Singularly Perturbed Problems with Boundary and Interior Layers: Theory and Application

2007 ◽  
pp. 47-179 ◽  
Author(s):  
V. F. Butuzov ◽  
A. B. Vasilieva
Keyword(s):  
Singularly Perturbed ◽  
Singularly Perturbed Problems ◽  
Interior Layers ◽  
Boundary And Interior Layers

scholarly journals PARAMETER-UNIFORM IMPROVED HYBRID NUMERICAL SCHEME FOR SINGULARLY PERTURBED PROBLEMS WITH INTERIOR LAYERS

2018 ◽  
Vol 23 (2) ◽  
pp. 167-189 ◽  
Author(s):  
Kaushik Mukherjee
Keyword(s):  
Numerical Scheme ◽  
Perturbation Parameter ◽  
Singularly Perturbed ◽  
Supremum Norm ◽  
Hybrid Scheme ◽  
Singularly Perturbed Problems ◽  
Central Difference ◽  
Convection Diffusion ◽  
Interior Layers ◽  
Wave Phenomena

In this paper, we consider a class of singularly perturbed convection-diffusion boundary-value problems with discontinuous convection coefficient which often occur as mathematical models for analyzing shock wave phenomena in gas dynamics. In general, interior layers appear in the solutions of this class of problems and this gives rise to difficulty while solving such problems using the classical numerical methods (standard central difference or standard upwind scheme) on uniform meshes when the perturbation parameter ε is small. To achieve better numerical approximation in solving this class of problems, we propose a new hybrid scheme utilizing a layer-resolving piecewise-uniform Shishkin mesh and the method is shown to be ε-uniformly stable. In addition to this, it is proved that the proposed numerical scheme is almost second-order uniformly convergent in the discrete supremum norm with respect to the parameter ε. Finally, extensive numerical experiments are conducted to support the theoretical results. Further, the numerical results obtained by the newly proposed scheme are also compared with the hybrid scheme developed in the paper [Z.Cen, Appl. Math. Comput., 169(1): 689-699, 2005]. It shows that the current hybrid scheme exhibits a significant improvement over the hybrid scheme developed by Cen, in terms of the parameter-uniform order of convergence.

scholarly journals Numerical solutions of a system of singularly perturbed reaction-diffusion problems

Filomat ◽  
2019 ◽  
Vol 33 (15) ◽  
pp. 4889-4905
Author(s):  
Ali Barati ◽  
Ali Atabaigi
Keyword(s):  
Numerical Solutions ◽  
Reaction Diffusion ◽  
Coupled System ◽  
Approximate Solutions ◽  
Perturbation Parameter ◽  
Reaction Diffusion Equations ◽  
Singularly Perturbed ◽  
Interior Layers ◽  
Boundary And Interior Layers ◽  
Diffusion Problems

This paper addresses the numerical approximation of solutions to a coupled system of singularly perturbed reaction-diffusion equations. The components of the solution exhibit overlapping boundary and interior layers. Sinc procedure can control the oscillations in computed solutions at boundary layer regions naturally because the distribution of Sinc points is denser at near the boundaries. Also the obtained results show that the proposed method is applicable even for small perturbation parameter as ? = 2-30. The convergence analysis of proposed technique is discussed, it is shown that the approximate solutions converge to the exact solutions at an exponential rate. Numerical experiments are carried out to demonstrate the accuracy and efficiency of the method.

Regularization of Integral Operators in Singularly Perturbed Problems Using Normal Forms

Vestnik MEI ◽  
2019 ◽  
Vol 6 ◽  
pp. 131-137
Author(s):  
Abdukhafiz A. Bobodzhanova ◽  
◽  
Valeriy F. Safonov ◽  
Keyword(s):  
Normal Forms ◽  
Integral Operators ◽  
Singularly Perturbed ◽  
Singularly Perturbed Problems
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