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A number of finite element formulations involving discontinuous weighting functions have been tested against analytic solutions for a steady scalar convection—diffusion problem at intermediate Peclet number, with a ‘hard’ downstream boundary condition. The emphasis is on extending these methods to isoparametric bilinear and biquadratic elements. In order to do this a procedure is given for the exact calculation of shape function Laplacians. Having confirmed the success of the Brooks—Hughes streamline upwind Petrov—Galerkin (SUPG) method for isoparametric bilinear elements, formulations for biquadratic elements are examined. Galerkin least squares offers little advantage over SUPG in the test problem. The generalized Galerkin method of Donea et al. gave excellent results, but because of concern over the possibility of cross‐streamline artificial diffusion in some cases, a strictly streamline formulation incorporating the optimal parameters of Donea et al. is proposed. This gave excellent results on a sufficiently refined mesh.

The purpose of this paper is to propose the generalized integral transform technique (GITT) to the solution of convection-diffusion problems with nonlinear boundary conditions by employing the corresponding nonlinear eigenvalue problem in the construction of the expansion basis.

The original nonlinear boundary condition coefficients in the problem formulation are all incorporated into the adopted eigenvalue problem, which may be itself integral transformed through a representative linear auxiliary problem, yielding a nonlinear algebraic eigenvalue problem for the associated eigenvalues and eigenvectors, to be solved along with the transformed ordinary differential system. The nonlinear eigenvalues computation may also be accomplished by rewriting the corresponding transcendental equation as an ordinary differential system for the eigenvalues, which is then simultaneously solved with the transformed potentials.

An application on one-dimensional transient diffusion with nonlinear boundary condition coefficients is selected for illustrating some important computational aspects and the convergence behavior of the proposed eigenfunction expansions. For comparison purposes, an alternative solution with a linear eigenvalue problem basis is also presented and implemented.

This novel approach can be further extended to various classes of nonlinear convection-diffusion problems, either already solved by the GITT with a linear coefficients basis, or new challenging applications with more involved nonlinearities.

A new finite volume (FV) method is proposed for the solution of convection‐diffusion equations defined on 2D convex domains of general shape. The domain is approximated by a polygonal region; a structured non‐uniform mesh is defined; the domain is partitioned in control volumes. The conservative form of the problem is solved by imposing the law to be verified on each control volume. The dependent variable is approximated to the second order by means of a quadratic profile. When, for the hyperbolic equation, discontinuities are present, or when the gradient of the solution is very high, a cubic profile is defined in such a way that it enjoys unidirectional monotonicity. Numerical results are given.

The purpose of this paper is to propose a parallel partition of unity method to solve the time-dependent convection-diffusion equations.

This paper opted for the time-dependent convection-diffusion equations using the finite element method and the partition of unity method.

This paper provides one efficient parallel algorithm which reaches the same accuracy as the standard Galerkin method (SGM) but saves a lot of computational time.

In this paper, a parallel partition of unity method is proposed for the time-dependent convection-diffusion equations. At each time step, the authors only need to solve a series of independent local sub-problems in parallel instead of one global problem.

The purpose of this paper is to provide a robust second-order numerical scheme for singularly perturbed delay parabolic convection–diffusion initial boundary value problem.

For the parabolic convection-diffusion initial boundary value problem, the authors solve the problem numerically by discretizing the domain in the spatial direction using the Shishkin-type meshes (standard Shishkin mesh, Bakhvalov–Shishkin mesh) and in temporal direction using the uniform mesh. The time derivative is discretized by the implicit-trapezoidal scheme, and the spatial derivatives are discretized by the hybrid scheme, which is a combination of the midpoint upwind scheme and central difference scheme.

The authors find a parameter-uniform convergent scheme which is of second-order accurate globally with respect to space and time for the singularly perturbed delay parabolic convection–diffusion initial boundary value problem. Also, the Thomas algorithm is used which takes much less computational time.

A singularly perturbed delay parabolic convection–diffusion initial boundary value problem is considered. The solution of the problem possesses a regular boundary layer. The authors solve this problem numerically using a hybrid scheme. The method is parameter-uniform convergent and is of second order accurate globally with respect to space and time. Numerical results are carried out to verify the theoretical estimates.

The paper gives the description of boundary element method (BEM) with subdomains for the solution of convection—diffusion equations with variable coefficients and Burgers’ equations. At first, the whole domain is discretized into K subdomains, in which linearization of equations by representing convective velocity by the sum of constant and variable parts is carried out. Then using fundamental solutions for convection—diffusion linear equations for each subdomain the boundary integral equation (in which the part of the convective term with the constant convective velocity is not included into the pseudo‐body force) is formulated. Only part of the convective term with the variable velocity, which is, as a rule, more than one order less than convective velocity constant part contribution, is left as the pseudo‐source. On the one hand, this does not disturb the numerical BEM—algorithm stability and, on the other hand, this leads to significant improvement in the accuracy of solution. The global matrix, similar to the case of finite element method, has block band structure whereas its width depends only on the numeration order of nodes and subdomains. It is noted, that in comparison with the direct boundary element method the number of global matrix non‐zero elements is not proportional to the square of the number of nodes, but only to the total number of nodal points. This allows us to use the BEM for the solution of problems with very fine space discretization. The proposed BEM with subdomains technique has been used for the numerical solution of one‐dimensional linear steady‐state convective—diffusion problem with variable coefficients and one‐dimensional non‐linear Burgers’ equation for which exact analytical solutions are available. It made it possible to find out the BEM correctness according to both time and space. High precision of the numerical method is noted. The good point of the BEM is the high iteration convergence, which is disturbed neither by high Reynolds numbers nor by the presence of negative velocity zones.

This paper aims to provide a well-behaved nonlinear scheme and accelerating iteration for the nonlinear convection diffusion equation with fundamental properties illustrated.

A nonlinear finite difference scheme is studied with fully implicit (FI) discretization used to acquire accurate simulation. A Picard–Newton (PN) iteration with a quadratic convergent ratio is designed to realize fast solution. Theoretical analysis is performed using the discrete function analysis technique. By adopting a novel induction hypothesis reasoning technique, the L^∞ (H¹) convergence of the scheme is proved despite the difficulty because of the combination of conservative diffusion and convection operator. Other properties are established consequently. Furthermore, the algorithm is extended from first-order temporal accuracy to second-order temporal accuracy.

Theoretical analysis shows that each of the two FI schemes is stable, its solution exists uniquely and has second-order spatial and first/second-order temporal accuracy. The corresponding PN iteration has the same order of accuracy and quadratic convergent speed. Numerical tests verify the conclusions and demonstrate the high accuracy and efficiency of the algorithms. Remarkable acceleration is gained.

The numerical method provides theoretical and technical support to accelerate resolving convection diffusion, non-equilibrium radiation diffusion and radiation transport problems.

The FI schemes and iterations for the convection diffusion problem are proposed with their properties rigorously analyzed. The induction hypothesis reasoning method here differs with those for linearization schemes and is applicable to other nonlinear problems.

This paper seeks to analyze transient convection‐diffusion by employing the generalized integral transform technique (GITT) combined with an arbitrary transient filtering solution, aimed at enhancing the convergence behavior of the associated eigenfunction expansions. The idea is to consider analytical approximations of the original problem as filtering solutions, defined within specific ranges of the time variable, which act diminishing the importance of the source terms in the original formulation and yielding a filtered problem for which the integral transformation procedure results in faster converging eigenfunction expansions. An analytical local instantaneous filtering is then more closely considered to offer a hybrid numerical‐analytical solution scheme for linear or nonlinear convection‐diffusion problems.

The approach is illustrated for a test‐case related to transient laminar convection within a parallel‐plates channel with axial diffusion effects.

The developing thermal problem is solved for the fully developed flow situation and a step change in inlet temperature. An analysis is performed on the variation of Peclet number, so as to investigate the importance of the axial heat or mass diffusion on convergence rates.

This paper succeeds in analyzing transient convection‐diffusion via GITT, combined with an arbitrary transient filtering solution, aimed at enhancing the convergence behaviour of the associated eigenfunction expansions.

A generalized Galerkin technique originally developed by Donea, Belytschko and Smolinski for solving the steady convection—diffusion equation using elements with quadratic interpolation has been modified to extend its application to the case of geometrically distorted 1D and 2D elements. The numerical results indicate that the modified scheme gives accurate results and presents a rather small sensitivity to element distortions.

The purpose of this paper is to present a simple meshless solution method for challenging engineering problems such as those with high wave numbers or convection-diffusion ones with high Peclet number. The method uses a set of residual-free bases in a local form.

The residual-free bases, called here as exponential basis functions, are found so that they satisfy the governing equations within each subdomain. The compatibility between the subdomains is weakly satisfied by enforcing the local approximation of the main state variables pass through the data at nodes surrounding the central node of the subdomain. The central state variable is first recovered from the approximation and then re-assigned to the central node to construct the associated equation. This leads to the least compatibility required in the solution, e.g. C0 continuity in Laplace problems.

The authors shall show that one can solve a variety of problems with regular and irregular point distribution with high convergence rate. The authors demonstrate that this is impossible to achieve using finite element method. Problems with Laplace and Helmholtz operators as well as elasto-static problems are solved to demonstrate the effectiveness of the method. A convection-diffusion problem with high Peclet number and problems with high wave numbers are among the examples. In all cases, results with high rate of convergence are obtained with moderate number of nodes per cloud.

The paper presents a simple meshless method which not only is capable of solving a variety of challenging engineering problems but also yields results with high convergence rate.