Development and validation of a SUPG finite element scheme for the compressible Navier–Stokes equations using a modified inviscid flux discretization

This paper considers the streamline‐upwind Petrov–Galerkin (SUPG) method applied to the unsteady compressible Navier–Stokes equations in conservation‐variable form. The spatial discretization, including a modified approach for interpolating the inviscid flux terms in the SUPG finite element formulation, and the second‐order accurate time discretization are presented. The numerical method is discussed in detail. The performance of the algorithm is then investigated by considering inviscid flow past a circular cylinder. Validation of the finite element formulation via comparisons with experimental data for high‐Mach number perfect gas laminar flows is presented, with a specific focus on comparisons with experimentally measured skin friction and convective heat transfer on a 15° compression ramp. Copyright © 2007 John Wiley & Sons, Ltd.     

Edge‐based finite element method for shallow water equations

This paper describes an edge‐based implementation of the generalized residual minimum (GMRES) solver for the fully coupled solution of non‐linear systems arising from finite element discretization of shallow water equations (SWEs). The gain in terms of memory, floating point operations and indirect addressing is quantified for semi‐discrete and space–time analyses. Stabilized formulations, including Petrov–Galerkin models and discontinuity‐capturing operators, are also discussed for both types of discretization. Results illustrating the quality of the stabilized solutions and the advantages of using the edge‐based approach are presented at the end of the paper. Copyright © 2001 John Wiley & Sons, Ltd.     

On streamline diffusion arising in Galerkin FEM with predictor/multi‐corrector time integration

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.     

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.     

A spectral‐element discontinuous Galerkin thermal lattice Boltzmann method for conjugate heat transfer applications

We present a spectral‐element discontinuous Galerkin thermal lattice Boltzmann method for fluid–solid conjugate heat transfer applications. Using the discrete Boltzmann equation, we propose a numerical scheme for conjugate heat transfer applications on unstructured, non‐uniform grids. We employ a double‐distribution thermal lattice Boltzmann model to resolve flows with variable Prandtl (Pr) number. Based upon its finite element heritage, the spectral‐element discontinuous Galerkin discretization provides an effective means to model and investigate thermal transport in applications with complex geometries. Our solutions are represented by the tensor product basis of the one‐dimensional Legendre–Lagrange interpolation polynomials. A high‐order discretization is employed on body‐conforming hexahedral elements with Gauss–Lobatto–Legendre quadrature nodes. Thermal and hydrodynamic bounce‐back boundary conditions are imposed via the numerical flux formulation that arises because of the discontinuous Galerkin approach. As a result, our scheme does not require tedious extrapolation at the boundaries, which may cause loss of mass conservation. We compare solutions of the proposed scheme with an analytical solution for a solid–solid conjugate heat transfer problem in a 2D annulus and illustrate the capture of temperature continuities across interfaces for conductivity ratio γ > 1. We also investigate the effect of Reynolds (Re) and Grashof (Gr) number on the conjugate heat transfer between a heat‐generating solid and a surrounding fluid. Steady‐state results are presented for Re = 5?40 and Gr = 10⁵?10⁶. In each case, we discuss the effect of Re and Gr on the heat flux (i.e. Nusselt number Nu) at the fluid–solid interface. Our results are validated against previous studies that employ finite‐difference and continuous spectral‐element methods to solve the Navier–Stokes equations. Copyright © 2016 John Wiley & Sons, Ltd.     

Generalized Fourier analyses of the advection–diffusion equation—Part I: one‐dimensional domains

This paper presents a detailed multi‐methods comparison of the spatial errors associated with finite difference, finite element and finite volume semi‐discretizations of the scalar advection–diffusion equation. The errors are reported in terms of non‐dimensional phase and group speed, discrete diffusivity, artificial diffusivity, and grid‐induced anisotropy. It is demonstrated that Fourier analysis provides an automatic process for separating the discrete advective operator into its symmetric and skew‐symmetric components and characterizing the spectral behaviour of each operator. For each of the numerical methods considered, asymptotic truncation error and resolution estimates are presented for the limiting cases of pure advection and pure diffusion. It is demonstrated that streamline upwind Petrov–Galerkin and its control‐volume finite element analogue, the streamline upwind control‐volume method, produce both an artificial diffusivity and a concomitant phase speed adjustment in addition to the usual semi‐discrete artifacts observed in the phase speed, group speed and diffusivity. The Galerkin finite element method and its streamline upwind derivatives are shown to exhibit super‐convergent behaviour in terms of phase and group speed when a consistent mass matrix is used in the formulation. In contrast, the CVFEM method and its streamline upwind derivatives yield strictly second‐order behaviour. In Part II of this paper, we consider two‐dimensional semi‐discretizations of the advection–diffusion equation and also assess the affects of grid‐induced anisotropy observed in the non‐dimensional phase speed, and the discrete and artificial diffusivities. Although this work can only be considered a first step in a comprehensive multi‐methods analysis and comparison, it serves to identify some of the relative strengths and weaknesses of multiple numerical methods in a common analysis framework. Published in 2004 by 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.     

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.     

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.     

A discontinuous Galerkin method for the Navier–Stokes equations

The foundations of a new discontinuous Galerkin method for simulating compressible viscous flows with shocks on standard unstructured grids are presented in this paper. The new method is based on a discontinuous Galerkin formulation both for the advective and the diffusive contributions. High‐order accuracy is achieved by using a recently developed hierarchical spectral basis. This basis is formed by combining Jacobi polynomials of high‐order weights written in a new co‐ordinate system. It retains a tensor‐product property, and provides accurate numerical quadrature. The formulation is conservative, and monotonicity is enforced by appropriately lowering the basis order and performing h‐refinement around discontinuities. Convergence results are shown for analytical two‐ and three‐dimensional solutions of diffusion and Navier–Stokes equations that demonstrate exponential convergence of the new method, even for highly distorted elements. Flow simulations for subsonic, transonic and supersonic flows are also presented that demonstrate discretization flexibility using hp‐type refinement. Unlike other high‐order methods, the new method uses standard finite volume grids consisting of arbitrary triangulizations and tetrahedrizations. Copyright © 1999 John Wiley & Sons, Ltd.     

Adiabatic shock capturing in perfect gas hypersonic flows

This paper considers the streamline‐upwind Petrov/Galerkin (SUPG) method applied to the compressible Euler and Navier–Stokes equations in conservation‐variable form. The spatial discretization, including a modified approach for interpolating the inviscid flux terms in the SUPG finite element formulation, is briefly reviewed. Of particular interest is the behavior of the shock‐capturing operator, which is required to regularize the scheme in the presence of strong, shock‐induced gradients. A standard shock‐capturing operator that has been widely used in previous studies by several authors is presented and discussed. Specific modifications are then made to this standard operator that is designed to produce a more physically consistent discretization in the presence of strong shock waves. The actual implementation of the term in a finite‐dimensional approximation is also discussed. The behavior of the standard and modified scheme is then compared for several supersonic/hypersonic flows. The modified shock‐capturing operator is found to preserve enthalpy in the inviscid portion of the flowfield substantially better than the standard operator. Published in 2009 by 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.     

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 preconditioned semi‐staggered dilation‐free finite volume method for the incompressible Navier–Stokes equations on all‐hexahedral elements

A new semi‐staggered finite volume method is presented for the solution of the incompressible Navier–Stokes equations on all‐quadrilateral (2D)/hexahedral (3D) meshes. The velocity components are defined at element node points while the pressure term is defined at element centroids. The continuity equation is satisfied exactly within each elements. The checkerboard pressure oscillations are prevented using a special filtering matrix as a preconditioner for the saddle‐point problem resulting from second‐order discretization of the incompressible Navier–Stokes equations. The preconditioned saddle‐point problem is solved using block preconditioners with GMRES solver. In order to achieve higher performance FORTRAN source code is based on highly efficient PETSc and HYPRE libraries. As test cases the 2D/3D lid‐driven cavity flow problem and the 3D flow past array of circular cylinders are solved in order to verify the accuracy of the proposed method. Copyright © 2005 John Wiley & Sons, Ltd.     

An efficient numerical solution for linear stability of circular jet: A combination of Petrov–Galerkin spectral method and exponential coordinate transformation based on Fornberg's treatment

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.     

Numerical modelling of shear-dependent mass transfer in large arteries

A numerical scheme for the simulation of blood flow and transport processes in large arteries is presented. Blood flow is described by the unsteady 3D incompressible Navier–Stokes equations for Newtonian fluids; solute transport is modelled by the advection–diffusion equation. The resistance of the arterial wall to transmural transport is described by a shear-dependent wall permeability model. The finite element formulation of the Navier–Stokes equations is based on an operator-splitting method and implicit time discretization. The streamline upwind/Petrov–Galerkin (SUPG) method is applied for stabilization of the advective terms in the transport equation and in the flow equations. A numerical simulation is carried out for pulsatile mass transport in a 3D arterial bend to demonstrate the influence of arterial flow patterns on wall permeability characteristics and transmural mass transfer. The main result is a substantial wall flux reduction at the inner side of the curved region. © 1997 John Wiley & Sons, Ltd.     

Stabilized Bubble Function Method for Shallow Water Long Wave Equation

A relationship between the stabilized bubble function method and the stabilized finite element method is shown in this paper. The Petrov–Galerkin formulation with bubble function, i.e. a stabilized bubble function method, is proposed for the shallow water long wave equation. The Petrov–Galerkin formulation with the bubble function formulation possesses better stability than the Bubnov–Galerkin formulation with the bubble function.     

High‐order 𝒞1 finite‐element interpolating schemes—Part I: Semi‐Lagrangian linear advection

This paper is devoted to the development of accurate high‐order interpolating schemes for semi‐Lagrangian advection. The characteristic‐Galerkin formulation is obtained by using a semi‐Lagrangian temporal discretization of the total derivative. The semi‐Lagrangian method requires high‐order interpolators for accuracy. A class of ??¹ finite‐element interpolating schemes is developed and two semi‐Lagrangian methods are considered by tracking the feet of the characteristic lines either from the interpolation or from the integration nodes. Numerical stability and analytical results quantifying the amount of artificial viscosity induced by the two methods are presented in the case of the one‐dimensional linear advection equation, based on the modified equation approach. Results of test problems to simulate the linear advection of a cosine hill illustrate the performance of the proposed approach. Copyright © 2007 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.