A Class of Singularly Perturbed Convection-Diffusion Problems with a Moving Interior Layer. An a Posteriori Adaptive Mesh Technique

Shishkin, G. I.; Shishkina, Lidia P.; Hemker, Piet

doi:10.2478/cmam-2004-0007

Cited by 19 publications

(10 citation statements)

References 13 publications

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“…The function z 0 .10/ .x; t/, .x; t/ 2 G 0 h , is the solution of the difference scheme of the Richardson method (10), (9). Thus, setting basic grid G 1 h .9a/ , we define the solution of the problem (10), (9):…”

Section: Richardson Methods For a Classical Schemementioning

confidence: 99%

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Improved Scheme on Adapted Locally-Uniform Meshes for a Singularly Perturbed Parabolic Convection-Diffusion Problem

Шишкин

2011

Lecture Notes in Computational Science and Engineering

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For a Dirichlet problem for a singularly perturbed parabolic convectiondiffusion equation with small parameter " multiplying the highest-order derivative, a finite difference scheme with improved accuracy is constructed that converges almost "-uniformly with order of the convergence rate close to 2 for fixed values of ". When constructing the scheme, monotone classical approximations of the differential equation on a priori adapted locally-uniform meshes are used. To improve accuracy of the scheme, the Richardson technique on embedded grids is applied.

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Section: Richardson Methods For a Classical Schemementioning

confidence: 99%

“…Difference schemes in [5][6][7][8][9] on adapted locally-uniform meshes (uniform on these subsets where the more precise solution is computed) converge almost "-uniformly, but with order of the convergence rate not higher than one for fixed values of the parameter ".…”

Section: Problem Formulation: Aim Of Researchmentioning

confidence: 99%

Improved Scheme on Adapted Locally-Uniform Meshes for a Singularly Perturbed Parabolic Convection-Diffusion Problem

Шишкин

2011

Lecture Notes in Computational Science and Engineering

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“…In the case of moving interior layers, fairly complicated difference schemes has been constructed in the papers [2,8,9,10] and [11] where one example of periodic problem was considered.…”

Section: Introductionmentioning

confidence: 99%

Analytic-Numerical Approach to Solving Singularly Perturbed Parabolic Equations with the Use of Dynamic Adapted Meshes

Lukyanenko¹,

Volkov²,

Nefedov³

et al. 2016

Model. anal. inf. sist.

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Abstract. The main objective of the paper is to present a new analytic-numerical approach to singularly perturbed reaction-diffusion-advection models with solutions containing moving interior layers (fronts). We describe some methods to generate the dynamic adapted meshes for an efficient numerical solution of such problems. It is based on a priori information about the moving front properties provided by the asymptotic analysis. In particular, for the mesh construction we take into account a priori asymptotic evaluation of the location and speed of the moving front, its width and structure. Our algorithms significantly reduce the CPU time and enhance the stability of the numerical process compared with classical approaches.The article is published in the authors' wording.

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“…al. [21], on the basis of posteriori adaptive mesh parameter uniform scheme for problems having moving interior layers is constructed, for a class of singularly perturbed partial differential equation of convection diffusion type. Dunne et.…”

Section: Introductionmentioning

confidence: 99%

A Fitted Mesh Method For Partial Differential Equations With A Time Lag

Sharma¹

2017

IJRASET

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A class of initial boundary value problem for singularly perturbed parabolic partial differential equation of convection diffusion type having dominating delayed convection term is examined on a rectangular domain. When the perturbation parameter specifying the problem tends to zero, a breakdown of singular perturbation occurs in narrow intervals of space and short interval of time and the solution of the perturbed problem will often behave analytically quite differently. In these narrow regions which are usually referred as boundary regions, the solution changes rapidly and form parabolic boundary layers in the neighborhood of the outflow boundary regions. Due to the presence of the perturbation parameter and in particular time delay classical numerical method on uniform meshes are known to be inadequate for the solution of such type of problems, because the error in the numerical solution depends appreciably on the value of the perturbation parameter and is comparable with the solution itself for small value of perturbation parameter. Thus in connection with the stiff behavior it is of interest to develop special numerical methods whose errors would be independent of the perturbation parameter, i.e. parameter-uniformly convergent methods. In this paper, a numerical method consisting of standard upwind finite difference operator on a piecewise uniform mesh is constructed, in order to overcome the difficulties due to the presence of perturbation parameter and the time delay. The first step in this direction consists of discretizing the time variable with the backward Euler's method with constant time step. This produces a set of stationary singularly perturbed semidiscrete problem class which is further discretized in space using upwind finite difference operator on a piecewise uniform mesh. An extensive amount of analysis is carried out in order to establish the convergence and stability of the method proposed. The analysis in this paper uses a suitable decomposition of the error into smooth and singular component and a comparison principle combined with appropriate barrier functions. The error estimates are obtained which proves uniform convergence of the method.

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A Class of Singularly Perturbed Convection-Diffusion Problems with a Moving Interior Layer. An a Posteriori Adaptive Mesh Technique

Cited by 19 publications

References 13 publications

Improved Scheme on Adapted Locally-Uniform Meshes for a Singularly Perturbed Parabolic Convection-Diffusion Problem

Improved Scheme on Adapted Locally-Uniform Meshes for a Singularly Perturbed Parabolic Convection-Diffusion Problem

Analytic-Numerical Approach to Solving Singularly Perturbed Parabolic Equations with the Use of Dynamic Adapted Meshes

A Fitted Mesh Method For Partial Differential Equations With A Time Lag

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