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Volume 1, Issue 2
Finite-Difference Methods for a Class of Strongly Nonlinear Singular Perturbation Problems

Relja Vulanović

Numer. Math. Theor. Meth. Appl., 1 (2008), pp. 235-244.

Published online: 2008-01

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  • Abstract

The paper is concerned with strongly nonlinear singularly perturbed boundary value problems in one dimension. The problems are solved numerically by finite-difference schemes on special meshes which are dense in the boundary layers. The Bakhvalov mesh and a special piecewise equidistant mesh are analyzed. For the central scheme, error estimates are derived in a discrete $L^1$ norm. They are of second order and decrease together with the perturbation parameter ε. The fourth-order Numerov scheme and the Shishkin mesh are also tested numerically. Numerical results show ε-uniform pointwise convergence on the Bakhvalov and Shishkin meshes.

  • Keywords

Boundary-value problem, singular perturbation, finite differences, Bakhvalov and piecewise equidistant meshes, $L^1$ stability.

  • AMS Subject Headings

65L10, 65L12

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{NMTMA-1-235, author = {Vulanović , Relja}, title = {Finite-Difference Methods for a Class of Strongly Nonlinear Singular Perturbation Problems}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2008}, volume = {1}, number = {2}, pages = {235--244}, abstract = {

The paper is concerned with strongly nonlinear singularly perturbed boundary value problems in one dimension. The problems are solved numerically by finite-difference schemes on special meshes which are dense in the boundary layers. The Bakhvalov mesh and a special piecewise equidistant mesh are analyzed. For the central scheme, error estimates are derived in a discrete $L^1$ norm. They are of second order and decrease together with the perturbation parameter ε. The fourth-order Numerov scheme and the Shishkin mesh are also tested numerically. Numerical results show ε-uniform pointwise convergence on the Bakhvalov and Shishkin meshes.

}, issn = {2079-7338}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/nmtma/6050.html} }
TY - JOUR T1 - Finite-Difference Methods for a Class of Strongly Nonlinear Singular Perturbation Problems AU - Vulanović , Relja JO - Numerical Mathematics: Theory, Methods and Applications VL - 2 SP - 235 EP - 244 PY - 2008 DA - 2008/01 SN - 1 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/nmtma/6050.html KW - Boundary-value problem, singular perturbation, finite differences, Bakhvalov and piecewise equidistant meshes, $L^1$ stability. AB -

The paper is concerned with strongly nonlinear singularly perturbed boundary value problems in one dimension. The problems are solved numerically by finite-difference schemes on special meshes which are dense in the boundary layers. The Bakhvalov mesh and a special piecewise equidistant mesh are analyzed. For the central scheme, error estimates are derived in a discrete $L^1$ norm. They are of second order and decrease together with the perturbation parameter ε. The fourth-order Numerov scheme and the Shishkin mesh are also tested numerically. Numerical results show ε-uniform pointwise convergence on the Bakhvalov and Shishkin meshes.

Vulanović , Relja. (2008). Finite-Difference Methods for a Class of Strongly Nonlinear Singular Perturbation Problems. Numerical Mathematics: Theory, Methods and Applications. 1 (2). 235-244. doi:
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