Singularly perturbed convection–diffusion problems with boundary and weak interior layers

Farrell, Paul A.; Hegarty, Alan F.; Miller, John J. H.; O’Riordan, Eugene; Шишкин, Г. И.

doi:10.1016/j.cam.2003.09.033

Cited by 84 publications

(60 citation statements)

References 3 publications

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“…In the case of boundary value problems for parabolic convectiondiffusion equations, the order of ε-uniform convergence with respect to the spatial and time variables does not exceed 1 (see, e.g., [1,4,8,13,14,17] and also their bibliographies). Thus for parabolic convection-diffusion problems it is important to construct special schemes whose ε-uniform convergence rate is greater than 1 with respect to both variables.…”

Section: Introductionmentioning

confidence: 99%

The Richardson extrapolation technique for quasilinear parabolic singularly perturbed convection-diffusion equations

Shishkin

Shishkina

2006

J. Phys.: Conf. Ser.

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Abstract. The Dirichlet problem is considered for a quasilinear singularly perturbed parabolic convection-diffusion equation on a rectangular domain. For this problem classical finite difference (nonlinear) schemes on piecewise uniform meshes condensing in the boundary layer converge ε-uniformly at a rate that is at best first-order. Using a Richardson extrapolation technique, we construct an improved (nonlinear) scheme that is ε-uniformly convergent at the, where N and N0 define the number of nodes in the spatial and time meshes, respectively. This nonlinear scheme is used in the construction of a linearized scheme where at each time level the nonlinear term is evaluated using the computed solution from the previous time level. Furthermore, using the linearized and nonlinear improved schemes, we construct a linearized improved Richardson scheme that converges ε-uniformly at the rate O((N −1 ln N ) 2 + N −q 0 ), where q ≥ 2.

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Section: Introductionmentioning

confidence: 99%

The Richardson extrapolation technique for quasilinear parabolic singularly perturbed convection-diffusion equations

Shishkin

Shishkina

2006

J. Phys.: Conf. Ser.

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“…Some authors (Valanarasu & Ramanujam, 2007;Miller, O'Riordan & Wang, 2000;Roos & Zarin, 2002;Farrel, Hegarty, Miller, O'Riordan & Shishkin, 2004) have considered the singularly perturbed problems with discontinuous source terms, which lead to a weak interior layer in the singular solution. It is interesting that the present weak-form integral equation method can handle this type problem without inducing any difficulty because the influence of the source term f (x) on the solution is given through the integral term…”

Section: A Weak Form Integral Equation Methodsmentioning

confidence: 99%

Solving Third-Order Singularly Perturbed Problems: Exponentially and Polynomially Fitted Trial Functions

Liu¹

2016

JMR

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For the third-order linearly singularly perturbed problems under four different types boundary conditions, we develop a weak-form integral equation method (WFIEM) to find the singular solution. In the WFIEM the exponentially and polynomially fitted trial functions are designed to satisfy the boundary conditions automatically, while the test functions satisfy the adjoint boundary conditions exactly. The WFIEM provides accurate and stable solutions to the highly singular third-order problems.

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“…Our goal is to construct an ε uniform numerical method for solving this problem, that is a numerical method which generates ε uniformly convergent numerical approximations to the solution and its derivatives. Note that problems with discontinuous data were treated theoretically, in the case of the solution of the convection diffusion with Dirichlet case problem [3,4]. In [5][6][7][8] the authors discussed a self-adjoint Dirichlet type problem with discontinuous source term.…”

Section: Introductionmentioning

confidence: 99%

“…Motivated by the works of [3][4][5][6] we, in the present paper, develop a computational method to solve SPBVPs for second order equations of the type:…”

Section: Introductionmentioning

confidence: 99%

Fitted Mesh Method for Singularly Perturbed Robin Type Boundary Value Problem with Discontinuous Source Term

Chandru

Shanthi

2015

Int. J. Appl. Comput. Math

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In this paper, second order singularly perturbed convection-diffusion Robin type problem with a discontinuous source term is considered. Due to the discontinuity interior layers appears in the solution. A numerical method is constructed for this problem which involves an appropriate piecewise-uniform mesh for the boundary and interior layers. The method is shown to be parameter uniformly convergent with respect to the singular perturbation parameter. Numerical examples are presented to illustrate the theoretical results.

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Singularly perturbed convection–diffusion problems with boundary and weak interior layers

Cited by 84 publications

References 3 publications

The Richardson extrapolation technique for quasilinear parabolic singularly perturbed convection-diffusion equations

The Richardson extrapolation technique for quasilinear parabolic singularly perturbed convection-diffusion equations

Solving Third-Order Singularly Perturbed Problems: Exponentially and Polynomially Fitted Trial Functions

Fitted Mesh Method for Singularly Perturbed Robin Type Boundary Value Problem with Discontinuous Source Term

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