Generalized Wave Continuity Equation Research Articles

Improvements for the Western North Atlantic, Caribbean and Gulf of Mexico ADCIRC Tidal Database (EC2015)

This research details the development and validation of an updated constituent tidal database for the Western North Atlantic, Caribbean and Gulf of Mexico (WNAT) region, referred to as the EC2015 database. Regional databases, such as EC2015, provide much higher resolution than global databases allowing users to more accurately define the tidal forcing on smaller sub-region domains. The database last underwent major updates in 2001 and was developed using the two-dimensional, depth-integrated form of the coastal hydrodynamic model, ADvanced CIRCulation (ADCIRC), which solves the shallow-water equations in the generalized wave continuity equation form. Six main areas of improvement are examined: (1) placement of the open ocean boundary; (2) higher coastal resolution using Vertical Datum (VDatum) models; (3) updated bathymetry from global databases; (4) updated boundary forcing compared using two global tidal databases; (5) updated bottom friction formulations; and (6) improved model physics by incorporating the advective terms in ADCIRC. The skill of the improved database is compared to that of its predecessor and is calculated using harmonic data from the National Oceanic and Atmospheric Administration Center for Operational Oceanographic Products and Services (NOAA CO-OPS) stations and historic International Hydrographic Organization (IHO) data. Overall, the EC2015 database significantly reduces errors realized in the EC2001 database and improves the quality of coastal tidal constituents available for smaller sub-regional models in the Western North Atlantic, Caribbean and Gulf of Mexico (WNAT) region.

Implementation and analysis of a partial‐element wetting and drying framework for generalized wave continuity equation‐based hydrodynamic models

SUMMARYThe ADCIRC hydrodynamic model has been used extensively to predict the effects of storm surge in coastal regions. In an effort to create a more holistic representation of coastal inundation (i.e., the total water level), some recent applications of ADCIRC have focused on modeling coastal river systems. Initial attempts to model river systems with ADCIRC have encountered complications associated with mass balance (artificial increase in mass in the downstream direction) and model instabilities when flow leaves the main channel, which suggests that complications may be attributed to the current treatment of the wet/dry boundary in CG ADCIRC. A new wetting/drying framework has been developed for, and implemented in, the 1D continuous Galerkin ADCIRC code, which is based on the generalized wave continuity equation (GWCE) formulation of the shallow water equations. The new algorithm allows for the inclusion of partially wet elements within the computational domain of ADCIRC, and the focus of this manuscript is flooding‐type wetting/drying processes (e.g., tidal fluctuations and storm surge inundation). The new framework uses a scaling factor, η, placed on the time derivative term in the primitive continuity equation, prior to formulation of the GWCE. Benefits of the new approach include better mass balance and more accurate tracking of the location of the wet/dry interface, compared with the current wetting and drying algorithm.Copyright © 2013 John Wiley & Sons, Ltd.

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

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.

Scalability of an Unstructured Grid Continuous Galerkin Based Hurricane Storm Surge Model

This paper evaluates the parallel performance and scalability of an unstructured grid Shallow Water Equation (SWE) hurricane storm surge model. We use the ADCIRC model, which is based on the generalized wave continuity equation continuous Galerkin method, within a parallel computational framework based on domain decomposition and the MPI (Message Passing Interface) library. We measure the performance of the model run implicitly and explicitly on various grids. We analyze the performance as well as accuracy with various spatial and temporal discretizations. We improve the output writing performance by introducing sets of dedicated writer cores. Performance is measured on the Texas Advanced Computing Center Ranger machine. A high resolution 9,314,706 finite element node grid with 1 s time steps can complete a day of real time hurricane storm surge simulation in less than 20 min of computer wall clock time, using 16,384 cores with sets of dedicated writer cores.

A Performance Comparison of Continuous and Discontinuous Finite Element Shallow Water Models

We present a comparative study of two finite element shallow water equation (SWE) models: a generalized wave continuity equation based continuous Galerkin (CG) model--an approach used by several existing SWE models--and a recently developed discontinuous Galerkin (DG) model. While DG methods are known to possess a number of favorable properties, such as local mass conservation, one commonly cited disadvantage is the larger number of degrees of freedom associated with the methods, which naturally translates into a greater computational cost compared to CG methods. However, in a series of numerical tests, we demonstrate that the DG SWE model is generally more efficient than the CG model (i) in terms of achieving a specified error level for a given computational cost and (ii) on large-scale parallel machines because of the inherently local structure of the method. Both models are verified on a series of analytic test cases and validated on a field-scale application.

Nonlinear effects in the propagation properties of numerical schemes employed in computational hydraulics and computational hydrology

Fourier analysis is a powerful tool for the determination of the propagation properties of numerical schemes employed in computational hydraulics and computational hydrology. Nevertheless, its use is restricted to cases that involve linear and constant coefficient equations, subject to periodic boundary conditions, or to linear and constant coefficient pure initial value problems occurring in infinite spatial domains. Most equations that describe hydraulic and hydrologic phenomena are nonlinear and, therefore, such limitation prevents the rigorous application of Fourier analysis and may even lead to erroneous conclusions when a so-called “frozen coefficients approach” is used. The purpose of this paper is to present a general methodology that removes the above described limitation. In order to accomplish this task, the differential or difference operator of the evolution equation or system of equations, are expanded in a Taylor–Fréchet series around a reference solution, which is assumed to be slowly varying. When only the zeroth and first order terms in the series are kept, the resulting equation is linear in a perturbation function, even though it depends nonlinearly on the reference solution, thereby capturing the dominant nonlinear behavior of the system under analysis. The ensuing step consists of performing a local analysis of the resulting equation or system of equations. Such local analysis is asymptotic in a conveniently defined small parameter. The zeroth order asymptotic approximation of the solution is governed by a linear, pure initial value problem with constant coefficients. Thus, Fourier analysis is applicable to such problem. Examples drawn from computational hydrology, such as the stability and iteration convergence of schemes employed for the solution of Richards equation, that describes the motion of water in partially saturated porous media; and from computational hydraulics, such as the stability of the generalized Preissman scheme, and the consistency of the generalized wave continuity equation formulation of the shallow water equations, are included, showing the advantages of the proposed methodology to explain observations that have defied understanding, when a frozen coefficient approach is employed.

In search of a consistent and conservative mass flux for the GWCE

Two methods for computing local mass flux for a continuous Galerkin finite element formulation of the Generalized Wave Continuity Equation (GWCE) are derived and a third method is discussed in light of the first two. The GWCE is shown to not conserve mass locally, while it can be shown to conserve a certain quantity locally. The two derived methods are demonstrated for a realistic tidal flow problem in the Bight of Abaco, Bahamas.

On the form of the momentum equation for shallow water models based on the generalized wave continuity equation

Nearly all generalized wave continuity (GWC)-based models utilize the velocity-based, non-conservative form of the momentum equation to obtain the depth-averaged changes in velocity. It has been hypothesized that a flux-based, conservative form of the momentum equation may improve accuracy and stability. Herein, we study the impact of the choice of dependent variable and form of the momentum equation in a GWC-based finite element shallow water model. The impact of this change on mass balance, stability, and accuracy (spatial and temporal) is rigorously assessed, first for 1D barotropic flows and then for 2D barotropic flows in a variety of basins. Both 1D and 2D results indicate that the conservative form improves mass balance on both global and local scales, with the most significant gains found in local mass balance in areas with steep bathymetry gradients. This is also the region where the conservative form shows an increase in local spatial accuracy. Taylor series analysis and numerical simulations indicate a strong correlation between local spatial truncation errors and local mass balance errors. Stability, temporal accuracy and global spatial accuracy do not show statistically significant changes between the two algorithms in both 1D and 2D studies.

Application of a coupled discontinuous–continuous Galerkin finite element shallow water model to coastal ocean dynamics

A coupled discontinuous–continuous Galerkin (DG–CG) shallow water model is compared to a continuous Galerkin generalized wave-continuity equation (GWCE) based model for the coastal ocean, whereby local mass imbalance typical of GWCE-based solutions is eliminated using the coupled DG–CG approach. Two mass imbalance indicators for the GWCE-based model are presented and analyzed. The indicators motivate discussion on the suitability of using a GWCE-based model versus the locally conservative coupled DG–CG model. Both realistic and idealized test problems for tide, wind, and wave-driven circulation form the basis of the study. For the problems studied, coupled DG–CG solutions retain the robustness of well-documented solutions from GWCE-based models and also capture the dynamics driven by small-scale, highly advective processes which are problematic for GWCE-based models. Issues associated with the coupled DG–CG model are explored, including increased cost due to increased degrees of freedom, the necessary application of slope limiters, as well as the actual coupling process.

Similarities between the quasi‐bubble and the generalized wave continuity equation solutions to the shallow water equations

AbstractTwo common strategies for solving the shallow water equations in the finite element community are the generalized wave continuity equation (GWCE) reformulation and the quasi‐bubble velocity approximation. The GWCE approach has been widely analysed in the literature. In this work, the quasi‐bubble equations are analysed and comparisons are made between the quasi‐bubble approximation of the primitive form of the shallow water equations and a linear finite element approximation of the GWCE reformulation of the shallow water equations. The discrete condensed quasi‐bubble continuity equation is shown to be identical to a discrete wave equation for a specific GWCE weighting parameter value. The discrete momentum equations are slightly different due to the bubble function. In addition, the dispersion relationships are shown to be almost identical and numerical experiments confirm that the two schemes compute almost identical results. Analysis of the quasi‐bubble formulation suggests a relationship that may guide selection of the optimal GWCE weighting parameter. Copyright © 2004 John Wiley & Sons, Ltd.

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

AbstractDispersion 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.

A 2D implicit time‐marching algorithm for shallow water models based on the generalized wave continuity equation

AbstractThis paper builds upon earlier work that developed and evaluated a 1D predictor–corrector time‐marching algorithm for wave equation models and extends it to 2D. Typically, the generalized wave continuity equation (GWCE) utilizes a three time‐level semi‐implicit scheme centred at k, and the momentum equation uses a two time‐level scheme centred at k+12. It has been shown that in highly non‐linear applications, the algorithm becomes unstable at even moderate Courant numbers. This work implements and analyses an implicit treatment of the non‐linear terms through the use of an iterative time‐marching algorithm in the two‐dimensional framework. Stability results show at least an eight‐fold increase in the maximum time step, depending on the domain. Studies also examined the sensitivity of the G parameter (a numerical weighting parameter in the GWCE) with results showing the greatest increase in stability occurs when 1⩽G/τmax⩽10, a range that coincides with the recommended range to minimize errors. Convergence studies indicate an increase in temporal accuracy from first order to second order, while overall error is less than the original algorithm, even at higher time steps. Finally, a parallel implementation of the new algorithm shows that it scales well. Copyright © 2004 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.

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

AbstractWave 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.

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.

Finite Element Approximations to the System of Shallow Water Equations, Part III: On the Treatment of Boundary Conditions

We continue our investigation of finite element approximations to the system of shallow water equations, based on the generalized wave continuity equation (GWCE) formulation. In previous work, we analyzed this systemassuming Dirichlet boundary conditions on both elevation and velocity. Based on physical grounds, it is possible to not impose boundary conditions on elevation. Thus we examine the formulation for the case of Dirichlet conditions on velocity only. The changes required to the finite element method are presented, and stability and error estimates are derived for an approximate linear model, assuming continuous time. Stability for a discrete time method is also shown.

A heterogenous level method for three-dimensional hydrodynamics and salinity bay modelling

The development and application of a heterogenous level method for three-dimensional hydrodynamics and salinity modelling is described. In this model the number of levels in the vertical dimension is adjusted according to the bathymetry at each horizontal node. The generalized wave continuity equation is solved first for the free surface elevation, then the horizontal velocities are obtained by solving the horizontal momentum equations. Next the vertical velocity is solved with the three-dimensional primitive equation. Finally the transport equation is solved for the salinity. Two examples are presented for the effectiveness of this model.

Finite Element Approximations to the System of Shallow Water Equations I: Continuous-Time A Priori Error Estimates

Various sophisticated finite element models for surface water flow based on the shallow water equations exist in the literature. Gray, Kolar, Luettich, Lynch, and Westerink have developed a hydrodynamic model based on the generalized wave continuity equation (GWCE) formulation and have formulated a Galerkin finite element procedure based on combining the GWCE with the nonconservative momentum equations. Numerical experiments suggest that this method is robust and accurate and suppresses spurious oscillations which plague other models. We analyze a slightly modified Galerkin model which uses the conservative momentum equations (CME). For this GWCE-CME system of equations, we present a continuous-time a priori error estimate based on an $\asl^{2}$ projection.

A filtered solution of the primitive shallow-water equations

It is well documented that the direct solution of the primitive equation form (PE) of the shallow-water equations by conventional Galerkin finite elements fails to control numerical noise. While some alternative numerical techniques applied directly to the primitive equations have been successfully developed, a reformulation of the fundamental equations into the generalized wave continuity equation form (GWCE) provides an alternative noise-free procedure that has been used in numerous field applications. The selection of penalty parameter, ‘G’ for GWCE must, however, be carefully adjusted to achieve both mass balance and noise control. In this paper, a numerical method for solving the primitive equation with a filter is presented. It is shown that a filtered primitive equation method (FPE) controls numerical noise and is mass conservative. Since the FPE method derives its basis from observations about the behavior of GWCE, we present comparisons of the solution of the primitive equations via FPE with the solution of the wave equation via the GWCE form.

Shallow water modeling in spherical coordinates: equation formulation, numerical implementation, and application

One class of surface water models is the shallow water models obtained by depth-averaging the microscale mass and momentum balances. Application of shallow water models to large scale problems (on the order of 1000's of km) requires the use of spherical coordinates. Traditionally, balance laws in spherical coordinates are derived by simply expanding the spatial operators in the standard depth-averaged equations. However, the equations themselves are based on an assumed planar surface so that an inconsistency exists between the derivation and the interpretation. In this article, a method is presented that properly accounts for the curvature of the Earth during the depth-averaging procedure. The derivation gives rise to new terms in both the continuity and momentum balances, terms that we refer to as curvature terms. A scaling analysis evaluates the magnitude of the terms. It is shown that the curvature term in the continuity balance is insignificant when the vertical velocity is small, i.e., at least four orders of magnitude less than lateral components. Similarly, when the vertical velocity is small, the curvature term in the momentum balance that comes from the convective terms is insignificant. The second curvature term in the momentum balance is associated with the macroscopic stress tensor; it is on the same order of magnitude as other stress terms and should be retained when momentum dissipation is modeled. Finite element discretization of the resulting balance laws utilizes the generalized wave continuity equation which, because of its monotonie dispersion relation, eliminates noise in the solution without resorting to artificial damping. A cartographic mapping simplifies finite element implementation in spherical coordinates so that Cartesian master elements can be used. Finally, an application to the Western North Atlantic and Gulf of Mexico illustrates the importance of using equations in spherical coordinates for large scale applications.