A bounded convection scheme for the overlapping control volume approach

This paper introduces a flux-limited scheme FLOCV for the overlapping control volume (OCV) approach to 2D steady and unsteady convection–diffusion problems on structured non-orthogonal grids. FLOCV switches from second- to first-order interpolation in the presence of extrema. Smooth switching between the two is ensured by weighted average second- and first-order upwind differencing, with the weights being dynamically determined. Five convective test problems are solved using this scheme and results are compared with known analytical solutions. It is found that FLOCV approximately retains second-order accuracy of the base discretization scheme on uniform grids and smooth non-uniform orthogonal grids. It is also found effective in removing oscillations for problems with discontinuities on both orthogonal and non-orthogonal grids, with little degradation of accuracy. © 1997 John Wiley & Sons, Ltd.     

MULTIDIMENSIONAL UPWIND RESIDUAL DISTRIBUTION SCHEMES FOR THE CONVECTION–DIFFUSION EQUATION

Multidimensional residual distribution schemes for the convection–diffusion equation are described. Compact upwind cell vertex schemes are used for the discretization of the convective term. For the diffusive term, two approaches are compared: the classical finite element Galerkin formulation, which preserves the compactness of the stencil used for the convective part, and various residual-based approaches in which the diffusive term, evaluated after a reconstruction step, is upwinded along with the convective term.     

FINITE ELEMENT RESOLUTION OF CONVECTION–DIFFUSION EQUATIONS WITH INTERIOR AND BOUNDARY LAYERS

The purpose of this paper is to present a new algorithm for the resolution of both interior and boundary layers present in the convection–diffusion equation in laminar regimes, based on the formulation of a family of polynomial– exponential elements. We have carried out an adaptation of the standard variational methods (finite element method and spectral element method), obtaining an algorithm which supplies non-oscillatory and accurate solutions. The algorithm consists of generating a coupled grid of polynomial standard elements and polynomial–exponential elements. The latter are able to represent the high gradients of the solution, while the standard elements represent the solution in the areas of smooth variation.     

对流扩散方程的摄动有限体积(PFV)方法及讨论

在有限体积(FV)方法的重构近似中,引入数值摄动处理,即把界面数值通量摄动展开成网格间距的幂级数,并利用积分方程自身的性质求出幂级数的系数,同时获得高精度迎风和中心型摄动有限体积(PFV)格式.对标量输运方程给出积分近似为二阶、重构近似为二、三和四阶迎风和中心型PFV格式,这些PFV格式的结构形式及使用基点数与一阶迎风格式完全一致,迎风PFV格式满足对流有界准则;二阶和四阶中心PFV格式对网格Peclet数的任意值均为正型格式,比常用的二阶中心格式优越.用一维标量输运和方腔流动算例说明PFV格式的优良性能,并把PFV方法与性质相近的摄动有限差分(PFD)方法及相关的高精度方法作了对比分析.     

An upwind finite‐volume element scheme and its maximum‐principle‐preserving property for nonlinear convection–diffusion problem

For a class of nonlinear convection–diffusion equation in multiple space dimensions, a kind of upwind finite‐volume element (UFVE) scheme is put forward. Some techniques, such as calculus of variations, commutating operators and prior estimates, are adopted. It is proved that the UFVE scheme is unconditionally stable and satisfies maximum principle. Optimal‐order estimates in H¹‐norm are derived to determine the error in the approximate solution. Numerical results are presented to observe the performance of the scheme. Copyright © 2007 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 CELL VERTEX ALGORITHM FOR THE INCOMPRESSIBLE NAVIER–STOKES EQUATIONS ON NON-ORTHOGONAL GRIDS

The steady, incompressible Navier–Stokes (N–S) equations are discretized using a cell vertex, finite volume method. Quadrilateral and hexahedral meshes are used to represent two- and three-dimensional geometries respectively. The dependent variables include the Cartesian components of velocity and pressure. Advective fluxes are calculated using bounded, high-resolution schemes with a deferred correction procedure to maintain a compact stencil. This treatment insures bounded, non-oscillatory solutions while maintaining low numerical diffusion. The mass and momentum equations are solved with the projection method on a non-staggered grid. The coupling of the pressure and velocity fields is achieved using the Rhie and Chow interpolation scheme modified to provide solutions independent of time steps or relaxation factors. An algebraic multigrid solver is used for the solution of the implicit, linearized equations. A number of test cases are anlaysed and presented. The standard benchmark cases include a lid-driven cavity, flow through a gradual expansion and laminar flow in a three-dimensional curved duct. Predictions are compared with data, results of other workers and with predictions from a structured, cell-centred, control volume algorithm whenever applicable. Sensitivity of results to the advection differencing scheme is investigated by applying a number of higher-order flux limiters: the MINMOD, MUSCL, OSHER, CLAM and SMART schemes. As expected, studies indicate that higher-order schemes largely mitigate the diffusion effects of first-order schemes but also shown no clear preference among the higher-order schemes themselves with respect to accuracy. The effect of the deferred correction procedure on global convergence is discussed.     

Control strategies for timestep selection in finite element simulation of incompressible flows and coupled reaction–convection–diffusion processes

We propose two timestep selection algorithms, based on feedback control theory, for finite element simulation of steady state and transient 2D viscous flow and coupled reaction–convection–diffusion processes. To illustrate performance of the schemes in practice, we solve Rayleigh–Benard–Marangoni flows, flow across a backward‐facing step, unsteady flow around a circular cylinder and chemical reaction systems. Numerical experiments confirm that the feedback controllers produce in some cases a very smooth stepsize variation, suggesting that robust control algorithms are possible. These experiments also show that parameter selection can improve timesteps when co‐ordinated with the convergence control of non‐linear iterations. Further, computational cost of the selection procedures is negligible, since they involve only storing a few extra vectors, computation of norms and evaluation of kinetic energy. Copyright © 2004 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.     

SOLUTION OF THE ADVECTION–DIFFUSION EQUATION USING A COMBINATION OF DISCONTINUOUS AND MIXED FINITE ELEMENTS

When transport is advection-dominated, classical numerical methods introduce excessive artificial diffusion and spurious oscillations. Special methods are required to overcome these phenomena. To solve the advection‒diffusion equation, a numerical method is developed using a discontinuous finite element method for the discretization of the advective terms. At the discontinuities of the approximate solution, numerical advective fluxes are calculated using one-dimensional approximate Riemann solvers. The method is stabilized with a multidimensional slope limiter which introduces small amounts of numerical diffusion when sharp concentration fronts occur. In addition, the diffusive term is discretized using a mixed hybrid finite element method. With this approach, numerical oscillations are completely avoided for a full range of cell Peclet numbers. The combination of discontinuous and mixed finite elements can be easily applied to 2D and 3D models using various types of elements in regular and irregular meshes. Numerical tests show good agreement with 1D and 2D analytical solutions. This approach is compared at the same time with two different numerical methods, a standard mixed finite method and a finite volume approach with high-resolution upwind terms. Regular and irregular meshes are used for the numerical tests to study the mesh effects on the numerical results. Our data show that in all cases this approach performs well. © 1997 by John Wiley & Sons, Ltd.     

A finite volume scheme preserving extremum principle for convection–diffusion equations on polygonal meshes

We propose a nonlinear finite volume scheme for convection–diffusion equation on polygonal meshes and prove that the discrete solution of the scheme satisfies the discrete extremum principle. The approximation of diffusive flux is based on an adaptive approach of choosing stencil in the construction of discrete normal flux, and the approximation of convection flux is based on the second‐order upwind method with proper slope limiter. Our scheme is locally conservative and has only cell‐centered unknowns. Numerical results show that our scheme can preserve discrete extremum principle and has almost second‐order accuracy. Copyright © 2017 John Wiley & Sons, Ltd.     

Bubble‐based stabilized finite element methods for time‐dependent convection–diffusion–reaction problems

In this paper, we propose a numerical algorithm for time‐dependent convection–diffusion–reaction problems and compare its performance with the well‐known numerical methods in the literature. Time discretization is performed by using fractional‐step θ‐scheme, while an economical form of the residual‐free bubble method is used for the space discretization. We compare the proposed algorithm with the classical stabilized finite element methods over several benchmark problems for a wide range of problem configurations. The effect of the order in the sequence of discretization (in time and in space) to the quality of the approximation is also investigated. Numerical experiments show the improvement through the proposed algorithm over the classical methods in either cases. Copyright © 2016 John Wiley & Sons, Ltd.     

A multiscale control volume finite element method for advection–diffusion equations

We present a new stabilized method for advection–diffusion equations, which combines a control volume FEM formulation of the governing equations with a novel multiscale approximation of the total flux. The latter incorporates information about the exact solution that cannot be represented on the mesh. To define this flux, we solve the governing equations along suitable mesh segments under the assumption that the flux varies linearly along these segments. This procedure yields second‐order accurate fluxes on the edges of the mesh. Then, we use curl‐conforming elements of the same order to lift these edge fluxes into the mesh elements. In so doing, we obtain a stabilized control volume FEM formulation that is second‐order accurate and does not require mesh‐dependent stabilization parameters. Numerical convergence studies on uniform and nonuniform grids along with several standard advection tests illustrate the computational properties of the new method. Published 2015. This article is a U.S. Government work and is in the public domain in the USA.     

Fully computable a posteriori error bounds for stabilised FEM approximations of convection–reaction–diffusion problems in three dimensions

Fully computable upper bounds are developed for the discretisation error measured in the natural (energy) norm for convection–reaction–diffusion problems in three dimensions. The upper bounds are genuine upper bounds in the sense that the numerical value of the estimated error exceeds the actual numerical value of the true error regardless of the coarseness of the mesh or the nature of the data for the problem. All constants appearing in the bounds are fully specified. Examples show the estimator to be reliable and accurate even in the case of complicated three‐dimensional problems. Copyright © 2013 John Wiley & Sons, Ltd.     

A note on two upwind strategies for RBF‐based grid‐free schemes to solve steady convection–diffusion equations

In this paper, two radial basis function (RBF)‐based local grid‐free upwind schemes have been discussed for convection–diffusion equations. The schemes have been validated over some convection–diffusion problems with sharp boundary layers. It is found that one of the upwind schemes realizes the boundary layers more accurately than the rest. Comparisons with the analytical solutions demonstrate that the local RBF grid‐free upwind schemes based on the exact velocity direction are stable and produce accurate results on domains discretized even with scattered distribution of nodal points. Copyright © 2009 John Wiley & Sons, Ltd.     

NUMERICAL SOLUTIONS OF OPTIMAL CONTROL FOR THERMALLY CONVECTIVE FLOWS

We study the numerical solution of optimal control problems associated with two-dimensional viscous incompressible thermally convective flows. Although the techniques apply to more general settings, the presentation is confined to the objectives of minimizing the vorticity in the steady state case and tracking the velocity field in the non-stationary case with boundary temperature controls. In the steady state case we develop a systematic way to use the Lagrange multiplier rules to derive an optimality system of equations from which an optimal solution can be computed; finite element methods are used to find approximate solutions for the optimality system of equations. In the time-dependent case a piecewise-in-time optimal control approach is proposed and the fully discrete approximation algorithm for solving the piecewise optimal control problem is defined. Numerical results are presented for both the steady state and time-dependent optimal control problems. © 1997 John Wiley & Sons, Ltd.     

ADVECTION–DIFFUSION PROCESSES AND RESIDENCE TIMES IN SEMIENCLOSED MARINE BASINS

This paper addresses the problem of estimating the residence times in a marine basin of a passive constituent released in the sea. The dispersion process is described by an advection–diffusion model and the hydrodynamics is assumed to be known. We have performed the analysis of two different scenarios: (i) basins with unidirectional flows, in three space dimensions and under the rigid lid approximation, and (ii) basins with flows forced by the tide, under the shallow water approximation. Let the random variable τ be defined as the time spent in the basin by a particle released at a given point. The probability distribution of τ is obtained from the solution of the advection–diffusion problem and the residence time of a particle is defined as the mean value of τ. Two different numerical approximations have been used to solve the continuous problem: the finite volume and Monte Carlo methods. For both continuous and discrete formulations it is proved that if all the particles eventually leave the basin, then the residence time has a finite value. We present here the results obtained for two study cases: a two- dimensional basin with a steady flow and a one-dimensional channel with flow induced by the tide. The results obtained by the finite volume and Monte Carlo methods are in very good agreement for both scenarios.     

Time‐domain BEM solution of convection–diffusion‐type MHD equations

The two‐dimensional convection–diffusion‐type equations are solved by using the boundary element method (BEM) based on the time‐dependent fundamental solution. The emphasis is given on the solution of magnetohydrodynamic (MHD) duct flow problems with arbitrary wall conductivity. The boundary and time integrals in the BEM formulation are computed numerically assuming constant variations of the unknowns on both the boundary elements and the time intervals. Then, the solution is advanced to the steady‐state iteratively. Thus, it is possible to use quite large time increments and stability problems are not encountered. The time‐domain BEM solution procedure is tested on some convection–diffusion problems and the MHD duct flow problem with insulated walls to establish the validity of the approach. The numerical results for these sample problems compare very well to analytical results. Then, the BEM formulation of the MHD duct flow problem with arbitrary wall conductivity is obtained for the first time in such a way that the equations are solved together with the coupled boundary conditions. The use of time‐dependent fundamental solution enables us to obtain numerical solutions for this problem for the Hartmann number values up to 300 and for several values of conductivity parameter. Copyright © 2007 John Wiley & Sons, Ltd.     

AN EXPLICIT FINITE VOLUME SPATIAL MARCHING METHOD FOR REDUCED NAVIER–STOKES EQUATIONS

This paper develops a spatial marching method for high-speed flows based on a finite volume approach. The method employs the reduced Navier– Stokes equations and a pressure splitting in the streamwise direction based on the Vigneron strategy. For marching from an upstream station to one downstream the modified five-level Runge–Kutta integration scheme due to Jameson and Schmidt is used. In addition, for shock handling and for good convergence properties the method employs a matrix form of the artificial dissipation terms, which has been shown to improve the accuracy of predictions. To achieve a fast rate of convergence, a local time-stepping concept is used. The method retains the time derivative in the governing equations and the solution at every spatial station is obtained in an iterative manner. The developed method is validated against two test cases: (a) supersonic flow past a flat plate; and (b) hypersonic flow past a compression corner involving a strong viscous–inviscid interaction. The computed wall pressure and wall heat transfer coefficients exhibit good general agreement with previous computations by other investigators and with experiments.