-uniform schemes with high-order time-accuracy for parabolic singular perturbation problems

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

doi:10.1093/imanum/20.1.99

Cited by 85 publications

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“…Therefore it is of interest to develop methods for which the order of convergence with respect to the time variable is increased. For equations without convective terms the improvement of the accuracy in time, preserving E·uniform convergence, by means of a defect-correction technique was also studied in [ 4,5). In this paper we develop schemes for which the order of *cwr,…”

Section: Numerical Analysismentioning

confidence: 99%

“…In [1][2][3][4][5] we introduced and analysed E-uniformly convergent difference schemes for singularly perturbed boundary value problems for elliptic and parabolic equations.…”

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confidence: 99%

“…The highest derivative in the equation is mul· tiplied by an arbitrarily small parameter E. When the parameter E tends to zero, boundary layers may appear, which leads to difficulties when classical discretization methods are applied, because the error in the approximate solution depends on the value of E. The appropriate location of the nodes is needed to ensure that the er· ror is independent of the parameter value and depends only on the number of nodes in the mesh. Special schemes with this property are called E-uniformly convergent.In [1][2][3][4][5] we introduced and analysed E-uniformly convergent difference schemes for singularly perturbed boundary value problems for elliptic and parabolic equations.If the problem data is sufficiently smooth, for the parabolic equations with convec· tion terms, the order of E-uniform convergence for the scheme studied is exactly one and up to a small logarithmic factor one with respect to the time and space variables, respectively, i.e. O(N-1 In 2 N +K-1 ), where N and K are the number of intervals in the space and time discretization.…”

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High-order time-accuracy schemes for parabolic singular perturbation problems with convection

Hemker

Шишкин

Shishkina

2002

Russian Journal of Numerical Analysis and Mathematical Modelling

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Aims and scopeThe journal provides English translations of selected new, original Russian papers on the theoretical aspects of numerical analysis and the application of mathematical methods to simulation and modelling. The editorial board, consisting of the most prominent Russian scientists in numerical analysis and mathematical modelling, select papers on the basis of their high scientific standard, innovative approach and topical interest.Papers are published in the following areas: Numerical analysis• Abstract -The first boundary value problem for a singularly perturbed parabolic PDE with convection is considered on un interval. For the case of sufficiently smooth data, it is easy to construct a standard finite difference operator and a piecewise uniform mesh condensing in the boundary layer, which gives an e-uniformly convergent difference scheme. The order of convergence for such a scheme is exactly one and close to one up to a small logarithmic factor with respect to the time and space variables, respectively. In this paper we construct high-order time-accurate E-uniformly convergent schemes by a defect-correction technique. The efficiency of the new defect-correction scheme is confirmed by numerical experiments.We consider the first boundary value problem for a singularly perturbed parabolic PDE with convection on an interval. The highest derivative in the equation is mul· tiplied by an arbitrarily small parameter E. When the parameter E tends to zero, boundary layers may appear, which leads to difficulties when classical discretization methods are applied, because the error in the approximate solution depends on the value of E. The appropriate location of the nodes is needed to ensure that the er· ror is independent of the parameter value and depends only on the number of nodes in the mesh. Special schemes with this property are called E-uniformly convergent.In [1][2][3][4][5] we introduced and analysed E-uniformly convergent difference schemes for singularly perturbed boundary value problems for elliptic and parabolic equations.If the problem data is sufficiently smooth, for the parabolic equations with convec· tion terms, the order of E-uniform convergence for the scheme studied is exactly one and up to a small logarithmic factor one with respect to the time and space variables, respectively, i.e. O(N-1 In 2 N +K-1 ), where N and K are the number of intervals in the space and time discretization. Because the amount of the computational work is proportional to the number K, the higher-order accuracy in time can considerably reduce the computational cost. Therefore it is of interest to develop methods for which the order of convergence with respect to the time variable is increased. For equations without convective terms the improvement of the accuracy in time, preserving E·uniform convergence, by means of a defect-correction technique was also studied in [ 4,5). In this paper we develop schemes for which the order of *cwr,

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Section: Numerical Analysismentioning

confidence: 99%

“…In [1][2][3][4][5] we introduced and analysed E-uniformly convergent difference schemes for singularly perturbed boundary value problems for elliptic and parabolic equations.…”

Section: Subscription Informationmentioning

confidence: 99%

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confidence: 99%

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High-order time-accuracy schemes for parabolic singular perturbation problems with convection

Hemker

Шишкин

Shishkina

2002

Russian Journal of Numerical Analysis and Mathematical Modelling

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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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“…The solution of problem (1), (2) is such that Hemker et al (2000) where 0 ≤ k + 2m ≤ 4. Bounds (4) highlight the fact that the solution of (1), (2) has a boundary layer at x = 0 and x = 1.…”

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A fitted numerical method for singularly perturbed parabolic reaction-diffusion problems

Munyakazi

Patidar

2013

Comp. Appl. Math.

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This paper treats a time-dependent singularly perturbed reaction-diffusion problem. We semidiscretize the problem in time by means of the classical backward Euler method. We develop a fitted operator finite difference method (FOFDM) to solve the resulting set of linear problems (one at each time level). We prove that the underlying fitted operator satisfies the maximum principle. This result is then used in the error analysis of the FOFDM. The method is shown to be first order convergent in time and second order convergent in space, uniformly with respect to the perturbation parameter. We test the method on several numerical examples to confirm our theoretical findings. IntroductionWhen solving numerically time-dependent singularly perturbed problems, it is customary to consider dimension splitting: The problems are semidiscretized in time (for example by using the classical backward Euler or Crank-Nicolson scheme). Then, at each time level, a set of stationary problems is solved using a suitably designed numerical method. Singularly perturbed problems involve a (perturbation) parameter which multiplies the highest derivative term in the model-equation of the problem. The solution to such problems is characterized by layer regions which are narrow parts of the domain over which the solution undergoes abrupt changes. It is well known that classical methods are not appropriate when the perturbation parameter becomes small unless very fine meshes are used for spatial discretization. However, this approach has two side effects: it increases the round-off error and the computational cost. There is a vast literature about non-classical numerical methods. In the context of finite differences, we can group these methods into two classes: the class of fitted mesh methods and the class of fitted operator methods. Both these types of methods have been used to solve stationary singularly perturbed problems in one and several dimensions. As examples, see Linß and Stynes (1999), Lubuma andPatidar (2006), Miller et al. (1996), Munyakazi and Patidar (2010a,b, 2012), Patidar (2005, 2007, Roos et al. (1996), Shishkin (1986. It should be noted that the discovery/development of fitted mesh methods is anterior to that of the fitted operator ones. The analysis of the latter is simpler due to the fact that they are based on uniform meshes unlike the former where non-uniform meshes are designed.

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-uniform schemes with high-order time-accuracy for parabolic singular perturbation problems

Cited by 85 publications

References 0 publications

High-order time-accuracy schemes for parabolic singular perturbation problems with convection

High-order time-accuracy schemes for parabolic singular perturbation problems with convection

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

A fitted numerical method for singularly perturbed parabolic reaction-diffusion problems

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