Finite element methods for one-dimensional combustion problems

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.     

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.     

Development of a convection–diffusion‐reaction magnetohydrodynamic solver on non‐staggered grids

This paper presents a convection–diffusion‐reaction (CDR) model for solving magnetic induction equations and incompressible Navier–Stokes equations. For purposes of increasing the prediction accuracy, the general solution to the one‐dimensional constant‐coefficient CDR equation is employed. For purposes of extending this discrete formulation to two‐dimensional analysis, the alternating direction implicit solution algorithm is applied. Numerical tests that are amenable to analytic solutions were performed in order to validate the proposed scheme. Results show good agreement with the analytic solutions and high rate of convergence. Like many magnetohydrodynamic studies, the Hartmann–Poiseuille problem is considered as a benchmark test to validate the code. Copyright © 2004 John Wiley & Sons, Ltd.     

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.     

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.     

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.     

Upwind basis finite elements for convection-dominated problems

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 multimesh adaptive scheme for air quality modeling with the finite element method

A multimesh adaptive scheme for convection–diffusion–reaction problems for a large number of components is presented. The problem is solved by splitting transport and reaction processes. This way, the evaluation of the nonreactive part for each component and the reaction at each node constitute independent tasks. This allows to discretize each component of the solution on a distinct computational mesh, adapted on the basis of its error indicator. The standard single‐mesh strategy is used for comparison. Simulations of a point emission in a 3D domain are presented. Low remeshing periods of the adaptive scheme are found to be optimal, in terms of computational cost and accuracy, for the nonreactive problem. Examples with several reaction terms, with an increase of the complexity, are then presented. Results show that the accuracy of single‐mesh and multimesh strategies are similar. Instead, the computational cost of the multimesh strategy is lower than the single‐mesh in the majority of the examples; this process is controlled by the stiff behavior of the reactive term. The problem size of the multimesh scheme is much lower, and therefore, larger spatial discretizations can be simulated for a given available memory. The efficiency of the multimesh strategy increases with the number of species and the number of species that develop a plume. Finally, an example of a punctual emission considering realistic values of the initial concentrations and using the Community Multiscale Air Quality‐CBO5 reaction model, which involves 62 components, is presented; the small‐scale structure of the different nitrogen components near the emitter is captured. Copyright © 2013 John Wiley & Sons, Ltd.     

A variational multiscale Newton–Schur approach for the incompressible Navier–Stokes equations

In the following paper, we present a consistent Newton–Schur (NS) solution approach for variational multiscale formulations of the time‐dependent Navier–Stokes equations in three dimensions. The main contributions of this work are a systematic study of the variational multiscale method for three‐dimensional problems and an implementation of a consistent formulation suitable for large problems with high nonlinearity, unstructured meshes, and non‐symmetric matrices. In addition to the quadratic convergence characteristics of a Newton–Raphson‐based scheme, the NS approach increases computational efficiency and parallel scalability by implementing the tangent stiffness matrix in Schur complement form. As a result, more computations are performed at the element level. Using a variational multiscale framework, we construct a two‐level approach to stabilizing the incompressible Navier–Stokes equations based on a coarse and fine‐scale subproblem. We then derive the Schur complement form of the consistent tangent matrix. We demonstrate the performance of the method for a number of three‐dimensional problems for Reynolds number up to 1000 including steady and time‐dependent flows. Copyright © 2009 John Wiley & Sons, Ltd.     

The (9,5) HOC formulation for the transient Navier–Stokes equations in primitive variable

We recently proposed an improved (9,5) higher order compact (HOC) scheme for the unsteady two‐dimensional (2‐D) convection–diffusion equations. Because of using only five points at the current time level in the discretization procedure, the scheme was seen to be computationally more efficient than its predecessors. It was also seen to capture very accurately the solution of the unsteady 2‐D Navier–Stokes (N–S) equations for incompressible viscous flows in the stream function–vorticity (ψ – ω) formulation. In this paper, we extend the scope of the scheme for solving the unsteady incompressible N–S equations based on primitive variable formulation on a collocated grid. The parabolic momentum equations are solved for the velocity field by a time‐marching strategy and the pressure is obtained by discretizing the elliptic pressure Poisson equation by the steady‐state form of the (9,5) scheme with the Neumann boundary conditions. In particular, for pressure, we adopt a strategy on the collocated grid in conjunction with ideas borrowed from the staggered grid approach in finite volume. We first apply this extension to a problem having analytical solution and then to the famous lid‐driven square cavity problem. We also apply our formulation to the backward‐facing step problem to see how the method performs for external flow problems. The results are presented and are compared with established numerical results. This new approach is seen to produce excellent comparison in all the cases. Copyright © 2007 John Wiley & Sons, Ltd.     

Newton linearization of the incompressible Navier–Stokes equations

The present study aims to accelerate the convergence to incompressible Navier–Stokes solution. For the sake of computational efficiency, Newton linearization of equations is invoked on non‐staggered grids to shorten the sequence to the final solution of the non‐linear differential system of equations. For the sake of accuracy, the resulting convection–diffusion–reaction finite‐difference equation is solved line‐by‐line using the proposed nodally exact one‐dimensional scheme. The matrix size is reduced and, at the same time, the CPU time is considerably saved due to the decrease of stencil points. The effectiveness of the implemented Newton linearization is demonstrated through computational exercises. Copyright © 2004 John Wiley & Sons, Ltd.     

The solution of a convection–diffusion–reaction equation arising from cathodic reduction in a pulsating crack

This paper derives the convection–diffusion-reaction equation governing the reaction between the dissolved oxygen in sea-water and the steel walls of a pulsating crack. By the neglect of the diffusion term it is shown that an exact solution of the convection-reaction equation can be obtained. A numerical method for the solution of the complete convection–diffusion-reaction equation is derived by the use of finite differences. The numerical computation of the initial transient and the final periodic steady-state values is also discussed.     

Solution to transient Navier–Stokes equations by the coupling of differential quadrature time integration scheme with dual reciprocity boundary element method

The two‐dimensional time‐dependent Navier–Stokes equations in terms of the vorticity and the stream function are solved numerically by using the coupling of the dual reciprocity boundary element method (DRBEM) in space with the differential quadrature method (DQM) in time. In DRBEM application, the convective and the time derivative terms in the vorticity transport equation are considered as the nonhomogeneity in the equation and are approximated by radial basis functions. The solution to the Poisson equation, which links stream function and vorticity with an initial vorticity guess, produces velocity components in turn for the solution to vorticity transport equation. The DRBEM formulation of the vorticity transport equation results in an initial value problem represented by a system of first‐order ordinary differential equations in time. When the DQM discretizes this system in time direction, we obtain a system of linear algebraic equations, which gives the solution vector for vorticity at any required time level. The procedure outlined here is also applied to solve the problem of two‐dimensional natural convection in a cavity by utilizing an iteration among the stream function, the vorticity transport and the energy equations as well. The test problems include two‐dimensional flow in a cavity when a force is present, the lid‐driven cavity and the natural convection in a square cavity. The numerical results are visualized in terms of stream function, vorticity and temperature contours for several values of Reynolds (Re) and Rayleigh (Ra) numbers. Copyright © 2008 John Wiley & Sons, Ltd.     

A transformation‐free HOC scheme for incompressible viscous flows on nonuniform polar grids

We recently proposed a transformation‐free higher‐order compact (HOC) scheme for two‐dimensional (2‐D) steady convection–diffusion equations on nonuniform Cartesian grids (Int. J. Numer. Meth. Fluids 2004; 44 :33–53). As the scheme was equipped to handle only constant coefficients for the second‐order derivatives, it could not be extended directly to curvilinear coordinates, where they invariably occur as variables. In this paper, we extend the scheme to cylindrical polar coordinates for the 2‐D convection–diffusion equations and more specifically to the 2‐D incompressible viscous flows governed by the Navier–Stokes (N–S) equations. We first apply the formulation to a problem having analytical solution and demonstrate its fourth‐order spatial accuracy. We then apply it to the flow past an impulsively started circular cylinder problem and finally to the driven polar cavity problem. We present our numerical results and compare them with established numerical and analytical and experimental results whenever available. This new approach is seen to produce excellent comparison in all the cases. Copyright © 2009 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.     

Development of a convection–diffusion–reaction model for solving Maxwell's equations in frequency domain

We propose in this study a numerically accurate and computationally efficient convection–diffusion–reaction finite difference scheme to discretize the full‐vector and semi‐vector optical waveguide equations. The scheme formulated in a grid stencil of five nodal points for solving the three‐dimensional waveguide equations employs the locally analytic solution. In this three‐dimensional study, calculations were carried out for the investigation of wave propagation in diffused channel, rectangular and rib types of optical waveguide. Copyright © 2011 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.     

An immersed boundary method for the incompressible Navier–Stokes equations in complex geometry

A nodally exact convection–diffusion–reaction scheme developed in Cartesian grids is applied to solve the flow equations in irregular domains within the framework of immersed boundary (IB) method. The artificial momentum forcing term applied at certain points in the flow and inside the body of any shape allows the imposition of no‐slip velocity condition to account for the body of complex boundary. Development of an interpolation scheme that can accurately lead to no‐slip velocity condition along the IB is essential since Cartesian grid lines generally do not coincide with the IB. The results simulated from the proposed IB method agree well with other numerical and experimental results for several chosen benchmark problems. The accuracy and fidelity of the IB flow solver to predict flows with irregular IBs are therefore demonstrated. Copyright © 2007 John Wiley & Sons, Ltd.     

A comparison of numerical methods applied to non-linear adsorption columns

Eight numerical schemes (first-order upstream finite difference, MacCormack, explicit Taylor–Galerkin, random choice, flux-corrected transport, ENO, TVD, and Euler–Lagrange methods) are compared on the basis of their computational efficiency for one-dimensional non-linear convection–diffusion problems. For the ideal chromatographic equation for which an exact solution exists, errors plotted against computational times show that the best methods are the random choice, Euler–Lagrange and flux-corrected MacCormack methods. Even when significant diffusion is added to the model, steep gradients are possible because of non-linearities. In such an instance, the random choice and flux-corrected transport methods give the best performance. One can now tackle more complicated problems and refer to this comparative study in order to choose an adequate numerical method which will provide sufficiently accurate results at a reasonable cost.     

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.