A high‐order Petrov–Galerkin finite element method for the classical Boussinesq wave model

A high‐order Petrov–Galerkin finite element scheme is presented to solve the one‐dimensional depth‐integrated classical Boussinesq equations for weakly non‐linear and weakly dispersive waves. Finite elements are used both in the space and the time domains. The shape functions are bilinear in space–time, whereas the weighting functions are linear in space and quadratic in time, with C⁰‐continuity. Dispersion correction and a highly selective dissipation mechanism are introduced through additional streamline upwind terms in the weighting functions. An implicit, conditionally stable, one‐step predictor–corrector time integration scheme results. The accuracy and stability of the non‐linear discrete equations are investigated by means of a local Taylor series expansion. A linear spectral analysis is used for the full characterization of the predictor–corrector inner iterations. Based on the order of the analytical terms of the Boussinesq model and on the order of the numerical discretization, it is concluded that the scheme is fourth‐order accurate in terms of phase velocity. The dissipation term is third order only affecting the shortest wavelengths. A numerical convergence analysis showed a second‐order convergence rate in terms of both element size and time step. Four numerical experiments are addressed and their results are compared with analytical solutions or experimental data available in the literature: the propagation of a solitary wave, the oscillation of a flat bottom closed basin, the oscillation of a non‐flat bottom closed basin, and the propagation of a periodic wave over a submerged bar. Copyright © 2008 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.     

A new stable space–time formulation for two‐dimensional and three‐dimensional incompressible viscous flow

A space–time finite element method for the incompressible Navier–Stokes equations in a bounded domain in ?^d (with d=2 or 3) is presented. The method is based on the time‐discontinuous Galerkin method with the use of simplex‐type meshes together with the requirement that the space–time finite element discretization for the velocity and the pressure satisfy the inf–sup stability condition of Brezzi and Babu?ka. The finite element discretization for the pressure consists of piecewise linear functions, while piecewise linear functions enriched with a bubble function are used for the velocity. The stability proof and numerical results for some two‐dimensional problems are presented. Copyright © 2001 John Wiley & Sons, Ltd.     

Augmented mixed finite element methods for a vorticity‐based velocity–pressure–stress formulation of the Stokes problem in 2D

In this paper, we consider an augmented velocity–pressure–stress formulation of the 2D Stokes problem, in which the stress is defined in terms of the vorticity and the pressure, and then we introduce and analyze stable mixed finite element methods to solve the associated Galerkin scheme. In this way, we further extend similar procedures applied recently to linear elasticity and to other mixed formulations for incompressible fluid flows. Indeed, our approach is based on the introduction of the Galerkin least‐squares‐type terms arising from the corresponding constitutive and equilibrium equations, and from the Dirichlet boundary condition for the velocity, all of them multiplied by stabilization parameters. Then, we show that these parameters can be suitably chosen so that the resulting operator equation induces a strongly coercive bilinear form, whence the associated Galerkin scheme becomes well posed for any choice of finite element subspaces. In particular, we can use continuous piecewise linear velocities, piecewise constant pressures, and rotated Raviart–Thomas elements for the stresses. Next, we derive reliable and efficient residual‐based a posteriori error estimators for the augmented mixed finite element schemes. In addition, several numerical experiments illustrating the performance of the augmented mixed finite element methods, confirming the properties of the a posteriori estimators, and showing the behavior of the associated adaptive algorithms are reported. The present work should be considered as a first step aiming finally to derive augmented mixed finite element methods for vorticity‐based formulations of the 3D Stokes problem. Copyright © 2010 John Wiley & Sons, Ltd.     

Time‐accurate stabilized finite‐element model for weakly nonlinear and weakly dispersive water waves

Introduction of a time‐accurate stabilized finite‐element approximation for the numerical investigation of weakly nonlinear and weakly dispersive water waves is presented in this paper. To make the time approximation match the order of accuracy of the spatial representation of the linear triangular elements by the Galerkin finite‐element method, the fourth‐order time integration of implicit multistage Padé method is used for the development of the numerical scheme. The streamline‐upwind Petrov–Galerkin (SUPG) method with crosswind diffusion is employed to stabilize the scheme and suppress the spurious oscillations, usually common in the numerical computation of convection‐dominated flow problems. The performance of numerical stabilization and accuracy is addressed. Treatments of various boundary conditions, including the open boundary conditions, the perfect reflecting boundary conditions along boundaries with irregular geometry, are also described. Numerical results showing the comparisons with analytical solutions, experimental measurements, and other published numerical results are presented and discussed. Copyright © 2007 John Wiley & Sons, Ltd.     

Accuracy and stability of a set of free‐surface time‐domain boundary element models based on B‐splines

An analysis is given for the accuracy and stability of some perturbation‐based time‐domain boundary element models (BEMs) with B‐spline basis functions, solving hydrodynamic free‐surface problems, including forward speed effects. The spatial convergence rate is found as a function of the order of the B‐spline basis. It is shown that for all the models examined the mixed implicit–explicit Euler time integration scheme is correct to second order. Stability diagrams are found for models based on B‐splines of orders third through to sixth for two different time integration schemes. The stability analysis can be regarded as an extension of the analysis by Vada and Nakos [Vada T, Nakos DE. Time marching schemes for ship motion simulations. In Proceedings of the 8th International Workshop on Water Waves and Floating Bodies, St. John's, Newfoundland, Canada, 1993; 155–158] to include B‐splines of orders higher than three (piecewise quadratic polynomials) and to include finite water depth and a current at an oblique angle to the model grid. Copyright © 2000 John Wiley & Sons, Ltd.     

Spectral/hp penalty least‐squares finite element formulation for unsteady incompressible flows

In this paper, we present spectral/hp penalty least‐squares finite element formulation for the numerical solution of unsteady incompressible Navier–Stokes equations. Pressure is eliminated from Navier–Stokes equations using penalty method, and finite element model is developed in terms of velocity, vorticity and dilatation. High‐order element expansions are used to construct discrete form. Unlike other penalty finite element formulations, equal‐order Gauss integration is used for both viscous and penalty terms of the coefficient matrix. For time integration, space–time decoupled schemes are implemented. Second‐order accuracy of the time integration scheme is established using the method of manufactured solution. Numerical results are presented for impulsively started lid‐driven cavity flow at Reynolds number of 5000 and transient flow over a backward‐facing step. The effect of penalty parameter on the accuracy is investigated thoroughly in this paper and results are presented for a range of penalty parameter. Present formulation produces very accurate results for even very low penalty parameters (10–50). Copyright © 2008 John Wiley & Sons, Ltd.     

A mixed‐interpolation finite element method for incompressible thermal flows of electrically conducting fluids

A new mixed‐interpolation finite element method is presented for the two‐dimensional numerical simulation of incompressible magnetohydrodynamic (MHD) flows which involve convective heat transfer. The proposed method applies the nodal shape functions, which are locally defined in nine‐node elements, for the discretization of the Navier–Stokes and energy equations, and the vector shape functions, which are locally defined in four‐node elements, for the discretization of the electromagnetic field equations. The use of the vector shape functions allows the solenoidal condition on the magnetic field to be automatically satisfied in each four‐node element. In addition, efficient approximation procedures for the calculation of the integrals in the discretized equations are adopted to achieve high‐speed computation. With the use of the proposed numerical scheme, MHD channel flow and MHD natural convection under a constant applied magnetic field are simulated at different Hartmann numbers. The accuracy and robustness of the method are verified through these numerical tests in which both undistorted and distorted meshes are employed for comparison of numerical solutions. Furthermore, it is shown that the calculation speed for the proposed scheme is much higher compared with that for a conventional numerical integration scheme under the condition of almost the same memory consumption. Copyright © 2016 John Wiley & Sons, Ltd.     

Positivity‐preserving,flux‐limited finite‐difference and finite‐element methods for reactive transport

A new class of positivity‐preserving, flux‐limited finite‐difference and Petrov–Galerkin (PG) finite‐element methods are devised for reactive transport problems.The methods are similar to classical TVD flux‐limited schemes with the main difference being that the flux‐limiter constraint is designed to preserve positivity for problems involving diffusion and reaction. In the finite‐element formulation, we also consider the effect of numerical quadrature in the lumped and consistent mass matrix forms on the positivity‐preserving property. Analysis of the latter scheme shows that positivity‐preserving solutions of the resulting difference equations can only be guaranteed if the flux‐limited scheme is both implicit and satisfies an additional lower‐bound condition on time‐step size. We show that this condition also applies to standard Galerkin linear finite‐element approximations to the linear diffusion equation. Numerical experiments are provided to demonstrate the behavior of the methods and confirm the theoretical conditions on time‐step size, mesh spacing, and flux limiting for transport problems with and without nonlinear reaction. Copyright © 2003 John Wiley & Sons, Ltd.     

A multi‐energy‐level lattice Boltzmann model for the compressible Navier–Stokes equations

In this paper, we propose a new lattice Boltzmann model for the compressible Navier–Stokes equations. The new model is based on a three‐energy‐level and three‐speed lattice Boltzmann equation by using a method of higher moments of the equilibrium distribution functions. As the 25‐bit model, we obtained the equilibrium distribution functions and the compressible Navier–Stokes equations with the second accuracy of the truncation errors. The numerical examples show that the model can be used to simulate the shock waves, contact discontinuities and supersonic flows around circular cylinder. The numerical results are compared with those obtained by traditional method. Copyright © 2007 John Wiley & Sons, Ltd.     

A fully non‐linear model for three‐dimensional overturning waves over an arbitrary bottom

An accurate three‐dimensional numerical model, applicable to strongly non‐linear waves, is proposed. The model solves fully non‐linear potential flow equations with a free surface using a higher‐order three‐dimensional boundary element method (BEM) and a mixed Eulerian–Lagrangian time updating, based on second‐order explicit Taylor series expansions with adaptive time steps. The model is applicable to non‐linear wave transformations from deep to shallow water over complex bottom topography up to overturning and breaking. Arbitrary waves can be generated in the model, and reflective or absorbing boundary conditions specified on lateral boundaries. In the BEM, boundary geometry and field variables are represented by 16‐node cubic ‘sliding’ quadrilateral elements, providing local inter‐element continuity of the first and second derivatives. Accurate and efficient numerical integrations are developed for these elements. Discretized boundary conditions at intersections (corner/edges) between the free surface or the bottom and lateral boundaries are well‐posed in all cases of mixed boundary conditions. Higher‐order tangential derivatives, required for the time updating, are calculated in a local curvilinear co‐ordinate system, using 25‐node ‘sliding’ fourth‐order quadrilateral elements. Very high accuracy is achieved in the model for mass and energy conservation. No smoothing of the solution is required, but regridding to a higher resolution can be specified at any time over selected areas of the free surface. Applications are presented for the propagation of numerically exact solitary waves. Model properties of accuracy and convergence with a refined spatio‐temporal discretization are assessed by propagating such a wave over constant depth. The shoaling of solitary waves up to overturning is then calculated over a 1:15 plane slope, and results show good agreement with a two‐dimensional solution proposed earlier. Finally, three‐dimensional overturning waves are generated over a 1:15 sloping bottom having a ridge in the middle, thus focusing wave energy. The node regridding method is used to refine the discretization around the overturning wave. Convergence of the solution with grid size is also verified for this case. Copyright © 2001 John Wiley & Sons, Ltd.     

A numerical study of projection algorithms in the finite element simulation of three‐dimensional viscous incompressible flow

In this work a comparative study of two versions of the projection algorithm used either for time integration or as an iterative method to solve the three‐dimensional incompressible Navier–Stokes equations is presented. It is also shown that these projection algorithms combined with the finite element method are particularly suited for the treatment of outflow boundary conditions in the context of external flows. This assertion is illustrated by means of some numerical examples in which five types of boundary conditions are compared. The scheme is applied to simulate the flow past a cylinder clamped on two fixed parallel solid walls. Comparison with experimental data available for this problem shows good agreement of the velocity and pressure fields, both computed with continuous piecewise linear elements. Copyright © 1999 John Wiley & Sons, Ltd.     

A non‐conforming least‐squares finite element method for incompressible fluid flow problems

In this paper, we develop least‐squares finite element methods (LSFEMs) for incompressible fluid flows with improved mass conservation. Specifically, we formulate a new locally conservative LSFEM for the velocity–vorticity–pressure Stokes system, which uses a piecewise divergence‐free basis for the velocity and standard C⁰ elements for the vorticity and the pressure. The new method, which we term dV‐VP improves upon our previous discontinuous stream‐function formulation in several ways. The use of a velocity basis, instead of a stream function, simplifies the imposition and implementation of the velocity boundary condition, and eliminates second‐order terms from the least‐squares functional. Moreover, the size of the resulting discrete problem is reduced because the piecewise solenoidal velocity element is approximately one‐half of the dimension of a stream‐function element of equal accuracy. In two dimensions, the discontinuous stream‐function LSFEM [1] motivates modification of our functional, which further improves the conservation of mass. We briefly discuss the extension of this modification to three dimensions. Computational studies demonstrate that the new formulation achieves optimal convergence rates and yields high conservation of mass. We also propose a simple diagonal preconditioner for the dV‐VP formulation, which significantly reduces the condition number of the LSFEM problem. Published 2012. This article is a US Government work and is in the public domain in the USA.     

High‐order semi‐implicit time‐integrators for a triangular discontinuous Galerkin oceanic shallow water model

We extend the explicit in time high‐order triangular discontinuous Galerkin (DG) method to semi‐implicit (SI) and then apply the algorithm to the two‐dimensional oceanic shallow water equations; we implement high‐order SI time‐integrators using the backward difference formulas from orders one to six. The reason for changing the time‐integration method from explicit to SI is that explicit methods require a very small time step in order to maintain stability, especially for high‐order DG methods. Changing the time‐integration method to SI allows one to circumvent the stability criterion due to the gravity waves, which for most shallow water applications are the fastest waves in the system (the exception being supercritical flow where the Froude number is greater than one). The challenge of constructing a SI method for a DG model is that the DG machinery requires not only the standard finite element‐type area integrals, but also the finite volume‐type boundary integrals as well. These boundary integrals pose the biggest challenge in a SI discretization because they require the construction of a Riemann solver that is the true linear representation of the nonlinear Riemann problem; if this condition is not satisfied then the resulting numerical method will not be consistent with the continuous equations. In this paper we couple the SI time‐integrators with the DG method while maintaining most of the usual attributes associated with DG methods such as: high‐order accuracy (in both space and time), parallel efficiency, excellent stability, and conservation. The only property lost is that of a compact communication stencil typical of time‐explicit DG methods; implicit methods will always require a much larger communication stencil. We apply the new high‐order SI DG method to the shallow water equations and show results for many standard test cases of oceanic interest such as: standing, Kelvin and Rossby soliton waves, and the Stommel problem. The results show that the new high‐order SI DG model, that has already been shown to yield exponentially convergent solutions in space for smooth problems, results in a more efficient model than its explicit counterpart. Furthermore, for those problems where the spatial resolution is sufficiently high compared with the length scales of the flow, the capacity to use high‐order (HO) time‐integrators is a necessary complement to the employment of HO space discretizations, since the total numerical error would be otherwise dominated by the time discretization error. In fact, in the limit of increasing spatial resolution, it makes little sense to use HO spatial discretizations coupled with low‐order time discretizations. Published in 2009 by John Wiley & Sons, Ltd.     

Fully non‐linear free‐surface simulations by a 3D viscous numerical wave tank

A finite difference scheme using a modified marker‐and‐cell (MAC) method is applied to investigate the characteristics of non‐linear wave motions and their interactions with a stationary three‐dimensional body inside a numerical wave tank (NWT). The Navier–Stokes (NS) equation is solved for two fluid layers, and the boundary values are updated at each time step by a finite difference time marching scheme in the frame of a rectangular co‐ordinate system. The viscous stresses and surface tension are neglected in the dynamic free‐surface condition, and the fully non‐linear kinematic free‐surface condition is satisfied by the density function method developed for two fluid layers. The incident waves are generated from the inflow boundary by prescribing a velocity profile resembling flexible flap wavemaker motions, and the outgoing waves are numerically dissipated inside an artificial damping zone located at the end of the tank. The present NS–MAC NWT simulations for a vertical truncated circular cylinder inside a rectangular wave tank are compared with the experimental results of Mercier and Niedzwecki, an independently developed potential‐based fully non‐linear NWT, and the second‐order diffraction computation. Copyright © 1999 John Wiley & Sons, Ltd.     

Least‐squares meshfree method for incompressible Navier–Stokes problems

A least‐squares meshfree method based on the first‐order velocity–pressure–vorticity formulation for two‐dimensional incompressible Navier–Stokes problem is presented. The convective term is linearized by successive substitution or Newton's method. The discretization of all governing equations is implemented by the least‐squares method. Equal‐order moving least‐squares approximation is employed with Gauss quadrature in the background cells. The boundary conditions are enforced by the penalty method. The matrix‐free element‐by‐element Jacobi preconditioned conjugate method is applied to solve the discretized linear systems. Cavity flow for steady Navier–Stokes problem and the flow over a square obstacle for time‐dependent Navier–Stokes problem are investigated for the presented least‐squares meshfree method. The effects of inaccurate integration on the accuracy of the solution are investigated. Copyright © 2004 John Wiley & Sons, Ltd.     

A PIECEWISE–LINEARIZED METHOD FOR ORDINARY DIFFERENTIAL EQUATIONS: TWO–POINT BOUNDARY–VALUE PROBLEMS

Piecewise-linearized methods for the solution of two-point boundary value problems in ordinary differential equations are presented. These problems are approximated by piecewise linear ones which have analytical solutions and reduced to finding the slope of the solution at the left boundary so that the boundary conditions at the right end of the interval are satisfied. This results in a rather complex system of non-linear algebraic equations which may be reduced to a single non-linear equation whose unknown is the slope of the solution at the left boundary of the interval and whose solution may be obtained by means of the Newton–Raphson method. This is equivalent to solving the boundary value problem as an initial value one using the piecewise-linearized technique and a shooting method. It is shown that for problems characterized by a linear operator a technique based on the superposition principle and the piecewise-linearized method may be employed. For these problems the accuracy of piecewise-linearized methods is of second order. It is also shown that for linear problems the accuracy of the piecewise-linearized method is superior to that of fourth-order-accurate techniques. For the linear singular perturbation problems considered in this paper the accuracy of global piecewise linearizat ion is higher than that of finite difference and finite element methods. For non-linear problems the accuracy of piecewise-linearized methods is in most cases lower than that of fourth-order methods but comparable with that of second-order techniques owing to the linearization of the non-linear terms.     

A high‐order accurate method for two‐dimensional incompressible viscous flows

A high‐order accurate solution method for complex geometries is developed for two‐dimensional flows using the stream function–vorticity formulation. High‐order accurate spectrally optimized compact schemes along with appropriate boundary schemes are used for spatial discretization while a two‐level backward Euler implicit scheme is used for the time integration. The linear system of equations for stream function and vorticity are solved by an inner iteration while contravariant velocities constitute outer iterations. The effect of curvilinear grids on the solution accuracy is studied. The method is used to compute Cartesian and inclined driven cavity, flow in a triangular cavity and viscous flow in constricted channel. Benchmark‐like accuracy is obtained in all the problems with fewer grid points compared to reported studies. Copyright © 2006 John Wiley & Sons, Ltd.     

High‐order time integrators for front‐tracking finite‐element analysis of viscous free‐surface flows

This paper proposes implicit Runge–Kutta (IRK) time integrators to improve the accuracy of a front‐tracking finite‐element method for viscous free‐surface flow predictions. In the front‐tracking approach, the modeling equations must be solved on a moving domain, which is usually performed using an arbitrary Lagrangian–Eulerian (ALE) frame of reference. One of the main difficulties associated with the ALE formulation is related to the accuracy of the time integration procedure. Indeed, most formulations reported in the literature are limited to second‐order accurate time integrators at best. In this paper, we present a finite‐element ALE formulation in which a consistent evaluation of the mesh velocity and its divergence guarantees satisfaction of the discrete geometrical conservation law. More importantly, it also ensures that the high‐order fixed mesh temporal accuracy of time integrators is preserved on deforming grids. It is combined with the use of a family of L‐stable IRK time integrators for the incompressible Navier–Stokes equations to yield high‐order time‐accurate free‐surface simulations. This is demonstrated in the paper using the method of manufactured solution in space and time as recommended in Verification and Validation. In particular, we report up to fifth‐order accuracy in time. The proposed free‐surface front‐tracking approach is then validated against cases of practical interest such as sloshing in a tank, solitary waves propagation, and coupled interaction between a wave and a submerged cylinder. Copyright © 2015 John Wiley & Sons, Ltd.     

Fourth‐order exponential finite difference methods for boundary value problems of convective diffusion type

Methods based on exponential finite difference approximations of h⁴ accuracy are developed to solve one and two‐dimensional convection–diffusion type differential equations with constant and variable convection coefficients. In the one‐dimensional case, the numerical scheme developed uses three points. For the two‐dimensional case, even though nine points are used, the successive line overrelaxation approach with alternating direction implicit procedure enables us to deal with tri‐diagonal systems. The methods are applied on a number of linear and non‐linear problems, mostly with large first derivative terms, in particular, fluid flow problems with boundary layers. Better accuracy is obtained in all the problems, compared with the available results in the literature. Application of an exponential scheme with a non‐uniform mesh is also illustrated. The h⁴ accuracy of the schemes is also computationally demonstrated. Copyright © 2001 John Wiley & Sons, Ltd.