Numerical solutions of singularly perturbed convection-diffusion problems

2014 ◽  
Vol 24 (6) ◽  
pp. 1268-1274 ◽  
Author(s):  
Fazhan Geng ◽  
Suping Qian ◽  
Shuai Li
Keyword(s):  
Asymptotic Expansion ◽  
Present Method ◽  
Numerical Solutions ◽  
Variational Iteration Method ◽  
Expansion Method ◽  
Singularly Perturbed ◽  
Content Type ◽  
Convection Diffusion ◽  
Asymptotic Expansion Method ◽  
Diffusion Problems

Purpose – The purpose of this paper is to find an effective numerical method for solving singularly perturbed convection-diffusion problems. Design/methodology/approach – The present method is based on the asymptotic expansion method and the variational iteration method (VIM). First a zeroth order asymptotic expansion for the solution of the given singularly perturbed convection-diffusion problem is constructed. Then the reduced terminal value problem is solved by using the VIM. Findings – Two numerical examples are introduced to show the validity of the present method. Obtained numerical results show that the present method can provide very accurate analytical approximate solutions not only in the boundary layer, but also away from the layer. Originality/value – The combination of the asymptotic expansion method and the VIM is applied to singularly perturbed convection-diffusion problems.

scholarly journals A Special Multigrid Strategy on Non-Uniform Grids for Solving 3D Convection–Diffusion Problems with Boundary/Interior Layers

Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1123
Author(s):  
Tianlong Ma ◽  
Lin Zhang ◽  
Fujun Cao ◽  
Yongbin Ge
Keyword(s):  
Present Method ◽  
Multigrid Method ◽  
Numerical Solutions ◽  
Three Dimensional ◽  
Finite Difference Schemes ◽  
Interior Layer ◽  
Convection Diffusion ◽  
Uniform Grids ◽  
Interior Layers ◽  
Diffusion Problems

Boundary or interior layer problems of high-dimensional convection–diffusion equations have distinct asymmetry. Consequently, computational grid distributions and linear algebraic systems arising from finite difference schemes for them are also asymmetric. Numerical solutions for these kinds of problems are more complicated than those symmetric problems. In this paper, we extended our previous work on the partial semi-coarsening multigrid method combined with the high-order compact (HOC) difference scheme for solving the two-dimensional (2D) convection–diffusion problems on non-uniform grids to the three-dimensional (3D) cases. The main merit of the present method is that the multigrid method on non-uniform grids can be performed with a different number of grids in different coordinate axes, which is more efficient than the multigrid method on non-uniform grids with the same number of grids in different coordinate axes. Numerical experiments are carried out to validate the accuracy and efficiency of the present method. It is shown that, without losing the high precision, the present method is very effective to reduce computing cost by cutting down the number of grids in the direction(s) which does/do not contain boundary or interior layer(s).

Numerical modelling of convection-diffusion problems with first-order chemical reaction using the dual reciprocity boundary element method

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Salam Adel Al-Bayati ◽  
Luiz C. Wrobel
Keyword(s):  
Steady State ◽  
Boundary Element Method ◽  
Boundary Element ◽  
Content Type ◽  
Convection Diffusion ◽  
First Order ◽  
Order Chemical Reaction ◽  
First Order Chemical Reaction ◽  
Diffusion Problems ◽  
Element Method

Purpose The purpose of this paper is to describe an extension of the boundary element method (BEM) and the dual reciprocity boundary element method (DRBEM) formulations developed for one- and two-dimensional steady-state problems, to analyse transient convection–diffusion problems associated with first-order chemical reaction. Design/methodology/approach The mathematical modelling has used a dual reciprocity approximation to transform the domain integrals arising in the transient equation into equivalent boundary integrals. The integral representation formula for the corresponding problem is obtained from the Green’s second identity, using the fundamental solution of the corresponding steady-state equation with constant coefficients. The finite difference method is used to simulate the time evolution procedure for solving the resulting system of equations. Three different radial basis functions have been successfully implemented to increase the accuracy of the solution and improving the rate of convergence. Findings The numerical results obtained demonstrate the excellent agreement with the analytical solutions to establish the validity of the proposed approach and to confirm its efficiency. Originality/value Finally, the proposed BEM and DRBEM numerical solutions have not displayed any artificial diffusion, oscillatory behaviour or damping of the wave front, as appears in other different numerical methods.

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