Journal ArticleDOI
Approximation of the solutions of singularly perturbed boundary-value problems with a parabolic boundary layer
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In this paper, the authors considered boundary-value problems for a parabolic equation with mixed derivatives, where the coefficient of the highest-order derivatives involves a parameter varying in the half-open interval (0, 1), and showed that the attempt to use adjustive methods to construct difference schemes that are uniformly convergent (with respect to the parameter) for such systems meets certain difficulties.Abstract:
Boundary-value problems for an equation of parabolic type, in which the coefficient of the highest-order derivatives involves a parameter varying in the half-open interval (0,1], are considered. As the parameter approaches zero, parabolic boundary layers develop near the boundary of the domain. It is shown that the attempt to use adjustive methods to construct difference schemes that are uniformly convergent (with respect to the parameter) for such systems meets certain difficulties; in fact, for uniform grids there is no such adjustive scheme. A study is presented of two problems for a parabolic equation with mixed derivatives: a periodic boundary-value problem in a strip and the Dirichlet problem in a two-dimensional domain whose boundary is a smooth curve. In both cases it is possible to construct difference schemes that converge uniformly in the parameter throughout the domain.read more
Citations
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Journal ArticleDOI
Steady-state convection-diffusion problems
TL;DR: The survey begins by examining the asymptotic nature of solutions to stationary convection-diffusion problems, which provides a suitable framework for the understanding of these solutions and the difficulties that numerical techniques will face.
Journal ArticleDOI
A brief survey on numerical methods for solving singularly perturbed problems
Mohan K. Kadalbajoo,Vikas Gupta +1 more
TL;DR: This survey paper contains a surprisingly large amount of literature on singularly perturbed problems and indeed can serve as an introduction to some of the ideas and methods for the singular perturbation problems.
Journal ArticleDOI
A parameter-robust finite difference method for singularly perturbed delay parabolic partial differential equations
TL;DR: In this article, a Dirichlet boundary value problem for a delay parabolic differential equation is studied on a rectangular domain in the x-t plane, where the second-order space derivative is multiplied by a small singular perturbation parameter, which gives rise to parabolic boundary layers on the two lateral sides of the rectangle.
Fitted mesh methods for problems with parabolic boundary layers
TL;DR: In this article, a Dirichlet boundary value problem for a linear parabolic dierential equation is studied on a rectangular domain in the x t plane, where the coecient of the second order space derivative is a small singular perturbation parameter, which gives rise to parabolic boundary layers on the two lateral sides of the rectangle.
Shishkin meshes in the numerical solution of singularly perturbed differential equations
Natalia Kopteva,Eugene O'Riordan +1 more
TL;DR: In this paper, the authors reviewed some of the salient features of the piecewise-uniform Shishkin mesh and the central analytical techniques involved in the associated numerical analysis are explained via a particular class of singularly perturbed differential equations.
References
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Book
Linear and Quasilinear Equations of Parabolic Type
TL;DR: In this article, the authors considered a hyperbolic parabolic singular perturbation problem for a quasilinear equation of kirchhoff type and obtained parameter dependent time decay estimates of the difference between the solutions of the solution of a quasi-linear parabolic equation and the corresponding linear parabolic equations.
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