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This paper presents the linear stability analysis of a round jet in a radially unbounded domain using a spectral Petrov–Galerkin scheme coped with exponential coordinate transformation based on Fornberg's treatment. A Fourier–Chebyshev Petrov–Galerkin spectral method is described for the computation of the linear stability equations based on half a Gauss–Lobatto mesh. Complex basis functions presented here are exponentially mapped as Chebyshev functions, which satisfy the pole condition exactly at the origin, and can be used to expand vector functions efficiently by using the solenoidal condition. The mathematical formulation is presented in detail focusing on the solenoidal vector field used for the approximation of the flow. The scheme provides spectral accuracy in the present cases and the numerical results are in agreement with former works. Copyright © 2008 John Wiley & Sons, Ltd.

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.

In this paper, we present the 'conformal Petrov-Galerkin' (CPG) method in order to solve the 2D convection-diffusion equation on meshes composed of triangular elements. By 'conformal' it is meant that the discrete system is obtained front the continuous weak formulation by appropriately selecting different finite-dimensional subspaces for the shape and test functions without any additional stabilizing term. Our approach is based on searching continuous test functions that provide exact nodal values for a selected class Of Solutions. This method induces a stabilizing upwinding effect that removes the wiggles obtained with the Galerkin method. Copyright (C) 2008 John Wiley & Sons, Ltd.

A finite element model for incompressible laminar two-phase flows is presented. A two-fluid model, describing the laminar non-equilibrium flow of two incompressible phases, is discretized by means of a properly designed streamline upwind Petrov–Galerkin (SUPG) finite element procedure. Such a procedure is consistent with a continuous pressure equation. The design and the implementation of the algorithm are presented together with its validation throughout a comparison with simulations available in the literature. Copyright © 2006 John Wiley & Sons, Ltd.

A time-marching formulation is derived from the space–time integrated least squares (STILS) method for solving a pure hyperbolic convection equation and is numerically compared to various known methods. Copyright © 2004 John Wiley & Sons, Ltd.

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.

In the present paper, the author shows that the predictor/multi-corrector (PMC) time integration for the advection–diffusion equations induces numerical diffusivity acting only in the streamline direction, even though the equations are spatially discretized by the conventional Galerkin finite element method (GFEM). The transient 2-D and 3-D advection problems are solved with the PMC scheme using both the GFEM and the streamline upwind/Petrov Galerkin (SUPG) as the spatial discretization methods for comparison. The solutions of the SUPG-PMC turned out to be overly diffusive due to the additional PMC streamline diffusion, while the solutions of the GFEM-PMC were comparatively accurate without significant damping and phase error. A similar tendency was seen also in the quasi-steady solutions to the incompressible viscous flow problems: 2-D driven cavity flow and natural convection in a square cavity. Copyright © 2002 John Wiley & Sons, Ltd.

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.

A new finite element method is presented to solve one-dimensional depth-integrated equations for fully non-linear and weakly dispersive waves. For spatial integration, the Petrov–Galerkin weighted residual method is used. The weak forms of the governing equations are arranged in such a way that the shape functions can be piecewise linear, while the weighting functions are piecewise cubic with C2-continuity. For the time integration an implicit predictor–corrector iterative scheme is employed. Within the framework of linear theory, the accuracy of the scheme is discussed by considering the truncation error at a node. The leading truncation error is fourth-order in terms of element size. Numerical stability of the scheme is also investigated. If the Courant number is less than 0.5, the scheme is unconditionally stable. By increasing the number of iterations and/or decreasing the element size, the stability characteristics are improved significantly. Both Dirichlet boundary condition (for incident waves) and Neumann boundary condition (for a reflecting wall) are implemented. Several examples are presented to demonstrate the range of applicabilities and the accuracy of the model. Copyright © 2001 John Wiley & Sons, Ltd.

SUMMARY This paper is concerned with the development of the finite element method in simulating scalar transport, governed by the convection‐reaction (CR) equation. A feature of the proposed finite element model is its ability to provide nodally exact solutions in the one-dimensional case. Details of the derivation of the upwind scheme on quadratic elements are given. Extension of the one-dimensional nodally exact scheme to the two-dimensional model equation involves the use of a streamline upwind operator. As the modified equations show in the four types of element, physically relevant discretization error terms are added to the flow direction and help stabilize the discrete system. The proposed method is referred to as the streamline upwind Petrov‐Galerkin finite element model. This model has been validated against test problems that are amenable to analytical solutions. In addition to a fundamental study of the scheme, numerical results that demonstrate the validity of the method are presented. Copyright © 2001 John Wiley & Sons, Ltd.

SUMMARY The Petrov‐Galerkin method has been developed with the primary goal of damping spurious oscillations near discontinuities in advection dominated flows. For time-dependent problems, the typical Petrov‐ Galerkin method is based on the minimization of the dispersion error and the simultaneous selective addition of dissipation. This optimal design helps to dampen the oscillations prevalent near discontinuities in standard Bubnov‐Galerkin solutions. However, it is demonstrated that when the Courant number is less than 1, the Petrov‐Galerkin method actually amplifies undershoots at the base of discontinuities. This is shown in an heuristic manner, and is demonstrated with numerical experiments with the scalar advection and Richards’ equations. A discussion of monotonicity preservation as a design criterion, as opposed to phase or amplitude error minimization, is also presented. The Petrov‐Galerkin method is further linked to the high-resolution, total variation diminishing (TVD) finite volume method in order to obtain a monotonicity preserving Petrov‐Galerkin method. Copyright © 2000 John Wiley & Sons, Ltd.

The prediction of the flow field in a novel spiral casing has been accomplished. Hydraulic turbine manufacturers are considering the potential of using a special type of spiral casing because of the easier manufacturing process involved in its fabrication. These special spiral casings are known as plate-spirals. Numerical simulation of complex three-dimensional flow through such spiral casings has been accomplished using a finite element method (FEM). An explicit Eulerian velocity correction scheme has been deployed to solve the Reynolds-average Navier–Stokes equations. The simulation has been performed to describe the flow in high Reynolds number (106) regimes. For spatial discretization, a streamline upwind Petrov–Galerkin (SUPG) technique has been used. The velocity field and the pressure distribution inside the spiral casing reveal meaningful results. Copyright © 2000 John Wiley & Sons, Ltd.

Navier-Stokes fluid flows in curved channels are considered. Upstream of the backward-facing step, there exists a channel with a 90° bend and a fixed curvature of 2.5. The purpose of conducting this study was to apply a finite element code to study the effect of the distorted upstream velocity profile developing over the bend on laminar expansion flows behind the step. The size of the eddies formed downstream of the step is addressed. The present work employs primitive velocities, which stagger the pressure working variable, to assure satisfaction of the inf-sup stability condition. In quadratic elements, spatial derivatives are approximated within the consistent Petrov-Galerkin finite element framework. Use of this method aids stability in the sense that artificial damping is solely added to the direction parallel to the flow direction. Through analytical testing, in conjunction with two other benchmark tests, the integrity of applying the computer code in quadratic elements is verified

Three adaptive finite element methods based on equidistribution, elliptic grid generation and hybrid techniques are used to study a system of reaction–diffusion equations. It is shown that these techniques must employ sub-equidistributing meshes in order to avoid ill-conditioned matrices and ensure the convergence of the Newton method. It is also shown that elliptic grid generation methods require much longer computer times than hybrid and static rezoning procedures. The paper also includes characteristic, Petrov–Galerkin and flux-corrected transport algorithms which are used to study a linear convection–reaction–diffusion equation that has an analytical solution. The flux-corrected transport technique yields monotonic solutions in good agreement with the analytical solution, whereas the Petrov–Galerkin method with quadratic upstream-weighted functions results in very diffused temperature profiles. The characteristic finite element method which uses a Lagrangian–Eulerian formulation overpredicts the flame front location and exhibits overshoots and undershoots near the temperature discontinuity. These overshoots and undershoots are due to the interpolation of the results of the Lagrangian operator onto the fixed Eulerian grid used to solve the reaction–diffusion operator, and indicate that characteristic finite element methods are not able to eliminate numerical diffusion entirely.

SUMMARY Different finite element schemes are investigated with respect to their application in numerical weather prediction. Different methods of staggering of variables are considered. The tests concern the accuracy of a Rossby wave prediction and the generation of noise in a geostrophic adjustment process. Theoretical results concerning the noise level of different schemes are confirmed by computations with a one-dimensional model. Favourable results were obtained by hybrid schemes, using different Galerkin treatments for different terms of the dynamic equations. In numerical models of the atmosphere an accurate treatment of transient features of the flow is essential. This leads to the requirement of a reasonable order of the discretization. The finite element method is able to obtain ahigh degree of accuracy, as shown in Reference 1. Atmospheric models based on this method are described in References 2-4. Another important aspect of atmospheric models is the occurrence of a stationary or very slowly varying part of the flow. This represents the climate of the model and is caused by a balance of forcing and different transport processes. Associated with this is the problem of geostrophic adjustment. In this respect the numerical properties of an atmospheric model may be analysed by similar considerations as aeronautical applications of finite elements, where the stationary flow is of interest. For such applications a number of finite element applications suffer from noise problems. This leads to the requirement of mixed interp~lation,~. ~ meaning different choices of finite element representations for the pressure and velocity variables. As pointed out in References 7 and 8, the requirement for mixed interpolation for finite element schemes in stationary flow applications is associated with the requirement for staggered grids in meteorological models.’- lo This latter requirement is the result both of practical experience in numerical weather prediction and of theoretical analysis based on the transfer function analysis of

The current status of streamline-upwind/Petrov-Galerkin (SUPG) methods for the analysis of flow problems is surveyed in an analytical review. Problem areas addressed include classical Galerkin, upwind, artificial-diffusion, SUPG, discontinuous Galerkin, space-time FEM, and discontinuity-capturing approaches to the scalar advection-diffusion equation; incompressible flows; advective-diffusive systems; and the compressible Euler and Navier-Stokes equations. Graphs and diagrams are provided, and the good stability properties of state-of-the-art SUPG methods are pointed out.

Finite elements using higher-order basis functions in the spirit of the QUICK method for convection-dominated fluid flow and transport problems are introduced and demonstrated. Instead of introducing new internal degrees of freedom, completeness is achieved by including functions based on nodal values exterior and upwind to the element domain. Applied with linear test functions to the weak statements for convection-dominated problems, a family of Petrov–Galerkin finite elements is developed. Quadratic and cubic versions are demonstrated for the one-dimensional convection–diffusion test problem. Elements of up to seventh degree are used for local solution refinement. The behaviour of these elements for one-dimensional linear and non-linear advection is investigated. A two-dimensional quadratic upwind element is demonstrated in a streamfunction–vorticity formulation of the Navier–Stokes equations for a driven cavity flow test problem. With some minor reservations, these elements are recommended for further study and application.

A Petrov-Galerkin finite element method is presented for calculation of the steady, axisymmetric thermosolutal convection and interface morphology in a model for vertical Bridgman crystal growth of nondilute binary alloys. The Petrov-Galerkin method is based on the formulation for biquadratic elements developed by Heinrich and Zienkiewicz and is introduced into the calculation of the velocity, temperature and concentration fields. The algebraic system is solved simultaneously for the field variables and interface shape by Newton's method. The results of the Petrov-Galerkin method are compared critically with those of Galerkin's method using the same finite element grids. Significant improvements in accuracy are found with the Petrov-Galerkin method only when the mesh is refined and when the formulation of the residual equations is modified to account for the mixed boundary conditions that arise at the solidification interface. Calculations for alloys with stable and unstable solute gradients show the occurrence of classical flow transitions and morphological instabilities in the solidification system.

SUMMARY In this paper we present streamline-upwind/Petrov-Galerkin finite element procedures for two-dimensional fluid dynamics computations based on the vorticity-stream function formulation of the incompressible Navier-Stokes equations. We address the difficulties associated with the convection term in the vorticity transport equation, lack of boundary condition for the vorticity at no-slip boundaries, and determination of the value of the stream function at the internal boundaries for multiply connected domains. The proposed techniques, implemented within the framework of block-iteration methods, have successfully been applied to various problems involving simply and multiply connected domains. There are some advantages in using the vorticity-stream function formulation of the incompressible Navier-Stokes equations for two-dimensional computations. Compared to the velocity-pressure formulation, the vorticity-stream function formulation leads to computed flow fields which satisfy the incompressibility condition automatically; also the number of unknown functions is reduced from three to two and the vorticity field is computed directly instead of being obtained by differentiation of the velocity field. The last advantage becomes important if one needs to study the vorticity field and therefore wants that this field be represented as accurately as possible. We propose suitable finite element procedures for the solution of the time-dependent vorticity transport equation and the Poisson’s equation which relates the stream function to the vorticity. The difficulties associated with the convection term in the vorticity transport equation, lack of

The Bradshaw-Ferriss-Atwell model for 2D constant property turbulent boundary layers is shown to be ill-posed with respect to numerical solution. It is shown that a simple modification to the model equations results in a well-posed system which is hyperbolic in nature. For this modified system a numerical algorithm is constructed by discretizing in space using the Petrov-Galerkin technique (of which the standard Galerkin method is a special case) and stepping in the timelike direction with the trapezoidal (Crank-Nicolson) rule. The algorithm is applied to a selection of test problems. It is found that the solutions produced by the standard Galerkin method exhibit oscillations. It is further shown that these oscillations may be eliminated by employing the Petrov-Galerkin method with the free parameters set to simple functions of the eigenvalues of the modified system.