number, and structure of spurious modes arising in typical finite difference and finite element schemes. The kernel concept is then used to characterize the smallest representable vortices for several representative discrete finite difference and finite element schemes. Both uniform and unstructured mesh situations are considered and compared. Numerical experiments are consistent with the analytic results.

The P1NC-P1 and RT0 finite element schemes are among the most promising low order elements for use in unstructured mesh marine and lake models. They are both free of spurious elevation modes, have good dispersive properties and have a relatively low computational cost. In this paper, we derive both finite element schemes in the same unified framework and discuss their respective qualities in terms of conservation, consistency, propagation factor and convergence rate. We also highlight the impact that the local variables placement can have on the model solution. The main conclusion that we can draw is that the choice between elements is highly application dependent. We suggest that the P1NC-P1 element is better suited to purely hydrodynamical applications while the RT0 element might perform better for hydrological applications that require scalar transport calculations.

We present an existence theorem of a two-dimensional sedimentation model coupling a shallow-water system with a sediment transport equation. A finite dimensional problem is solved using a Brouwer fix point theorem. We prove that the limits of the resulting solution sequences satisfy the model equations. To cite this article: B. Toumbou et al., C. R. Acad. Sci. Paris, Ser. I 344 (2007).