UNIFORMLY-CONVERGENT NUMERICAL METHODS FOR A SYSTEM OF COUPLED SINGULARLY PERTURBED CONVECTION–DIFFUSION EQUATIONS WITH MIXED TYPE BOUNDARY CONDITIONS

In this paper, two hybrid difference schemes on the Shishkin mesh are constructed for solving a weakly coupled system of two singularly perturbed convection - diffusion second order ordinary differential equations subject to the mixed type boundary conditions. We prove that the method has almost second order convergence in the supremum norm independent of the diffusion parameter. Error bounds for the numerical solution and also the numerical derivative are established. Numerical results are provided to illustrate the theoretical results.

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AN ALMOST-SECOND-ORDER METHOD FOR A SYSTEM OF SINGULARLY PERTURBED CONVECTION–DIFFUSION EQUATIONS WITH NONSMOOTH CONVECTION COEFFICIENTS AND SOURCE TERMS

International Journal of Computational Methods ◽

10.1142/s0219876210002167 ◽

2010 ◽

Vol 07 (02) ◽

pp. 261-277 ◽

Cited By ~ 3

Author(s):

A. TAMILSELVAN ◽

N. RAMANUJAM

Keyword(s):

Coupled System ◽

Second Order ◽

Singularly Perturbed ◽

Diffusion Equations ◽

Source Terms ◽

Convection Diffusion ◽

Weakly Coupled ◽

Weakly Coupled System ◽

Type Boundary ◽

Second Order Convergence

In this paper, a weakly coupled system of two singularly perturbed convection–diffusion equations with discontinuous convection coefficients and source terms with Dirichlet type boundary conditions is considered. A hybrid finite difference scheme on a Shishkin mesh generating almost-second-order convergence in the maximum norm is constructed for solving this problem. To illustrate the theoretical results, numerical experiments are performed.

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A parameter-uniform second order numerical method for a weakly coupled system of singularly perturbed convection–diffusion equations with discontinuous convection coefficients and source terms

CALCOLO ◽

10.1007/s10092-017-0218-3 ◽

2017 ◽

Vol 54 (3) ◽

pp. 1027-1053 ◽

Cited By ~ 1

Author(s):

Mahabub Basha Pathan ◽

Shanthi Vembu

Keyword(s):

Numerical Method ◽

Coupled System ◽

Second Order ◽

Singularly Perturbed ◽

Diffusion Equations ◽

Source Terms ◽

Convection Diffusion ◽

Weakly Coupled ◽

Weakly Coupled System

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A HYBRID DIFFERENCE SCHEME FOR SINGULARLY PERTURBED SECOND ORDER ORDINARY DIFFERENTIAL EQUATIONS WITH DISCONTINUOUS CONVECTION COEFFICIENT AND MIXED TYPE BOUNDARY CONDITIONS

International Journal of Computational Methods ◽

10.1142/s0219876208001637 ◽

2008 ◽

Vol 05 (04) ◽

pp. 575-593 ◽

Cited By ~ 3

Author(s):

R. MYTHILI PRIYADHARSHINI ◽

N. RAMANUJAM

Keyword(s):

Differential Equations ◽

Boundary Conditions ◽

Difference Scheme ◽

Ordinary Differential Equations ◽

Mixed Type ◽

Second Order ◽

Singularly Perturbed ◽

Convection Coefficient ◽

Type Boundary ◽

Numerical Derivative

This paper presents, a hybrid difference scheme for singularly perturbed second order ordinary differential equations with a small parameter multiplying the highest derivative with a discontinuous convection coefficient subject to mixed type boundary conditions. Error bounds for the numerical solution and numerical derivative are established. Numerical results are provided to illustrate the theoretical results.

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A Numerical Method for Singularly Perturbed Second Order Coupled System of Convection–Diffusion Robin Type Boundary Value Problems with Discontinuous Source Term

International Journal of Applied and Computational Mathematics ◽

10.1007/s40819-014-0021-7 ◽

2014 ◽

Vol 1 (3) ◽

pp. 381-397 ◽

Cited By ~ 3

Author(s):

P. Mahabub Basha ◽

V. Shanthi

Keyword(s):

Boundary Value Problems ◽

Numerical Method ◽

Source Term ◽

Boundary Value ◽

Coupled System ◽

Second Order ◽

Singularly Perturbed ◽

Convection Diffusion ◽

Discontinuous Source Term ◽

Type Boundary

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Mixed-type boundary conditions for second order elliptic differential equations

Journal of the Mathematical Society of Japan ◽

10.2969/jmsj/02630405 ◽

1974 ◽

Vol 26 (3) ◽

pp. 405-432 ◽

Cited By ~ 7

Author(s):

Yoshio KATO

Keyword(s):

Differential Equations ◽

Boundary Conditions ◽

Mixed Type ◽

Second Order ◽

Elliptic Differential Equations ◽

Type Boundary

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High-Order Method for a Singularly Perturbed Second-Order Ordinary Differential Equation with Discontinuous Source Term Subject to Mixed Type Boundary Conditions

Mathematical Modelling and Scientific Computation - Communications in Computer and Information Science ◽

10.1007/978-3-642-28926-2_38 ◽

2012 ◽

pp. 351-359 ◽

Cited By ~ 1

Author(s):

R. Mythili Priyadharshini ◽

N. Ramanujam

Keyword(s):

Differential Equation ◽

Ordinary Differential Equation ◽

Boundary Conditions ◽

Mixed Type ◽

Source Term ◽

Singularly Perturbed ◽

Order Method ◽

High Order Method ◽

Discontinuous Source Term ◽

Type Boundary

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An Asymptotic-Numerical Method for a Class of Weakly Coupled System of Singularly Perturbed Convection-Diffusion Equations

Numerical Functional Analysis and Optimization ◽

10.1080/01630563.2019.1615946 ◽

2019 ◽

Vol 40 (13) ◽

pp. 1550-1571

Author(s):

Aditya Kaushik ◽

Anil K. Vashishth ◽

Vijayant Kumar ◽

Manju Sharma

Keyword(s):

Numerical Method ◽

Coupled System ◽

Singularly Perturbed ◽

Diffusion Equations ◽

Convection Diffusion ◽

Asymptotic Numerical Method ◽

Weakly Coupled ◽

Weakly Coupled System

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An analysis of overlapping Schwarz method for a weakly coupled system of singularly perturbed convection‐diffusion equations

International Journal for Numerical Methods in Fluids ◽

10.1002/fld.4794 ◽

2019 ◽

Vol 92 (6) ◽

pp. 528-544

Author(s):

J. Christy Roja ◽

A. Tamilselvan ◽

N. Geetha

Keyword(s):

Coupled System ◽

Singularly Perturbed ◽

Diffusion Equations ◽

Convection Diffusion ◽

Schwarz Method ◽

Weakly Coupled ◽

Weakly Coupled System ◽

Overlapping Schwarz

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Solving Systems of Singularly Perturbed Convection Diffusion Problems via Initial Value Method

Journal of Applied Mathematics ◽

10.1155/2020/1062025 ◽

2020 ◽

Vol 2020 ◽

pp. 1-8 ◽

Cited By ~ 1

Author(s):

Wondwosen Gebeyaw Melesse ◽

Awoke Andargie Tiruneh ◽

Getachew Adamu Derese

Keyword(s):

Approximate Solution ◽

Coupled System ◽

Perturbation Parameter ◽

Singularly Perturbed ◽

Initial Value ◽

Convection Diffusion ◽

Weakly Coupled ◽

Weakly Coupled System ◽

The Given ◽

Diffusion Problems

In this paper, an initial value method for solving a weakly coupled system of two second-order singularly perturbed Convection–diffusion problems exhibiting a boundary layer at one end is proposed. In this approach, the approximate solution for the given problem is obtained by solving, a coupled system of initial value problem (namely, the reduced system), and two decoupled initial value problems (namely, the layer correction problems), which are easily deduced from the given system of equations. Both the reduced system and the layer correction problems are independent of perturbation parameter, ε. These problems are then solved analytically and/or numerically, and those solutions are combined to give an approximate solution to the problem. Further, error estimates are derived and examples are provided to illustrate the method.

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