scholarly journals Uniformly Convergent Hybrid Numerical Method for Singularly Perturbed Delay Convection-Diffusion Problems

2021 ◽  
Vol 2021 ◽  
pp. 1-20
Author(s):  
Mesfin Mekuria Woldaregay ◽  
Gemechis File Duressa
Keyword(s):  
Boundary Layer ◽  
Hybrid Method ◽  
Singularly Perturbed ◽  
Nonoscillatory Solutions ◽  
Convection Diffusion ◽  
Layer Region ◽  
The Stability ◽  
Regular Region ◽  
The Right ◽  
Diffusion Problems

This paper deals with numerical treatment of nonstationary singularly perturbed delay convection-diffusion problems. The solution of the considered problem exhibits boundary layer on the right side of the spatial domain. To approximate the term with the delay, Taylor’s series approximation is used. The resulting time-dependent singularly perturbed convection-diffusion problems are solved using Crank-Nicolson method for temporal discretization and hybrid method for spatial discretization. The hybrid method is designed using mid-point upwind in regular region with central finite difference in boundary layer region on piecewise uniform Shishkin mesh. Numerical examples are used to validate the theoretical findings and analysis of the proposed scheme. The present method gives accurate and nonoscillatory solutions in regular and boundary layer regions of the solution domain. The stability and the uniform convergence of the scheme are proved. The scheme converges uniformly with almost second-order rate of convergence.

Parameter free hybrid numerical method for solving modified Burgers’ equations on a nonuniform mesh

2016 ◽  
Vol 10 (02) ◽  
pp. 1750029
Author(s):  
Mohan K. Kadalbajoo ◽  
Ashish Awasthi
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Numerical Method ◽  
Point Of View ◽  
Hybrid Scheme ◽  
Nonuniform Mesh ◽  
Central Difference ◽  
Layer Region ◽  
Regular Region ◽  
The Right

In this paper, the modified Burgers’ equation is considered. These kinds of problems come from the field of sonic boom and explosions theories. At big Reynolds’ number there is a boundary layer in the right side of the domain. From numerical point of view, the major difficulty in dealing with this type of problem is that the smooth initial data can give rise to solution varying regions i.e. boundary layer regions. To tackle this situation, we propose a numerical method on nonuniform mesh of Shishkin type, which works well at high as well as low Reynolds number. The proposed method comprises of Euler implicit scheme and hybrid scheme in time and space direction, respectively. First, we discretize the continuous problem in temporal direction by Euler implicit method, which yields a set of ode’s at each time level. The resulting set of differential equations are approximated by a hybrid scheme on Shishkin mesh i.e. upwind in regular region (nonboundary layer region) and central difference in boundary layer regions. The convergence of proposed method has been shown parameter uniform. Some numerical experiments have been carried out to corroborate the theoretical results.

scholarly journals Novel Numerical Scheme for Singularly Perturbed Time Delay Convection-Diffusion Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-13
Author(s):  
Mesfin Mekuria Woldaregay ◽  
Worku Tilahun Aniley ◽  
Gemechis File Duressa
Keyword(s):  
Time Delay ◽  
Numerical Test ◽  
Perturbation Parameter ◽  
Singularly Perturbed ◽  
Parabolic Differential Equations ◽  
Convection Diffusion ◽  
Convection Diffusion Equation ◽  
Derivative Term ◽  
The Stability ◽  
The Right

This paper deals with numerical treatment of singularly perturbed parabolic differential equations having large time delay. The highest order derivative term in the equation is multiplied by a perturbation parameter ε , taking arbitrary value in the interval 0 , 1 . For small values of ε , solution of the problem exhibits an exponential boundary layer on the right side of the spatial domain. The properties and bounds of the solution and its derivatives are discussed. The considered singularly perturbed time delay problem is solved using the Crank-Nicolson method in temporal discretization and exponentially fitted operator finite difference method in spatial discretization. The stability of the scheme is investigated and analysed using comparison principle and solution bound. The uniform convergence of the scheme is discussed and proven. The formulated scheme converges uniformly with linear order of convergence. The theoretical analysis of the scheme is validated by considering numerical test examples for different values of ε .

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