An efficient and accurate non‐hydrostatic model with embedded Boussinesq‐type like equations for surface wave modeling

A novel approach that embeds the Boussinesq‐type like equations into an implicit non‐hydrostatic model (NHM) is developed. Instead of using an integration approach, Boussinesq‐type like equations with a reference velocity under a virtual grid system are introduced to analytically obtain an analytical form of pressure distribution at the top layer. To determine the size of vertical layers in the model, a top‐layer control technique is proposed and the reference location is employed to optimize linear wave dispersion property. The efficiency and accuracy of this NHM with Boussinesq‐type like equations (NHM‐BTE) are critically examined through four free‐surface wave examples. Overall model results show that NHM‐BTE using only two vertical layers is capable of accurately simulating highly dispersive wave motion and wave transformation over irregular bathymetry. Copyright © 2008 John Wiley & Sons, Ltd.     

A σ‐coordinate non‐hydrostatic model with embedded Boussinesq‐type‐like equations for modeling deep‐water waves

A σ‐coordinate non‐hydrostatic model, combined with the embedded Boussinesq‐type‐like equations, a reference velocity, and an adapted top‐layer control, is developed to study the evolution of deep‐water waves. The advantage of using the Boussinesq‐type‐like equations with the reference velocity is to provide an analytical‐based non‐hydrostatic pressure distribution at the top‐layer and to optimize wave dispersion property. The σ‐based non‐hydrostatic model naturally tackles the so‐called overshooting issue in the case of non‐linear steep waves. Efficiency and accuracy of this non‐hydrostatic model in terms of wave dispersion and nonlinearity are critically examined. Overall results show that the newly developed model using a few layers is capable of resolving the evolution of non‐linear deep‐water wave groups. Copyright © 2009 John Wiley & Sons, Ltd.     

A semi‐implicit finite difference method for non‐hydrostatic,free‐surface flows

In this paper a semi‐implicit finite difference model for non‐hydrostatic, free‐surface flows is analyzed and discussed. It is shown that the present algorithm is generally more accurate than recently developed models for quasi‐hydrostatic flows. The governing equations are the free‐surface Navier–Stokes equations defined on a general, irregular domain of arbitrary scale. The momentum equations, the incompressibility condition and the equation for the free‐surface are integrated by a semi‐implicit algorithm in such a fashion that the resulting numerical solution is mass conservative and unconditionally stable with respect to the gravity wave speed, wind stress, vertical viscosity and bottom friction. Copyright © 1999 John Wiley & Sons, Ltd.     

Three‐dimensional numerical modelling of free surface flows with non‐hydrostatic pressure

A three‐dimensional numerical model is developed for incompressible free surface flows. The model is based on the unsteady Reynolds‐averaged Navier–Stokes equations with a non‐hydrostatic pressure distribution being incorporated in the model. The governing equations are solved in the conventional sigma co‐ordinate system, with a semi‐implicit time discretization. A fractional step method is used to enable the pressure to be decomposed into its hydrostatic and hydrodynamic components. At every time step one five‐diagonal system of equations is solved to compute the water elevations and then the hydrodynamic pressure is determined from a pressure Poisson equation. The model is applied to three examples to simulate unsteady free surface flows where non‐hydrostatic pressures have a considerable effect on the velocity field. Emphasis is focused on applying the model to wave problems. Two of the examples are about modelling small amplitude waves where the hydrostatic approximation and long wave theory are not valid. The other example is the wind‐induced circulation in a closed basin. The numerical solutions are compared with the available analytical solutions for small amplitude wave theory and very good agreement is obtained. Copyright © 2002 John Wiley & Sons, Ltd.     

A two‐dimensional vertical non‐hydrostatic σ model with an implicit method for free‐surface flows

An implicit finite difference model in the σ co‐ordinate system is developed for non‐hydrostatic, two‐dimensional vertical plane free‐surface flows. To accurately simulate interaction of free‐surface flows with uneven bottoms, the unsteady Navier–Stokes equations and the free‐surface boundary condition are solved simultaneously in a regular transformed σ domain using a fully implicit method in two steps. First, the vertical velocity and pressure are expressed as functions of horizontal velocity. Second, substituting these relationship into the horizontal momentum equation provides a block tri‐diagonal matrix system with the unknown of horizontal velocity, which can be solved by a direct matrix solver without iteration. A new treatment of non‐hydrostatic pressure condition at the top‐layer cell is developed and found to be important for resolving the phase of wave propagation. Additional terms introduced by the σ co‐ordinate transformation are discretized appropriately in order to obtain accurate and stable numerical results. The developed model has been validated by several tests involving free‐surface flows with strong vertical accelerations and non‐linear waves interacting with uneven bottoms. Comparisons among numerical results, analytical solutions and experimental data show the capability of the model to simulate free‐surface flow problems. Copyright © 2004 John Wiley & Sons, Ltd.     

A 3‐D non‐hydrostatic pressure model for small amplitude free surface flows

A three‐dimensional, non‐hydrostatic pressure, numerical model with k–ε equations for small amplitude free surface flows is presented. By decomposing the pressure into hydrostatic and non‐hydrostatic parts, the numerical model uses an integrated time step with two fractional steps. In the first fractional step the momentum equations are solved without the non‐hydrostatic pressure term, using Newton's method in conjunction with the generalized minimal residual (GMRES) method so that most terms can be solved implicitly. This method only needs the product of a Jacobian matrix and a vector rather than the Jacobian matrix itself, limiting the amount of storage and significantly decreasing the overall computational time required. In the second step the pressure–Poisson equation is solved iteratively with a preconditioned linear GMRES method. It is shown that preconditioning reduces the central processing unit (CPU) time dramatically. In order to prevent pressure oscillations which may arise in collocated grid arrangements, transformed velocities are defined at cell faces by interpolating velocities at grid nodes. After the new pressure field is obtained, the intermediate velocities, which are calculated from the previous fractional step, are updated. The newly developed model is verified against analytical solutions, published results, and experimental data, with excellent agreement. Copyright © 2005 John Wiley & Sons, Ltd.     

An implicit numerical algorithm for solving non‐hydrostatic free‐surface flow problems

Details are given of the development of a two‐dimensional vertical numerical model for simulating unsteady free‐surface flows, using a non‐hydrostatic pressure distribution. In this model, the Reynolds equations and the kinematic free‐surface boundary condition are solved simultaneously, so that the water surface elevation can be integrated into the solution and solved for, together with the velocity and pressure fields. An efficient numerical algorithm has been developed, deploying implicit parameters similar to those used in the Crank–Nicholson method, and generating a block tri‐diagonal algebraic system of equations. The model has been applied to simulate a range of unsteady flow problems involving relatively strong vertical accelerations. The results show that the numerical algorithm described is able to produce accurate predictions and is also easy to apply. Copyright © 2001 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.     

An improved depth-averaged nonhydrostatic shallow water model with quadratic pressure approximation

Phase-resolved information is necessary for many coastal wave problems, for example, for the wave conditions in the vicinity of harbor structures. Two-dimensional (2D) depth-averaging shallow water models are commonly used to obtain a phase-resolved solution near the coast. These models are in general more computationally effective compared with computational fluid dynamics software and will be even more capable if equipped with a parallelized code. In the current article, a 2D wave model solving the depth-averaged continuity equation and the Euler equations is implemented in the open-source hydrodynamic code REEF3D. The model is based on a nonhydrostatic extension and a quadratic vertical pressure profile assumption, which provides a better approximation of the frequency dispersion. It is the first model of its kind to employ high-order discretization schemes and to be fully parallelized following the domain decomposition strategy. Wave generation and absorption are achieved with a relaxation method. The simulations of nonlinear long wave propagations and transformations over nonconstant bathymetries are presented. The results are compared with benchmark wave propagation cases. A large-scale wave propagation simulation over realistic irregular topography is shown to demonstrate the model's capability of solving operational large-scale problems.     

High‐order compact numerical schemes for non‐hydrostatic free surface flows

The development of a numerical scheme for non‐hydrostatic free surface flows is described with the objective of improving the resolution characteristics of existing solution methods. The model uses a high‐order compact finite difference method for spatial discretization on a collocated grid and the standard, explicit, single step, four‐stage, fourth‐order Runge–Kutta method for temporal discretization. The Cartesian coordinate system was used. The model requires the solution of two Poisson equations at each time‐step and tridiagonal matrices for each derivative at each of the four stages in a time‐step. Third‐ and fourth‐order accurate boundaries for the flow variables have been developed including the top non‐hydrostatic pressure boundary. The results demonstrate that numerical dissipation which has been a problem with many similar models that are second‐order accurate is practically eliminated. A high accuracy is obtained for the flow variables including the non‐hydrostatic pressure. The accuracy of the model has been tested in numerical experiments. In all cases where analytical solutions are available, both phase errors and amplitude errors are very small. Copyright © 2006 John Wiley & Sons, Ltd.     

A structured but non‐uniform Cartesian grid‐based model for the shallow water equations

In large‐scale shallow flow simulations, local high‐resolution predictions are often required in order to reduce the computational cost without losing the accuracy of the solution. This is normally achieved by solving the governing equations on grids refined only to those areas of interest. Grids with varying resolution can be generated by different approaches, e.g. nesting methods, patching algorithms and adaptive unstructured or quadtree gridding techniques. This work presents a new structured but non‐uniform Cartesian grid system as an alternative to the existing approaches to provide local high‐resolution mesh. On generating a structured but non‐uniform Cartesian grid, the whole computational domain is first discretized using a coarse background grid. Local refinement is then achieved by directly allocating a specific subdivision level to each background grid cell. The neighbour information is specified by simple mathematical relationships and no explicit storage is needed. Hence, the structured property of the uniform grid is maintained. After employing some simple interpolation formulae, the governing shallow water equations are solved using a second‐order finite volume Godunov‐type scheme in a similar way as that on a uniform grid. Copyright © 2010 John Wiley & Sons, Ltd.     

An implicit three‐dimensional fully non‐hydrostatic model for free‐surface flows

An implicit method is developed for solving the complete three‐dimensional (3D) Navier–Stokes equations. The algorithm is based upon a staggered finite difference Crank‐Nicholson scheme on a Cartesian grid. A new top‐layer pressure treatment and a partial cell bottom treatment are introduced so that the 3D model is fully non‐hydrostatic and is free of any hydrostatic assumption. A domain decomposition method is used to segregate the resulting 3D matrix system into a series of two‐dimensional vertical plane problems, for each of which a block tri‐diagonal system can be directly solved for the unknown horizontal velocity. Numerical tests including linear standing waves, nonlinear sloshing motions, and progressive wave interactions with uneven bottoms are performed. It is found that the model is capable to simulate accurately a range of free‐surface flow problems using a very small number of vertical layers (e.g. two–four layers). The developed model is second‐order accuracy in time and space and is unconditionally stable; and it can be effectively used to model 3D surface wave motions. Copyright © 2004 John Wiley & Sons, Ltd.     

A numerical method for the determination of weakly non‐hydrostatic non‐linear free surface wave propagation

A numerical method is described that may be used to determine the propagation characteristics of weakly non‐hydrostatic non‐linear free surface waves over a general, bottom topography. In shallow water of constant undisturbed depth, such waves are equivalent to the familiar cnoidal waves characterized by sharp crests and relatively flat troughs. For a certain range of parameters, these propagate without change of form by virtue of the weakly non‐hydrostatic balance in the vertical momentum equation. Effectively, this counters the tendency for the non‐linearity in a purely hydrostatic theory to lead to a continuously deforming surface wave profile. The realistic representation furnished by cnoidal wave theory of free surface waves in the shallow near‐shore zone has led to its utilization in evaluating their propagation characteristics. Nonetheless, the classic analytical theory is inapplicable to the case of wave propagation over a sloping beach or off‐shore sand bar topography. Under these conditions, a local change in form of the surface wave profile is anticipated before the waves break and knowing this is required in order to evaluate fully the propagation process. The efficacy of the numerical method is first demonstrated by comparing the solution for water of constant depth with the evaluation of the analytical solution expressed in terms of the Jacobian elliptic function cn. The general method described in the paper is then illustrated by experiments to determine the change in profile of weakly non‐hydrostatic non‐linear surface waves propagating over bed forms representative of those found in shallow coastal seas. Copyright © 2001 John Wiley & Sons, Ltd.     

Practical evaluation of five partly discontinuous finite element pairs for the non‐conservative shallow water equations

This paper provides a comparison of five finite element pairs for the shallow water equations. We consider continuous, discontinuous and partially discontinuous finite element formulations that are supposed to provide second‐order spatial accuracy. All of them rely on the same weak formulation, using Riemann solver to evaluate interface integrals. We define several asymptotic limit cases of the shallow water equations within their space of parameters. The idea is to develop a comparison of these numerical schemes in several relevant regimes of the subcritical shallow water flow. Finally, a new pair, using non‐conforming linear elements for both velocities and elevation (P?P), is presented, giving optimal rates of convergence in all test cases. P?P₁ and P?P₁ mixed formulations lack convergence for inviscid flows. P?P₂ pair is more expensive but provides accurate results for all benchmarks. P?P provides an efficient option, except for inviscid Coriolis‐dominated flows, where a small lack of convergence is observed. Copyright © 2009 John Wiley & Sons, Ltd.     

Further experiences with computing non‐hydrostatic free‐surface flows involving water waves

A semi‐implicit, staggered finite volume technique for non‐hydrostatic, free‐surface flow governed by the incompressible Euler equations is presented that has a proper balance between accuracy, robustness and computing time. The procedure is intended to be used for predicting wave propagation in coastal areas. The splitting of the pressure into hydrostatic and non‐hydrostatic components is utilized. To ease the task of discretization and to enhance the accuracy of the scheme, a vertical boundary‐fitted co‐ordinate system is employed, permitting more resolution near the bottom as well as near the free surface. The issue of the implementation of boundary conditions is addressed. As recently proposed by the present authors, the Keller‐box scheme for accurate approximation of frequency wave dispersion requiring a limited vertical resolution is incorporated. The both locally and globally mass conserved solution is achieved with the aid of a projection method in the discrete sense. An efficient preconditioned Krylov subspace technique to solve the discretized Poisson equation for pressure correction with an unsymmetric matrix is treated. Some numerical experiments to show the accuracy, robustness and efficiency of the proposed method are presented. Copyright © 2004 John Wiley & Sons, Ltd.     

A new well‐balanced Hermite weighted essentially non‐oscillatory scheme for shallow water equations

Hermite weighted essentially non‐oscillatory (HWENO) methods were introduced in the literature, in the context of Euler equations for gas dynamics, to obtain high‐order accuracy schemes characterized by high compactness (e.g. Qiu and Shu, J. Comput. Phys. 2003; 193 :115). For example, classical fifth‐order weighted essentially non‐oscillatory (WENO) reconstructions are based on a five‐cell stencil whereas the corresponding HWENO reconstructions are based on a narrower three‐cell stencil. The compactness of the schemes allows easier treatment of the boundary conditions and of the internal interfaces. To obtain this compactness in HWENO schemes both the conservative variables and their first derivatives are evolved in time, whereas in the original WENO schemes only the conservative variables are evolved. In this work, an HWENO method is applied for the first time to the shallow water equations (SWEs), including the source term due to the bottom slope, to obtain a fourth‐order accurate well‐balanced compact scheme. Time integration is performed by a strong stability preserving the Runge–Kutta method, which is a five‐step and fourth‐order accurate method. Besides the classical SWE, the non‐homogeneous equations describing the time and space evolution of the conservative variable derivatives are considered here. An original, well‐balanced treatment of the source term involved in such equations is developed and tested. Several standard one‐dimensional test cases are used to verify the high‐order accuracy, the C‐property and the good resolution properties of the model. Copyright © 2010 John Wiley & Sons, Ltd.