The Use of Defect Correction for the Solution of Parabolic Singular Perturbation Problems

Hemker, Piet; Шишкин, Г. И.; Shishkina, L. P.

doi:10.1002/zamm.19970770111

Cited by 26 publications

(29 citation statements)

References 6 publications

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“…The same methods are used to improve the ε-uniform rates of convergence of computed solutions for linear singularly perturbed problems (see, e.g., [5,6,7,16,20]). Recently, using Richardson extrapolation, ε-uniformly convergent finite difference schemes with improved accuracy were constructed also for quasilinear singularly perturbed reaction-diffusion parabolic [21] and elliptic [19] problems.…”

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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“…The location of the nodes is determined properly from the a priori estimates of the solution and its derivatives. The way to construct the mesh for problem (2.1) is the same as in [4,5,15]. More specifically, we take…”

Section: The ε-Uniformly Convergent Schemementioning

confidence: 99%

“…The technique used in this paper to improve accuracy is similar to that from [4][5][6]. For the difference scheme (3.2), (4.1) the error in the approximation of the partial derivative (∂/∂t) u(x, t) is caused by the divided difference δ t z(x, t) and is associated with the truncation error given by…”

Section: The ε-Uniformly Convergent Schemementioning

confidence: 99%

“…Defect correction techniques proved to be efficient for constructing ε-uniformly convergent schemes of high-order accuracy with respect to t in the case of singularly perturbed reaction-diffusion and convection-diffusion problems (see, for example, [4][5][6]. Therefore, this technique seems attractive because of its possible use in constructing high-order accurate schemes in x and t for singularly perturbed problems under consideration.…”

Section: Introductionmentioning

confidence: 99%

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Novel Defect-correction High-order, in Space and Time, Accurate Schemes for Parabolic Singularly Perturbed Convection-diffusion Problems

Hemker

Шишкин

Shishkina

2003

Computational Methods in Applied Mathematics

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-New high-order accurate finite difference schemes based on defect correction are considered for an initial boundary-value problem on an interval for singularly perturbed parabolic PDEs with convection; the highest space derivative in the equation is multiplied by the perturbation parameter ε, ε ∈ (0, 1]. Solutions of the well-known classical numerical schemes for such problems do not converge ε-uniformly (the errors of such schemes depend on the value of the parameter ε and are comparable with the solution itself for small values of ε). The convergence order of the existing ε-uniformly convergent schemes does not exceed 1 in space and time. In this paper, using a defect correction technique, we construct a special difference scheme that converges ε-uniformly with the second (up to a logarithmic factor) order of accuracy with respect to x and with the second order of accuracy and higher with respect to t. The conditions are given which ensure the ε-uniform convergence of the defect-correction schemes with a rate of 2,3, where N + 1 and K + 1 denote the number of the mesh points in x and t, respectively. Theoretical results and the efficiency of the newly constructed schemes are confirmed by numerical experiments.2000 Mathematics Subject Classification: 65M06; 65M12; 65M15.

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“…Wang [24] present a convergence analysis, for the exponentially fitted finite volume method in two dimension applied to a class of singularly perturbed convection diffusion equation with exponential boundary layers, which is then shown to be independent of the perturbation parameter.Another attempt is due to [25], in this paper singularly perturbed convection diffusion problem with a concentrated source and a discontinuous convection field is considered, using two upwind difference schemes on general meshes the epsilon uniform convergence of scheme is proved in the maximum norm on Shishkin and Bakhvalov meshes. A method of defect correction technique is also used by several authors for constructing the parameter uniform methods of higher order accuracy for convection diffusion problems [26,27,28], Hemker [29] presents an -uniform convergent scheme of higher order accuracy in time and space, based on the defect correction principle, in the case of boundary value problems for singularly perturbed parabolic convection diffusion equations. The fact that singular perturbation problems involves partial differential equation is of crucial importance as many additional technical issues arise with partial differential equations, such as the smoothness of the solution, the compatibility of the data, and the geometry of the domain.…”

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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The Use of Defect Correction for the Solution of Parabolic Singular Perturbation Problems

Cited by 26 publications

References 6 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

Novel Defect-correction High-order, in Space and Time, Accurate Schemes for Parabolic Singularly Perturbed Convection-diffusion Problems

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

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