Interior Layers in Singularly Perturbed Problems

2016 ◽  
pp. 25-40 ◽  
Author(s):  
Eugene O’Riordan
Keyword(s):  
Singularly Perturbed ◽  
Singularly Perturbed Problems ◽  
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.

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

A Parameter Robust Finite Element Method for Fourth Order Singularly Perturbed Problems

2017 ◽  
Vol 17 (2) ◽  
pp. 337-349 ◽  
Author(s):  
Christos Xenophontos
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Fourth Order ◽  
Hermite Polynomials ◽  
Perturbation Parameter ◽  
Singularly Perturbed ◽  
The Finite Element Method ◽  
Singularly Perturbed Problems ◽  
Exponentially Graded ◽  
Element Method

AbstractWe consider fourth order singularly perturbed problems in one-dimension and the approximation of their solution by the h version of the finite element method. In particular, we use piecewise Hermite polynomials of degree ${p\geq 3}$ defined on an exponentially graded mesh. We show that the method converges uniformly, with respect to the singular perturbation parameter, at the optimal rate when the error is measured in both the energy norm and a stronger, ‘balanced’ norm. Finally, we illustrate our theoretical findings through numerical computations, including a comparison with another scheme from the literature.

scholarly journals Robust numerical solutions of two singularly perturbed problems in mathematical biology

2015 ◽  
Vol 1 (2) ◽  
Author(s):  
Shivaranjani Nagarajan ◽  
John J. H. Miller ◽  
Valarmathi Sigamani
Keyword(s):  
Mathematical Biology ◽  
Numerical Solutions ◽  
Singularly Perturbed ◽  
Singularly Perturbed Problems
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