A parameter robust computational method for a weakly coupled system of singularly perturbed convection-diffusion boundary value problem with discontinuous source terms

2021 ◽  
Vol 14 (1) ◽  
pp. 54
Author(s):  
Pathan Mahabub Basha ◽  
Vembu Shanthi
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value ◽  
Coupled System ◽  
Computational Method ◽  
Singularly Perturbed ◽  
Source Terms ◽  
Diffusion Boundary ◽  
Convection Diffusion ◽  
Weakly Coupled ◽  
Weakly Coupled System

SHOOTING METHOD FOR SINGULARLY PERTURBED FOURTH-ORDER ORDINARY DIFFERENTIAL EQUATIONS OF REACTION-DIFFUSION TYPE

2013 ◽  
Vol 10 (06) ◽  
pp. 1350041 ◽  
Author(s):  
J. CHRISTY ROJA ◽  
A. TAMILSELVAN
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value ◽  
Outer Region ◽  
Coupled System ◽  
Shishkin Mesh ◽  
Singularly Perturbed ◽  
Initial Value ◽  
Weakly Coupled ◽  
Weakly Coupled System ◽  
The Given

A class of singularly perturbed boundary value problems (SPBVPs) for fourth-order ordinary differential equations (ODEs) is considered. The SPBVP is reduced into a weakly coupled system of two ODEs subject to suitable initial and boundary conditions. In order to solve them numerically, a method is suggested in which the given interval is divided into two inner regions (boundary layer regions) and one outer region. Two initial-value problems associated with inner regions and one boundary value problem corresponding to the outer region are derived from the given SPBVP. In each of the two inner regions, an initial value problem is solved by using fitted mesh finite difference (FMFD) scheme on Shishkin mesh and the boundary value problem corresponding to the outer region is solved by using classical finite difference (CFD) scheme on Shishkin mesh. A combination of the solution so obtained yields a numerical solution of the boundary value problem on the whole interval. First, in this method, we find the zeroth-order asymptotic expansion approximation of the solution of the weakly coupled system. Error estimates are derived. Examples are presented to illustrate the numerical method. This method is suitable for parallel computing.

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

2010 ◽  
Vol 07 (02) ◽  
pp. 261-277 ◽  
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.

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

2020 ◽  
Vol 2020 ◽  
pp. 1-8 ◽  
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.

A Robust Computational Method for System of Singularly Perturbed Differential Difference Equations with Discontinuous Convection Coefficients

2016 ◽  
Vol 13 (04) ◽  
pp. 1641008 ◽  
Author(s):  
V. Subburayan
Keyword(s):  
Difference Equations ◽  
Piecewise Linear ◽  
Coupled System ◽  
Linear Interpolation ◽  
Computational Method ◽  
Singularly Perturbed ◽  
Supremum Norm ◽  
Piecewise Linear Interpolation ◽  
Weakly Coupled ◽  
Weakly Coupled System

In this paper, a standard numerical method with piecewise linear interpolation on Shishkin mesh is suggested to solve a weakly coupled system of singularly perturbed boundary value problem for second-order ordinary differential difference equations with discontinuous convection coefficients and source terms. An error estimate is derived by using the supremum norm and it is of almost first-order convergence. Numerical results are provided to illustrate the theoretical results.

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