Two‐dimensional,unstructured mesh generation for tidal models

The successful implementation of a finite element model for computing shallow‐water flow requires the identification and spatial discretization of a surface water region. Since no robust criterion or node spacing routine exists, which incorporates physical characteristics and subsequent responses into the mesh generation process, modelers are left to rely on crude gridding criteria as well as their knowledge of particular domains and their intuition. Two separate methods to generate a finite element mesh are compared for the Gulf of Mexico. A wavelength‐based criterion and an alternative approach, which employs a localized truncation error analysis (LTEA), are presented. Both meshes have roughly the same number of nodes, although the distribution of these nodes is very different. Two‐dimensional depth‐averaged simulations of flow using a linearized form of the generalized wave continuity equation and momentum equations are performed with the LTEA‐based mesh and the wavelength‐to‐gridsize ratio mesh. All simulations are forced with a single tidal constituent, M₂. Use of the LTEA‐based procedure is shown to produce a superior (i.e., less error) two‐dimensional grid because the physics of shallow‐water flow, as represented by discrete equations, are incorporated into the mesh generation process. Copyright © 2001 John Wiley & Sons, Ltd.     

Two‐dimensional dispersion analyses of finite element approximations to the shallow water equations

Dispersion analysis of discrete solutions to the shallow water equations has been extensively used as a tool to define the relationships between frequency and wave number and to determine if an algorithm leads to a dual wave number response and near 2Δx oscillations. In this paper, we explore the application of two‐dimensional dispersion analysis to cluster based and Galerkin finite element‐based discretizations of the primitive shallow water equations and the generalized wave continuity equation (GWCE) reformulation of the harmonic shallow water equations on a number of grid configurations. It is demonstrated that for various algorithms and grid configurations, contradictions exist between the results of one‐dimensional and two‐dimensional dispersion analysis as a result of subtle changes in the mass matrix. Numerical experiments indicate that the two‐dimensional dispersion analysis correctly predicts the existence and onset of near 2Δx noise in the solution. Copyright © 2004 John Wiley & Sons, Ltd.     

Estimate of the truncation error of finite volume discretization of the Navier–Stokes equations on colocated grids

A methodology is proposed for the calculation of the truncation error of finite volume discretizations of the incompressible Navier–Stokes equations on colocated grids. The truncation error is estimated by restricting the solution obtained on a given grid to a coarser grid and calculating the image of the discrete Navier–Stokes operator of the coarse grid on the restricted velocity and pressure field. The proposed methodology is not a new concept but its application to colocated finite volume discretizations of the incompressible Navier–Stokes equations is made possible by the introduction of a variant of the momentum interpolation technique for mass fluxes where the pressure part of the mass fluxes is not dependent on the coefficients of the linearized momentum equations. The theory presented is supported by a number of numerical experiments. The methodology is developed for two‐dimensional flows, but extension to three‐dimensional cases should not pose problems. Copyright © 2005 John Wiley & Sons, Ltd.     

2D unstructured mesh generation for oceanic and coastal tidal models from a localized truncation error analysis with complex derivatives

A method for computing target element size for tidal, shallow water flow is developed and demonstrated. The method, Localized truncation error analysis with complex derivatives (LTEA-CD) utilizes localized truncation error estimates of the linearized shallow water momentum equations consisting of complex derivative terms. This application of complex derivatives is the chief way in which the method differs from a similar existing method, LTEA. It is shown that LTEA-CD produces results that are essentially equivalent to those of LTEA (which in turn has been demonstrated to be capable of producing practicable target element sizes) with reduced computational cost. Moreover, LTEA-CD is capable of computing truncation error and corresponding target element sizes at locations up to and including the boundary, whereas LTEA can be applied only on the interior of the model domain. We demonstrate the convergence of solutions over meshes generated with LTEA-CD using an idealized representation of the western North Atlantic Ocean, Caribbean Sea and Gulf of Mexico.     

Dispersion analysis of the least‐squares finite‐element shallow‐water system

The frequency or dispersion relation for the least‐squares mixed formulation of the shallow‐water equations is analysed. We consider the use of different approximation spaces corresponding to co‐located and staggered meshes, respectively. The study includes the effect of Coriolis, and the dispersion properties are compared analytically and graphically with those of the mixed Galerkin formulation. Numerical solutions of a test problem to simulate slow Rossby modes illustrate the theoretical results. Copyright © 2003 John Wiley & Sons, Ltd.     

A two‐dimensional finite volume morphodynamic model on unstructured triangular grids

We discuss the application of a finite volume method to morphodynamic models on unstructured triangular meshes. The model is based on coupling the shallow water equations for the hydrodynamics with a sediment transport equation for the morphodynamics. The finite volume method is formulated for the quasi‐steady approach and the coupled approach. In the first approach, the steady hydrodynamic state is calculated first and the corresponding water velocity is used in the sediment transport equation to be solved subsequently. The second approach solves the coupled hydrodynamics and sediment transport system within the same time step. The gradient fluxes are discretized using a modified Roe's scheme incorporating the sign of the Jacobian matrix in the morphodynamic system. A well‐balanced discretization is used for the treatment of source terms. We also describe an adaptive procedure in the finite volume method by monitoring the bed–load in the computational domain during its transport process. The method uses unstructured meshes, incorporates upwinded numerical fluxes and slope limiters to provide sharp resolution of steep bed gradients that may form in the approximate solution. Numerical results are shown for a test problem in the evolution of an initially hump‐shaped bed in a squared channel. For the considered morphodynamical regimes, the obtained results point out that the coupled approach performs better than the quasi‐steady approach only when the bed–load rapidly interacts with the hydrodynamics. Copyright © 2009 John Wiley & Sons, Ltd.     

Two‐dimensional dam break flow simulation

Numerical modelling of shallow water flow in two dimensions is presented in this work with the results obtained in dam break tests. Free surface flow in channels can be described mathematically by the shallow‐water system of equations. These equations have been discretized using an approach based on unstructured Delaunay triangles and applied to the simulation of two‐dimensional dam break flows. A cell centred finite volume method based on Roe's approximate Riemann solver across the edges of the cells is presented and the results are compared for first‐ and second‐order accuracy. Special treatment of the friction term has been adopted and will be described. The scheme is capable of handling complex flow domains as shown in the simulation corresponding to the test cases proposed, i.e. that of a dam break wave propagating into a 45° bend channel (UCL) and in a channel with a constriction (LNEC‐IST). Comparisons of experimental and numerical results are shown. Copyright © 2000 John Wiley & Sons, Ltd.     

A stabilized finite element method for the Saint‐Venant equations with application to irrigation

When the two‐dimensional shallow water equations are applied to solve practical irrigation problems, additional numerical difficulties arise. Large friction coefficients, dry bed conditions and singular infiltration terms engender new challenges which are addressed here to build a finite element method that is robust enough for this type of application. The proposed method is a stabilized formulation based on the symmetric quasi‐linear form and the set of entropy variables. The robustness of the method is increased with a discontinuity capturing operator. A predictor multi‐corrector algorithm is employed to solve the generalized trapezoidal rule. One of the novel features of the present technique is that an ‘explicit’ method has been developed with characteristics of implicit methods, so that the solution can be advanced at a convective CFL number of 1, regardless of the source terms. This leads to an economic procedure. Finally, an entropy production (in) equality is developed, which ensures the correct physical behaviour of the model and helps to determine the correct sign of the infiltration term. Copyright © 2002 John Wiley & Sons, Ltd.     

Edge‐based finite element method for shallow water equations

This paper describes an edge‐based implementation of the generalized residual minimum (GMRES) solver for the fully coupled solution of non‐linear systems arising from finite element discretization of shallow water equations (SWEs). The gain in terms of memory, floating point operations and indirect addressing is quantified for semi‐discrete and space–time analyses. Stabilized formulations, including Petrov–Galerkin models and discontinuity‐capturing operators, are also discussed for both types of discretization. Results illustrating the quality of the stabilized solutions and the advantages of using the edge‐based approach are presented at the end of the paper. Copyright © 2001 John Wiley & Sons, Ltd.     

Impact of mass lumping on gravity and Rossby waves in 2D finite‐element shallow‐water models

The goal of this study is to evaluate the effect of mass lumping on 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 P₀?P₁, RT₀ and P?P₁ 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 © 2008 John Wiley & Sons, Ltd.     

A comparison of the GWCE and mixed P − P1 formulations in finite‐element linearized shallow‐water models

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 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.     

High‐order 𝒞1 finite‐element interpolating schemes—Part II: Nonlinear semi‐Lagrangian shallow‐water models

The finite‐element, semi‐implicit, and semi‐Lagrangian methods are used on unstructured meshes to solve the nonlinear shallow‐water system. Several ??¹ approximation schemes are developed for an accurate treatment of the advection terms. The employed finite‐element discretization schemes are the P–P₁ and P₂–P₁ pairs. Triangular finite elements are attractive because of their flexibility for representing irregular boundaries and for local mesh refinement. By tracking the characteristics backward from both the interpolation and quadrature nodes and using ??¹ interpolating schemes, an accurate treatment of the nonlinear terms and, hence, of Rossby waves is obtained. Results of test problems to simulate slowly propagating Rossby modes illustrate the promise of the proposed approach in ocean modelling. Copyright © 2007 John Wiley & Sons, Ltd.     

An unstructured grid,three‐dimensional model based on the shallow water equations

A semi‐implicit finite difference model based on the three‐dimensional shallow water equations is modified to use unstructured grids. There are obvious advantages in using unstructured grids in problems with a complicated geometry. In this development, the concept of unstructured orthogonal grids is introduced and applied to this model. The governing differential equations are discretized by means of a semi‐implicit algorithm that is robust, stable and very efficient. The resulting model is relatively simple, conserves mass, can fit complicated boundaries and yet is sufficiently flexible to permit local mesh refinements in areas of interest. Moreover, the simulation of the flooding and drying is included in a natural and straightforward manner. These features are illustrated by a test case for studies of convergence rates and by examples of flooding on a river plain and flow in a shallow estuary. Copyright © 2000 John Wiley & Sons, Ltd.     

A finite element method for the one‐dimensional extended Boussinesq equations

A new finite element method for Nwogu's (O. Nwogu, ASCE J. Waterw., Port, Coast., Ocean Eng., 119 , 618–638 (1993)) one‐dimensional extended Boussinesq equations is presented using a linear element spatial discretisation method coupled with a sophisticated adaptive time integration package. The accuracy of the scheme is compared to that of an existing finite difference method (G. Wei and J.T. Kirby, ASCE J. Waterw., Port, Coast., Ocean Eng., 121 , 251–261 (1995)) by considering the truncation error at a node. Numerical tests with solitary and regular waves propagating in variable depth environments are compared with theoretical and experimental data. The accuracy of the results confirms the analytical prediction and shows that the new approach competes well with existing finite difference methods. The finite element formulation is shown to enable the method to be extended to irregular meshes in one dimension and has the potential to allow for extension to the important practical case of unstructured triangular meshes in two dimensions. This latter case is discussed. Copyright © 1999 John Wiley & Sons, Ltd.     

Stability/dispersion analysis of the discontinuous Galerkin linearized shallow‐water system

The frequency or dispersion relation for the discontinuous Galerkin mixed formulation of the 1‐D linearized shallow‐water equations is analysed, using several basic DG mixed schemes. The dispersion properties are compared analytically and graphically with those of the mixed continuous Galerkin formulation for piecewise‐linear bases on co‐located grids. Unlike the Galerkin case, the DG scheme does not exhibit spurious stationary pressure modes. However, spurious propagating modes have been identified in all the present discontinuous Galerkin formulations. Numerical solutions of a test problem to simulate fast gravity modes illustrate the theoretical results and confirm the presence of spurious propagating modes in the DG schemes. Copyright © 2005 John Wiley & Sons, Ltd.     

A finite element method for the two‐dimensional extended Boussinesq equations

A new numerical method for Nwogu's (ASCE Journal of Waterway, Port, Coastal and Ocean Engineering 1993; 119 :618)two‐dimensional extended Boussinesq equations is presented using a linear triangular finite element spatial discretization coupled with a sophisticated adaptive time integration package. The authors have previously presented a finite element method for the one‐dimensional form of these equations (M. Walkley and M. Berzins (International Journal for Numerical Methods in Fluids 1999; 29 (2):143)) and this paper describes the extension of these ideas to the two‐dimensional equations and the application of the method to complex geometries using unstructured triangular grids. Computational results are presented for two standard test problems and a realistic harbour model. Copyright © 2002 John Wiley & Sons, Ltd.     

A nodal‐based implementation of a stabilized finite element method for incompressible flow problems

The objective of this paper is twofold. First, a stabilized finite element method (FEM) for the incompressible Navier–Stokes is presented and several numerical experiments are conducted to check its performance. This method is capable of dealing with all the instabilities that the standard Galerkin method presents, namely the pressure instability, the instability arising in convection‐dominated situations and the less popular instabilities found when the Navier–Stokes equations have a dominant Coriolis force or when there is a dominant absorption term arising from the small permeability of the medium where the flow takes place. The second objective is to describe a nodal‐based implementation of the finite element formulation introduced. This implementation is based on an a priori calculation of the integrals appearing in the formulation and then the construction of the matrix and right‐hand side vector of the final algebraic system to be solved. After appropriate approximations, this matrix and this vector can be constructed directly for each nodal point, without the need to loop over the elements, thus making the calculations much faster. In order to be able to do this, all the variables have to be defined at the nodes of the finite element mesh, not on the elements. This is also so for the stabilization parameters of the formulation. However, doing this gives rise to questions regarding the consistency and the conservation properties of the final scheme, which are addressed in this paper. Copyright © 2000 John Wiley & Sons, Ltd.     

A modified Galerkin/finite element method for the numerical solution of the Serre‐Green‐Naghdi system

A new modified Galerkin/finite element method is proposed for the numerical solution of the fully nonlinear shallow water wave equations. The new numerical method allows the use of low‐order Lagrange finite element spaces, despite the fact that the system contains third order spatial partial derivatives for the depth averaged velocity of the fluid. After studying the efficacy and the conservation properties of the new numerical method, we proceed with the validation of the new numerical model and boundary conditions by comparing the numerical solutions with laboratory experiments and with available theoretical asymptotic results. Copyright © 2016 John Wiley & Sons, Ltd.     

An implicit time‐marching algorithm for shallow water models based on the generalized wave continuity equation

Wave equation models currently discretize the generalized wave continuity equation with a three‐time‐level scheme centered at k and the momentum equation with a two‐time‐level scheme centered at k+1/2; non‐linear terms are evaluated explicitly. However in highly non‐linear applications, the algorithm becomes unstable at even moderate Courant numbers. This paper examines an implicit treatment of the non‐linear terms using an iterative time‐marching algorithm. Depending on the domain, results from one‐dimensional experiments show up to a tenfold increase in stability and temporal accuracy. The sensitivity of stability to variations in the G‐parameter (a numerical weighting parameter in the generalized wave continuity equation) was examined; results show that the greatest increase in stability occurs with G/τ=2–50. In the one‐dimensional experiments, three different types of node spacing techniques—constant, variable, and LTEA (Localized Truncation Error Analysis)—were examined; stability is positively correlated to the uniformity of the node spacing. Lastly, a scaling analysis demonstrates that the magnitudes of the non‐linear terms are positively correlated to those that most influence stability, particularly the term containing the G‐parameter. It is evident that the new algorithm improves stability and temporal accuracy in a cost‐effective manner. Copyright © 2001 John Wiley & Sons, Ltd.