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.     

Some applications of the concept of minimized integrated exponential error for low dispersion and low dissipation

Several techniques to optimize parameters that regulate dispersion and dissipation effects in finite difference schemes have been devised in our previous works. They all use the notion that dissipation neutralizes dispersion. These techniques are the minimized integrated square difference error (MISDE) and the minimized integrated exponential error for low dispersion and low dissipation (MIEELDLD). It is shown in this work based on several numerical schemes tested that the technique of MIEELDLD is more accurate than MISDE to optimize the parameters that regulate dispersion and dissipation effects with the aim of improving the shock‐capturing properties of numerical methods. First, we consider the family of third‐order schemes proposed by Takacs. We use the techniques MISDE and MIEELDLD to optimize two parameters, namely, the cfl number and another variable which also controls dispersion and dissipation. Second, these two techniques are used to optimize a numerical scheme proposed by Gadd. Moreover, we compute the optimal cfl for some multi‐level schemes in 1D. Numerical tests for some of these numerical schemes mentioned above are performed at different cfl numbers and it is shown that the results obtained are dependent on the cfl number chosen. The errors from the numerical results have been quantified into dispersion and dissipation using a technique devised by Takacs. Finally, we make use of a composite scheme made of corrected Lax–Friedrichs and the two‐step Lax–Friedrichs schemes like the CFLF4 scheme at its optimal cfl number, to solve some problems in 2D, namely: solid body rotation test, acoustics and the circular Riemann problem. Copyright © 2011 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.     

Numerical simulation of pollutant transport acted by wave for a shallow water sea bay

A numerical model of pollutant transport acted by water waves on a shallow‐water mild‐slope beach is established in this study. The numerical model is combined with a wave propagation model, a multiple wave‐breaking model, a wave‐induced current model and a pollutant convection–dispersion model. The wave propagation model is based on the higher‐order approximation of parabolic mild‐slope equation which can be used to simulate the wave refraction, diffraction and breaking in a large area of near‐shore zone combined with the wave‐breaking model. The wave‐induced current model is established using the concept of the radiation stress and considering the effect of bottom resistance caused by waves. The numerical model is verified by laboratory experiment results of regular and irregular waves over two mild beaches with different slopes. The numerical results agree well with experimental results. The numerical model has been applied in the near‐shore zone of Bohai bay in China. It is concluded that pollutant transport parallel to the shoreline due to the action of waves, which will induce serious pollution on the beach. Copyright © 2005 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.     

Decomposition methods for time‐domain Maxwell's equations

Decomposition methods based on split operators are proposed for numerical integration of the time‐domain Maxwell's equations for the first time. The methods are obtained by splitting the Hamiltonian function of Maxwell's equations into two analytically computable exponential sub‐propagators in the time direction based on different order decomposition methods, and then the equations are evaluated in the spatial direction by the staggered fourth‐order finite‐difference approximations. The stability and numerical dispersion analysis for different order decomposition methods are also presented. The theoretical predictions are confirmed by our numerical results. Copyright © 2007 John Wiley & Sons, Ltd.     

A higher‐order accuracy lattice Boltzmann model for the wave equation

A lattice Boltzmann model with higher‐order accuracy for the wave motion is proposed. The new model is based on the technique of the higher‐order moment of equilibrium distribution functions and a series of lattice Boltzmann equations in different time scales. The forms of moments are derived from the binary wave equation by designing the higher‐order dissipation and dispersion terms. The numerical results agree well with classical ones. Copyright © 2008 John Wiley & Sons, Ltd.     

A PDF propagation approach to model turbulent dispersion in swirling flows

A computationally efficient approach that solves for the spatial covariance matrix along the dense particle ensemble-averaged trajectory has been successfully applied to describe turbulent dispersion in swirling flows. The procedure to solve for the spatial covariance matrix is based on turbulence isotropy assumption, and it is analogous to Taylor's approach for turbulent dispersion. Unlike stochastic dispersion models, this approach does not involve computing a large number of individual particle trajectories in order to adequately represent the particle phase; a few representative particle ensembles are sufficient to describe turbulent dispersion. The particle Lagrangian properties required in this method are based on a previous study (Shirolkar and McQuay, 1998). The fluid phase information available from practical turbulence models is sufficient to estimate the time and length scales in the model. In this study, two different turbulence models are used to solve for the fluid phase – the standard k–ε model, and a multiple-time-scale (MTS) model. The models developed here are evaluated with the experiments of Sommerfeld and Qiu (1991). A direct comparison between the dispersion model developed in this study and a stochastic dispersion model based on the eddy lifetime concept is also provided. Estimates for the Reynolds stresses required in the stochastic model are obtained from a set of second-order algebraic relations. The results presented in the study demonstrate the computational efficiency of the present dispersion modeling approach. The results also show that the MTS model provides improved single-phase results in comparison to the k–ε model. The particle statistics, which are computed based on the fundamentals of the present approach, compare favorably with the experimental data. Furthermore, these statistics closely compare to those obtained using a stochastic dispersion model. Finally, the results indicate that the particle predictions are relatively unaffected by whether the Reynolds stresses are based on algebraic relations or on the turbulence isotropy assumption.     

Combined compact difference method for solving the incompressible Navier–Stokes equations

This paper presents a numerical method for solving the two‐dimensional unsteady incompressible Navier–Stokes equations in a vorticity–velocity formulation. The method is applicable for simulating the nonlinear wave interaction in a two‐dimensional boundary layer flow. It is based on combined compact difference schemes of up to 12th order for discretization of the spatial derivatives on equidistant grids and a fourth‐order five‐ to six‐alternating‐stage Runge–Kutta method for temporal integration. The spatial and temporal schemes are optimized together for the first derivative in a downstream direction to achieve a better spectral resolution. In this method, the dispersion and dissipation errors have been minimized to simulate physical waves accurately. At the same time, the schemes can efficiently suppress numerical grid‐mesh oscillations. The results of test calculations on coarse grids are in good agreement with the linear stability theory and comparable with other works. The accuracy and the efficiency of the current code indicate its potential to be extended to three‐dimensional cases in which full boundary layer transition happens. Copyright © 2011 John Wiley & Sons, Ltd.     

Dispersion error reduction for acoustic problems using the smoothed finite element method (SFEM)

The smoothed finite element method (SFEM), which was recently introduced for solving the mechanics and acoustic problems, uses the gradient smoothing technique to operate over the cell‐based smoothing domains. On the basis of the previous work, this paper reports a detailed analysis on the numerical dispersion error in solving two‐dimensional acoustic problems governed by the Helmholtz equation using the SFEM, in comparison with the standard finite element method. Owing to the proper softening effects provided naturally by the cell‐based gradient smoothing operations, the SFEM model behaves much softer than the standard finite element method model. Therefore, the SFEM can significantly reduce the dispersion error in the numerical solution. Results of both theoretical and numerical experiments will support these important findings. It is shown clearly that the SFEM suits ideally well for solving acoustic problems, because of the crucial effectiveness in reducing the dispersion error. Copyright © 2015 John Wiley & Sons, Ltd.     

Acoustic simulation using a novel approach for reducing dispersion error

It is well‐known that the traditional finite element method (FEM) fails to provide accurate results to the Helmholtz equation with the increase of wave number because of the ‘pollution error’ caused by numerical dispersion. In order to overcome this deficiency, a gradient‐weighted finite element method (GW‐FEM) that combines Shepard interpolation and linear shape functions is proposed in this work. Three‐node triangular and four‐node tetrahedral elements that can be generated automatically are first used to discretize the problem domain in 2D and 3D spaces, respectively. For each independent element, a compacted support domain is then formed based on the element itself and its adjacent elements sharing common edges (or faces). With the aid of Shepard interpolation, a weighted acoustic gradient field is then formulated, which will be further used to construct the discretized system equations through the generalized Galerkin weak form. Numerical examples demonstrate that the present algorithm can significantly reduces the dispersion error in computational acoustics. Copyright © 2016 John Wiley & Sons, Ltd.     

Internal waves in a two‐layer system using fully nonlinear internal‐wave equations

In order to understand the nonlinear effect in a two‐layer system, fully nonlinear strongly dispersive internal‐wave equations, based on a variational principle, were proposed in this study. A simple iteration method was used to solve the internal‐wave equations in order to solve the equations stably. The applicability of the proposed numerical computation scheme was confirmed to agree with linear dispersion relation theoretically obtained from variational principle. The proposed computational scheme was also shown to reproduce internal waves including higher‐order nonlinear effect from the analysis of internal solitary waves in a two‐layer system. Furthermore, for the second‐order numerical analysis, the balance of nonlinearity and dispersion was found to be similar to the balance assumed in the KdV theory and the Boussinesq‐type equations. Copyright © 2009 John Wiley & Sons, Ltd.     

Robustness of the hybrid DRP‐WENO scheme for shock flow computations

This paper describes a new variant of hybrid scheme that is constructed by a wave‐capturing scheme and a nonoscillatory scheme for flow computations in the presence of shocks. The improved fifth‐order upwind weighted essentially nonoscillatory scheme is chosen to be conjugated with the seven‐point dispersion‐relation‐preserving scheme by means of an adaptive switch function of grid‐point type. The new hybrid scheme can achieve a better resolution than the hybrid scheme which is based on the classical weighted essentially scheme. Ami Harten's multiresolution analysis algorithm is applied to density field for detecting discontinuities and setting point values of the switch function adaptively. Moreover, the tenth‐order central filter is applied in smooth part of the flow field for damping dispersion errors. This scheme can promote overall computational efficiency and yield oscillation‐free results in shock flows. The resolution properties and robustness of the new hybrid scheme are tested in both 1D and 2D linear and nonlinear cases. It performs well for computing flow problems with rich structures of weak/strong shocks and large/small vortices, such as the shock‐boundary layer interaction problem in a shock tube, which illustrates that it is very robust and accurate for direct numerical simulation of gas‐dynamics flows. Copyright © 2011 John Wiley & Sons, Ltd.     

Inviscid flow modelling using asymmetric implicit finite difference schemes

Asymmetric spatial implicit high‐order schemes are introduced and, based on Fourier analysis, the dispersion and damping are calculated depending on the asymmetry parameter. The derived schemes are then applied to a number of inviscid problems. For incompressible convection problems the proposed asymmetric schemes (applied as upwind schemes) lead to stable and accurate results. To extend the applicability of the proposed schemes to compressible problems acoustic upwinding is used. In a two‐dimensional compressible flow example acoustic and conventional upwinding are combined. Evaluation of all presented results leads to the conclusion that, of the studied schemes, the implicit fifth order upwinding scheme with an asymmetry parameter of about 0.5 leads to the optimal results. Copyright © 2005 John Wiley & Sons, Ltd.     

Application of the lattice‐Boltzmann method to study flow and dispersion in channels with and without expansion and contraction geometry

The lattice‐Boltzmann (LB) method, derived from lattice gas automata, is a relatively new technique for studying transport problems. The LB method is investigated for its accuracy to study fluid dynamics and dispersion problems. Two problems of relevance to flow and dispersion in porous media are addressed: (i) Poiseuille flow between parallel plates (which is analogous to flow in pore throats in two‐dimensional porous networks), and (ii) flow through an expansion–contraction geometry (which is analogous to flow in pore bodies in two‐dimensional porous networks). The results obtained from the LB simulations are compared with analytical solutions when available, and with solutions obtained from a finite element code (FIDAP) when analytical results are not available. Excellent agreement is found between the LB results and the analytical/FIDAP solutions in most cases, indicating the utility of the lattice‐Boltzmann method for solving fluid dynamics and dispersion problems. Copyright © 1999 John Wiley & Sons, Ltd.     

Optimized sixth‐order monotonicity‐preserving scheme by nonlinear spectral analysis

In this paper, sixth‐order monotonicity‐preserving optimized scheme (OMP6) for the numerical solution of conservation laws is developed on the basis of the dispersion and dissipation optimization and monotonicity‐preserving technique. The nonlinear spectral analysis method is developed and is used for the purpose of minimizing the dispersion errors and controlling the dissipation errors. The new scheme (OMP6) is simple in expression and is easy for use in CFD codes. The suitability and accuracy of this new scheme have been tested through a set of one‐dimensional, two‐dimensional, and three‐dimensional tests, including the one‐dimensional Shu–Osher problem, the two‐dimensional double Mach reflection, and the Rayleigh–Taylor instability problem, and the three‐dimensional direct numerical simulation of decaying compressible isotropic turbulence. All numerical tests show that the new scheme has robust shock capturing capability and high resolution for the small‐scale waves due to fewer numerical dispersion and dissipation errors. Moreover, the new scheme has higher computational efficiency than the well‐used WENO schemes. Copyright © 2013 John Wiley & Sons, Ltd.     

A fast non‐Fickian particle‐tracking diffusion simulator and the effect of shear on the pollutant diffusion process

This paper presents a fast method for the generation of non‐Fickian particle paths within a particle‐tracking pollutant diffusion model based on a Fourier spectral representation of fractional Brownian motion (fBm), a generalization of ordinary Brownian motion. Correlated diffusive components in a particle‐tracking algorithm are modelled using fBm increments that have long‐range correlations over numerous spatial and/or temporal scales; hence producing non‐Fickian diffusion. A fast algorithm to generate fBm and its increment by using its power spectral density S(f) in a fast Fourier transform algorithm is given. A general equation for the scaling of fBm within a velocity flow field with simple linear shear is presented. An initial numerical study of the nature of fBm shear dispersion has been conducted by incorporating fBm increments into a non‐Fickian particle‐tracking algorithm. It is shown that the effect of simple (i.e. linear) shear on the diffusion process is to produce enhanced diffusive phenomena with the longitudinal spreading of the plume scaling with exponent ∼1+H, where H is the Hurst exponent used to describe fBm. Finally, a more complex shear zone at the entrance of a coastal bay model is investigated using both a traditional particle‐tracking method and the fBm‐based method. Copyright © 2000 John Wiley & Sons, Ltd.     

A Lagrangian–Eulerian model of particle dispersion in a turbulent plane mixing layer

A Lagrangian–Eulerian model for the dispersion of solid particles in a two‐dimensional, incompressible, turbulent flow is reported and validated. Prediction of the continuous phase is done by solving an Eulerian model using a control‐volume finite element method (CVFEM). A Lagrangian model is also applied, using a Runge–Kutta method to obtain the particle trajectories. The effect of fluid turbulence upon particle dispersion is taken into consideration through a simple stochastic approach. Validation tests are performed by comparing predictions for both phases in a particle‐laden, plane mixing layer airflow with corresponding measurements formerly reported by other authors. Even though some limitations are detected in the calculation of particle dispersion, on the whole the validation results are rather successful. Copyright © 2002 John Wiley & Sons, Ltd.     

Two‐dimensional non‐Newtonian injection molding with a new control volume FEM/volume of fluid method

A new method based on volume of fluid for interface tracking in the simulation of injection molding is presented. The proposed method is comprised of two main stages: accumulation and distribution of the volume fraction. In the first stage the equation for the volume fraction with a noninterfacial flux condition is solved. In the second stage the accumulated volume of fluid that arises as a consequence of the application of the first one is dispersed. This procedure guarantees that the fluid fills the available space without dispersion of the interface. The mathematical model is based on two‐phase transport equations that are numerically integrated through the control volume finite element method. The numerical results for the interface position are successfully verified with analytical results and numerical data available in the literature for one‐dimensional and two‐dimensional domains. The transient position of the advance fronts showed an effective and consistent simulation of an injection molding process. The nondispersive volume of fluid method here proposed is implemented for the simulation of nonisothermal injection molding in two‐dimensional cavities. The obtained results are represented as transient interface positions, isotherms and pressure distributions during the injection molding of low density polyethylene. Copyright © 2012 John Wiley & Sons, Ltd.     

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.