Steady-state Convection-diffusion Problems Research Articles

Numerical treatment of a singularly perturbed 2-D convection-diffusion elliptic problem with Robin-type boundary conditions

We consider a singularly perturbed two-dimensional steady-state convection-diffusion problem with Robin boundary conditions. The coefficient of the highest-order terms in the differential equation and in the boundary conditions, denoted by ϵ, is a positive perturbation parameter, and so it may be arbitrarily small. Solutions to such problems present regular (exponential) boundary layers as well as corner layers. In this article, a numerical approach is carried out using a finite-difference technique with an appropriate layer-adapted piecewise-uniform Shishkin mesh to provide a good approximation of the exact solution. Some numerical examples are presented that show that the approximations obtained are accurate and that they are in agreement with the theoretical results.

Побудова асимптотики розв’язку для системи сингулярно збурених рівнянь методом істотно особливих функцій

Singularly perturbed problems with turning points arise as mathematical models for various physical phenomena. The internal turning point problem is a one-dimensional version of the steady-state convection-diffusion problem with a dominant convective term and a velocity field that changes sign in the reservoir. Boundary turning point problems, on the other hand, arise in geophysics and in the modeling of thermal boundary layers in laminar flow. The paper analyzes the results from the asymptotic analysis of singularly perturbed problems with turning points. For a homogeneous system of singularly perturbed differential equations with a small parameter at the highest derivative and a turning point, the conditions for constructing a uniform asymptotic solution are obtained. We consider the case when the spectrum of the limit operator contains multiple and identically zero elements. The asymptotics are constructed by the method of essentially singular functions, which allows using the Airy model operator in the vicinity of the turning point. The construction of asymptotic solutions contains arbitrary constants, which are determined uniquely during the solution of the iterative equations. At the same time, the conditions for the existence of a solution of a system of differentials with a small parameter for the highest derivative and for the presence of a turning point are obtained, provided that the turning point is located on the interval [0; l]. An example of constructing the asymptotic of a homogeneous system of differential equations is given.

Asymptotic expansion for the solution of a convection-diffusion problem in a thin graph-like junction

A steady-state convection-diffusion problem with a small diffusion of order O ( ε ) is considered in a thin three-dimensional graph-like junction consisting of thin cylinders connected through a domain (node) of diameter O ( ε ), where ε is a small parameter. Using multiscale analysis, the asymptotic expansion for the solution is constructed and justified. The asymptotic estimates in the norm of Sobolev space H 1 as well as in the uniform norm are proved for the difference between the solution and proposed approximations with a predetermined accuracy with respect to the degree of ε.

A dual mesh finite domain method for steady-state convection–diffusion problems

The dual-mesh finite domain method (DMFDM) proposed by Reddy (2019)[1] is used to study the steady-state convection–diffusion problems in 1D and 2D. In the DMFDM, one mesh of finite elements for the approximation of the domain and primary variables and another mesh of control domains, which also covers the whole domain, to satisfy the governing differential equations are used. The approach is distinguished from both the finite element method and the finite volume method, although the method shares the desirable features of both methods. Numerical examples are presented to illustrate the methodology and accuracy compared to the finite element and finite volume solutions.

Imposing various boundary conditions on positive definite kernels

This work is motivated by the frequent occurrence of boundary value problems with various boundary conditions in the modeling of some problems in engineering and physical science. Here we propose a new technique to force the positive definite kernels such as some radial basis functions to satisfy the boundary conditions exactly. It can improve the applications of existing methods based on positive definite kernels and radial basis functions especially the kernel based pseudospectral method for handling the differential equations with more complicated boundary conditions. In the proposed technique some new kernels are constructed using the positive definite kernels in a manner that they satisfy the required conditions. In addition, we prove the positive definiteness of the newly constructed kernel, and also the non-singularity of the collocation matrix is proved under some conditions. The proposed method is verified through the numerical solution of some benchmark problems such as a singularly perturbed steady-state convection–diffusion problem, two and three dimensional Poissons equations with various boundary conditions.

Dual reciprocity boundary node method for convection-diffusion problems

This paper presents a dual reciprocity boundary node method(DHBNM) for steady-state convection-diffusion problems with variable velocities. In this formulation, the solutions are composed of a particular and a complementary solutions, where the former is solved by boundary node method, and the latter is obtained by dual reciprocity method(DRM). The velocity field is decomposed into an average and a perturbation ones, which is treated using DRM by some steps of approximating process. A reverse procedure for the particular solution is developed to overcome the difficulty of getting the analytical solution of the particular solution. Numerical studies have validated that the DHBNM can produce high accuracy and stability.

A New Nodal Expansion Method with High-Order Moments for the Reduction of Numerical Oscillation in Convection-Diffusion Problems

The nodal integral method (NIM) has been widely used to solve multidimensional steady-state convection-diffusion problems. However, unphysical oscillating behavior arises when NIM is applied to steep-gradient problems and discontinuous problems. In this paper, a new nodal expansion method (NEM) with high-order moments (NEM_HM) is developed to reduce the numerical oscillation drawback of NIM. High-order moments of transverse-integrated variables are introduced. Based on the definition of Legendre moments, all the expansion coefficients of NEM_HM can be defined as shared moments and unshared moments. Then, the calculation framework of the traditional NEM is extended to include the high-order moments. Additional nodal balance equations are introduced to ensure the uniqueness of all the shared variables such as node-average variables. Finally, coupled discrete equations are obtained in terms of various order moments on the surfaces of the nodes. The classical Smith-Hutton problem and a cross-flow problem are chosen to test the effectiveness of NEM_HM. Numerical results show that the accuracy of NEM_HM outperforms NIM for steep-gradient problems and discontinuous cases.

An adaptive SUPG method for evolutionary convection–diffusion equations

An adaptive algorithm for the numerical simulation of time-dependent convection–diffusion–reaction equations will be proposed and studied. The algorithm allows the use of the natural extension of any error estimator for the steady-state problem for controlling local refinement and coarsening. The main idea consists in considering the SUPG solution of the evolutionary problem as the SUPG solution of a particular steady-state convection–diffusion problem with data depending on the computed solution. The application of the error estimator is based on a heuristic argument by considering a certain term to be of higher order. This argument is supported in the one-dimensional case by numerical analysis. In the numerical studies, particularly the residual-based error estimator from [18] will be applied, which has proved to be robust in the SUPG norm. The effectivity of this error estimator will be studied and the numerical results (accuracy of the solution, fineness of the meshes) will be compared with results obtained by utilizing the adaptive algorithm proposed in [5].

A robust local polynomial collocation method

SUMMARYIn this paper, a robust local polynomial collocation method is presented. Based on collocation, this method is rather simple and straightforward. The present method is developed in a way that the governing equation is satisfied on boundaries as well as boundary conditions. This requirement makes the present method more accurate and robust than conventional collocation methods, especially in estimating the partial derivatives of the solution near the boundary. Studies about the sensitivity of the shape parameter and the local supporting range in the moving least square approach and the convergence of the nodal resolution are carried out by using some benchmark problems. This method is further verified by applying it to a steady‐state convection–diffusion problem. Finally, the present method is applied to calculate the velocity fields of two potential flow problems. More accurate numerical results are obtained.Copyright © 2012 John Wiley & Sons, Ltd.

A TLM method for steady‐state convection–diffusion: Some additions and refinements

AbstractA recent paper introduced a novel and efficient scheme, based on the transmission line modelling (TLM) method, for solving steady‐state convection–diffusion problems. This paper shows how this one‐dimensional scheme can be adapted to include reaction and source terms and how it can be implemented with non‐equidistant nodes. It introduces new ways of calculating the necessary model parameters which can improve the accuracy of the scheme, shows how steady‐state solutions can be obtained directly, and compares results with those from two finite difference (FD) methods. While the cost of implementation is higher than for the FD schemes, the new TLM scheme can be significantly more accurate, especially when convection dominates. Copyright © 2008 John Wiley & Sons, Ltd.

High-order compact exponential finite difference methods for convection–diffusion type problems

A class of high-order compact (HOC) exponential finite difference (FD) methods is proposed for solving one- and two-dimensional steady-state convection–diffusion problems. The newly proposed HOC exponential FD schemes have nonoscillation property and yield high accuracy approximation solution as well as are suitable for convection-dominated problems. The O( h 4) compact exponential FD schemes developed for the one-dimensional (1D) problems produce diagonally dominant tri-diagonal system of equations which can be solved by applying the tridiagonal Thomas algorithm. For the two-dimensional (2D) problems, O( h 4 + k 4) compact exponential FD schemes are formulated on the nine-point 2D stencil and the line iterative approach with alternating direction implicit (ADI) procedure enables us to deal with diagonally dominant tridiagonal matrix equations which can be solved by application of the one-dimensional tridiagonal Thomas algorithm with a considerable saving in computing time. To validate the present HOC exponential FD methods, three linear and nonlinear problems, mostly with boundary or internal layers where sharp gradients may appear due to high Peclet or Reynolds numbers, are numerically solved. Comparisons are made between analytical solutions and numerical results for the currently proposed HOC exponential FD methods and some previously published HOC methods. The present HOC exponential FD methods produce excellent results for all test problems. It is shown that, besides including the excellent performances in computational accuracy, efficiency and stability, the present method has the advantage of better scale resolution. The method developed in this article is easy to implement and has been applied to obtain the numerical solutions of the lid driven cavity flow problem governed by the 2D incompressible Navier–Stokes equations using the stream function-vorticity formulation.

Iterative Methods for Solving Convection-diffusion Problem

AbstractTo obtain an approximate solution of the steady-state convectiondiffusion problem, it is necessary to solve the corresponding system of linear algebraic equations. The basic peculiarity of these LA systems is connected with the fact that they have non-symmetric matrices. We discuss the questions of approximate solution of 2D convection-diffusion problems on the basis of two- and three-level iterative methods. The general theory of iterative methods of solving grid equations is used to present the material of the paper. The basic problems of constructing grid approximations for steady-state convection-diffusion problems are considered. We start with the consideration of the Dirichlet problem for the differential equation with a convective term in the divergent, nondivergent, and skew-symmetric forms. Next, the corresponding grid problems are constructed. And, finally, iterative methods are used to solve approximately the above grid problems. Primary consideration is given to the study of the dependence of the number of iteration on the Peclet number, which is the ratio of the convective transport to the diffusive one.

On the Construction of Deflation-Based Preconditioners

In this article we introduce new bounds on the effective condition number of deflated and preconditioned-deflated symmetric positive definite linear systems. For the case of a subdomain deflation such as that of Nicolaides [SIAM J. Numer. Anal., 24 (1987), pp. 355--365], these theorems can provide direction in choosing a proper decomposition into subdomains. If grid refinement is performed, keeping the subdomain grid resolution fixed, the condition number is insensitive to the grid size. Subdomain deflation is very easy to implement and has been parallelized on a distributed memory system with only a small amount of additional communication. Numerical experiments for a steady-state convection-diffusion problem are included.

Computational Electrochemistry. Simulations of Homogeneous Chemical Reactions in the Confluence Reactor and Channel Flow Cell

The finite element method (FEM) is employed to simulate steady-state convection diffusion problems involving electrode processes and coupled homogeneous kinetic reactions in the channel cell and the confluence reactor. Initial FEM simulations are reported for EC, ECE, and DISP reactions in the channel cell, and the results are shown to agree with previous art. Calculations reveal the FEM extends the kinetic range accessible for such mechanisms beyond those reported previously using the backward implicit finite difference method. The FEM is then applied to simulate, for the first time, the quantitative effects of coupled homogeneous reactivity on the voltammetric response of the confluence reactor. Specifically, a working surface is presented for the application of the confluence reactor to the investigation of the CE type reactions. This surface provides a quantitative relationship between the chemical reaction rate, volume flow rate, and inlet concentrations using this new device. In all simulations performed, the FEM is found to be a highly efficient and accurate alternative to the finite difference method when applied to hydrodynamic voltammetric measurements.

Numerical solution of convection-diffusion problems at high Péclet number using boundary elements

This paper describes a boundary element scheme for solving steady-state convection–diffusion problems at high Peclet numbers. A special treatment of the singular integrals is included which uses discontinuous elements and a regularization procedure. Transformations are performed to avoid directly evaluating Bessel functions for Cauchy principal value and hypersingular integrals. Test examples are solved with values of Peclet number up to 107 to assess the numerical scheme. © 1998 John Wiley & Sons, Ltd.

BOUNDARY ELEMENT SOLUTIONS FOR FREE BOUNDARY CONVECTION-DIFFUSION PROBLEMS

The boundary element technique is used to solve the steady state convection-diffusion problem with constant velocity in a two-dimensional domain with a free interface. These problems arise in a number of important heat transfer applications involving melting or solidification, such as bulk crystal growth in Bridgman furnaces. The boundary element approach reduces the dimension of the problem, thereby improving the computational efficiency, and is particularly well suited to free-surface problems in which the position and shape of the solid-liquid interface are of primary importance. Results are presented for a case study problem representing solidification in a two-dimensional, rectangular configuration.

A Boundary Element Method Formulation for Design Sensitivities in Steady-State Conduction-Convection Problems

A Boundary Element Method (BEM) formulation for the determination of design sensitivities of temperature distributions to various shape and process parameters in steady-state convection-diffusion problems is presented in this paper. The present formulation is valid for constant or piecewise-constant convective velocities. This approach is based on direct differentiation (DDA) of the relevant BEM formulation of the problem. It retains the advantages of the BEM regarding accuracy and efficiency while avoiding strongly singular kernels. The BEM formulation is also observed to avoid any false diffusion. This approach provides a new avenue toward efficient optimization of steady-state convection-diffusion problems and may be easily adapted to investigate the thermal aspects of various machining processes.

A dual reciprocity boundary element formulation for convection-diffusion problems with variable velocity fields

This paper presents a boundary element formulation for steady-state convection-diffusion problems with variable velocity fields. In this formulation, the velocity field is decomposed into an average, which is included into the fundamental solution of the corresponding equation with constant coefficients, and a perturbation which is treated using a Dual Reciprocity approximation. The formulation allows the numerical solution to be represented in terms of boundary values only, and produces accurate results for diffusion-dominated problems with low velocity values.

An analysis of viscous splitting and adaptivity for steady-state convection-diffusion problems

This paper analyzes a viscous splitting approach for the solution of steady-state convection-diffusion problems with constant coefficients. The main advantage of this approach is that the convective and diffusive parts of the equation can be treated separately. A Fourier analysis is provided for both the fully discrete problem in one space dimension and the semi-discrete problem in N space dimensions. Nearly optimal solutions and meshes were generated using an adaptive mesh refinement strategy for a model one-dimensional problem.