The two‐dimensional streamline upwind scheme for the convection–reaction equation

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.     

Analysis of flow in the plate‐spiral of a reaction turbine using a streamline upwind Petrov–Galerkin method

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.     

Anisotropic mesh adaption for time‐dependent problems

We propose a space–time adaptive procedure for a model parabolic problem based on a theoretically sound anisotropic a posteriori error analysis. A space–time finite element scheme (continuous in space but discontinuous in time) is employed to discretize this problem, thus allowing for non‐matching meshes at different time levels. Copyright © 2008 John Wiley & Sons, Ltd.     

The anti‐dissipative,non‐monotone behavior of Petrov–Galerkin upwinding

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.     

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.     

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.     

An optimizing reduced PLSMFE formulation for non‐stationary conduction–convection problems

In this paper, proper orthogonal decomposition (POD) is combined with the Petrov–Galerkin least squares mixed finite element (PLSMFE) method to derive an optimizing reduced PLSMFE formulation for the non‐stationary conduction–convection problems. Error estimates between the optimizing reduced PLSMFE solutions based on POD and classical PLSMFE solutions are presented. The optimizing reduced PLSMFE formulation can circumvent the constraint of Babu?ka–Brezzi condition so that the combination of finite element subspaces can be chosen freely and allow optimal‐order error estimates to be obtained. Numerical simulation examples have shown that the errors between the optimizing reduced PLSMFE solutions and the classical PLSMFE solutions are consistent with theoretical results. Moreover, they have also shown the feasibility and efficiency of the POD method. Copyright © 2008 John Wiley & Sons, Ltd.     

A Petrov–Galerkin finite element model for one‐dimensional fully non‐linear and weakly dispersive wave propagation

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.     

Finite element and implicit Runge–Kutta implementation of an acoustics–convection upstream resolution algorithm for the time‐dependent two‐dimensional Euler equations

The second of a two‐paper series, this paper details a solver for the characteristics‐bias system from the acoustics–convection upstream resolution algorithm for the Euler and Navier–Stokes equations. An integral formulation leads to several surface integrals that allow effective enforcement of boundary conditions. Also presented is a new multi‐dimensional procedure to enforce a pressure boundary condition at a subsonic outlet, a procedure that remains accurate and stable. A classical finite element Galerkin discretization of the integral formulation on any prescribed grid directly yields an optimal discretely conservative upstream approximation for the Euler and Navier–Stokes equations, an approximation that remains multi‐dimensional independently of the orientation of the reference axes and computational cells. The time‐dependent discrete equations are then integrated in time via an implicit Runge–Kutta procedure that in this paper is proven to remain absolutely non‐linearly stable for the spatially‐discrete Euler and Navier–Stokes equations and shown to converge rapidly to steady states, with maximum Courant number exceeding 100 for the linearized version. Even on relatively coarse grids, the acoustics–convection upstream resolution algorithm generates essentially non‐oscillatory solutions for subsonic, transonic and supersonic flows, encompassing oblique‐ and interacting‐shock fields that converge within 40 time steps and reflect reference exact solutions. Copyright © 2005 John Wiley & Sons, Ltd.     

Comparison of Galerkin and control volume finite element for advection–diffusion problems

The control volume finite element method (CVFEM) was developed to combine the local numerical conservation property of control volume methods with the unstructured grid and generality of finite element methods (FEMs). Most implementations of CVFEM include mass‐lumping and upwinding techniques typical of control volume schemes. In this work we compare, via numerical error analysis, CVFEM and FEM utilizing consistent and lumped mass implementations, and stabilized Petrov–Galerkin streamline upwind schemes in the context of advection–diffusion processes. For this type of problem, we find no apparent advantage to the local numerical conservation aspect of CVFEM as compared to FEM. The stabilized schemes improve accuracy and degree of positivity on coarse grids, and also reduce iteration counts for advection‐dominated problems. Published in 2005 by John Wiley & Sons, Ltd.     

A new weighted upwind finite volume element method based on non‐standard covolume for time‐dependent convection–diffusion problems

In modern numerical simulation of problems in energy resources and environmental science, it is important to develop efficient numerical methods for time‐dependent convection–diffusion problems. On the basis of nonstandard covolume grids, we propose a new kind of high‐order upwind finite volume element method for the problems. We first prove the stability and mass conservation in the discrete forms of the scheme. Optimal second‐order error estimate in L₂‐norm in spatial step is then proved strictly. The scheme is effective for avoiding numerical diffusion and nonphysical oscillations and has second‐order accuracy. Numerical experiments are given to verify the performance of the scheme. Copyright © 2013 John Wiley & Sons, Ltd.     

A multi‐dimensional monotonic finite element model for solving the convection–diffusion‐reaction equation

In this paper, we develop a finite element model for solving the convection–diffusion‐reaction equation in two dimensions with an aim to enhance the scheme stability without compromising consistency. Reducing errors of false diffusion type is achieved by adding an artificial term to get rid of three leading mixed derivative terms in the Petrov–Galerkin formulation. The finite element model of the Petrov–Galerkin type, while maintaining convective stability, is modified to suppress oscillations about the sharp layer by employing the M‐matrix theory. To validate this monotonic model, we consider test problems which are amenable to analytic solutions. Good agreement is obtained with both one‐ and two‐dimensional problems, thus validating the method. Other problems suitable for benchmarking the proposed model are also investigated. Copyright © 2002 John Wiley & Sons, Ltd.     

Goal‐oriented space–time adaptivity in the finite element Galerkin method for the computation of nonstationary incompressible flow

This paper presents a general strategy for designing adaptive space–time finite element discretizations of the nonstationary Navier–Stokes equations. The underlying framework is that of the dual weighted residual method for goal‐oriented a posteriori error estimation and automatic mesh adaptation. In this approach, the error in the approximation of certain quantities of physical interest, such as the drag coefficient, is estimated in terms of local residuals of the computed solution multiplied by sensitivity factors, which are obtained by numerically solving an associated dual problem. In the resulting local error indicators, the effects of spatial and temporal discretization are separated, which allows for the simultaneous adjustment of time step and spatial mesh size. The efficiency of the proposed method for the construction of economical meshes and the quantitative assessment of the error is illustrated by several test examples. Copyright © 2012 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 third‐order implicit discontinuous Galerkin method based on a Hermite WENO reconstruction for time‐accurate solution of the compressible Navier–Stokes equations

A space and time third‐order discontinuous Galerkin method based on a Hermite weighted essentially non‐oscillatory reconstruction is presented for the unsteady compressible Euler and Navier–Stokes equations. At each time step, a lower‐upper symmetric Gauss–Seidel preconditioned generalized minimal residual solver is used to solve the systems of linear equations arising from an explicit first stage, single diagonal coefficient, diagonally implicit Runge–Kutta time integration scheme. The performance of the developed method is assessed through a variety of unsteady flow problems. Numerical results indicate that this method is able to deliver the designed third‐order accuracy of convergence in both space and time, while requiring remarkably less storage than the standard third‐order discontinous Galerkin methods, and less computing time than the lower‐order discontinous Galerkin methods to achieve the same level of temporal accuracy for computing unsteady flow problems. Copyright © 2015 John Wiley & Sons, Ltd.     

Development of a class of multiple time‐stepping schemes for convection–diffusion equations in two dimensions

In this paper we present a class of semi‐discretization finite difference schemes for solving the transient convection–diffusion equation in two dimensions. The distinct feature of these scheme developments is to transform the unsteady convection–diffusion (CD) equation to the inhomogeneous steady convection–diffusion‐reaction (CDR) equation after using different time‐stepping schemes for the time derivative term. For the sake of saving memory, the alternating direction implicit scheme of Peaceman and Rachford is employed so that all calculations can be carried out within the one‐dimensional framework. For the sake of increasing accuracy, the exact solution for the one‐dimensional CDR equation is employed in the development of each scheme. Therefore, the numerical error is attributed primarily to the temporal approximation for the one‐dimensional problem. Development of the proposed time‐stepping schemes is rooted in the Taylor series expansion. All higher‐order time derivatives are replaced with spatial derivatives through use of the model differential equation under investigation. Spatial derivatives with orders higher than two are not taken into account for retaining the linear production term in the convection–diffusion‐reaction differential system. The proposed schemes with second, third and fourth temporal accuracy orders have been theoretically explored by conducting Fourier and dispersion analyses and numerically validated by solving three test problems with analytic solutions. Copyright © 2006 John Wiley & Sons, Ltd.     

Projection‐based variational multiscale method for incompressible Navier–Stokes equations in time‐dependent domains

A variational multiscale method for computations of incompressible Navier–Stokes equations in time‐dependent domains is presented. The proposed scheme is a three‐scale variational multiscale method with a projection‐based scale separation that uses an additional tensor valued space for the large scales. The resolved large and small scales are computed in a coupled way with the effects of unresolved scales confined to the resolved small scales. In particular, the Smagorinsky eddy viscosity model is used to model the effects of unresolved scales. The deforming domain is handled by the arbitrary Lagrangian–Eulerian approach and by using an elastic mesh update technique with a mesh‐dependent stiffness. Further, the choice of orthogonal finite element basis function for the resolved large scale leads to a computationally efficient scheme. Simulations of flow around a static beam attached to a square base, around an oscillating beam and around a plunging aerofoil are presented. Copyright © 2016 John Wiley & Sons, Ltd.     

Time‐linearized time‐harmonic 3‐D Navier–Stokes shock‐capturing schemes

In the present paper, a numerical method for the computation of time‐harmonic flows, using the time‐linearized compressible Reynolds‐averaged Navier–Stokes equations is developed and validated. The method is based on the linearization of the discretized nonlinear equations. The convective fluxes are discretized using an O(Δx) MUSCL scheme with van Leer flux‐vector‐splitting. Unsteady perturbations of the turbulent stresses are linearized using a frozen‐turbulence‐Reynolds‐number hypothesis, to approximate eddy‐viscosity perturbations. The resulting linear system is solved using a pseudo‐time‐marching implicit ADI‐AF (alternating‐directions‐implicit approximate‐factorization) procedure with local pseudo‐time‐steps, corresponding to a matrix‐successive‐underrelaxation procedure. The stability issues associated with the pseudo‐time‐marching solution of the time‐linearized Navier–Stokes equations are discussed. Comparison of computations with measurements and with time‐nonlinear computations for 3‐D shock‐wave oscillation in a square duct, for various back‐pressure fluctuation frequencies (180, 80, 20 and 10 Hz), assesses the shock‐capturing capability of the time‐linearized scheme. Copyright © 2007 John Wiley & Sons, Ltd.     

Derivation and characteristics analysis of an acoustics–convection upstream resolution algorithm for the two‐dimensional Euler and Navier–Stokes equations

The first of a two‐paper series, this paper introduces a new decomposition not of the hyperbolic flux vector but of the flux vector Jacobian. The paper then details for the Euler and Navier–Stokes equations an intrinsically infinite directional upstream‐bias formulation that rests on the mathematics and physics of multi‐dimensional acoustics and convection. Based upon characteristic velocities, this formulation introduces the upstream bias directly at the differential equation level, before the spatial discretization, within a characteristics‐bias governing system. Through a decomposition of the Euler flux divergence into multi‐dimensional acoustics and convection–acoustics components, this characteristics‐bias system induces consistent upstream bias along all directions of spatial wave propagation, with anisotropic variable‐strength upstreaming that correlates with the spatial distribution of characteristic velocities. Copyright © 2005 John Wiley & Sons, Ltd.