A higher order richardson scheme for a singularly perturbed semilinear elliptic convection-diffusion equation

Шишкин, Г. И.; Shishkina, L. P.

doi:10.1134/s0965542510030061

Cited by 8 publications

(10 citation statements)

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“…Let us present here a priori estimates of the solutions and derivatives for boundary value problem (2.2), (2.1); the derivation of the estimates is similar to those of [21,22]. where U(x) and V (x) are the regular and singular parts of the solution, respectively.…”

Section: A Priori Estimates Of Solutions and Derivativesmentioning

confidence: 99%

“…Other approaches to the construction of schemes of a higher order of accuracy different from Richardson's technique were considered in [19,21]. For singularly perturbed semilinear equations special ε-uniformly convergent schemes of the higher order of accuracy were constructed on the base of the Richardson technique for the parabolic reaction-diffusion equations in a strip [23] and for the convection-diffusion equations in a segment [24], for elliptic reaction-diffusion equations in a strip [20] and for convection-diffusion equations in a strip [22].…”

Section: Introductionmentioning

confidence: 99%

“…In this paper we also develop difference schemes of the higher order of accuracy for the semilinear singularly perturbed elliptic equation of the convection-diffusion type in a vertical strip considered in [22]. In contrast to [22] where iterative schemes of the Richardson method of higher order of accuracy were constructed with the use of lower and upper solutions as indicators, in this paper we construct ε-uniformly convergent schemes of the Newton method.…”

Section: Introductionmentioning

confidence: 99%

“…In contrast to [22] where iterative schemes of the Richardson method of higher order of accuracy were constructed with the use of lower and upper solutions as indicators, in this paper we construct ε-uniformly convergent schemes of the Newton method.…”

Section: Introductionmentioning

confidence: 99%

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Iterative Newton solution method for the Richardson scheme for a semilinear singular perturbed elliptic convection–diffusion equation

Шишкин

Shishkina

2011

Russian Journal of Numerical Analysis and Mathematical Modelling

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Grid approximations of the Dirichlet problem are considered in a vertical strip for the semilinear elliptic convection-diffusion equation; for this problem, the nonlinear difference scheme based on the classic approximation of the problem on a piecewise-uniform grid refined in a layer converges ε-uniformly in the uniform norm with the convergence rate order not higher than one.Using the Richardson technique, we construct a nonlinear scheme (the Richardson scheme on nested piecewise-uniform grids) convergent ε-uniformly with the improved convergence rate O(N −2 1 ln 2 N 1 + N −2 2 ), where N 1 + 1 and N 2 + 1 are the numbers of nodes on the axis x 1 and on the unit segment of the axis x 2 , respectively. In order to solve this scheme, we construct iterative schemes of higher accuracy, which are the linearized scheme (the nonlinear term is calculated based on the desired function taken from the previous iteration) and the scheme of the Newton method, as well as truncated variants of iterative schemes convergent at the rate O(N −2 1 ln 2 N 1 + N −2 2 ). The number T of iterations required in the truncated schemes for the solution of the problem does not depend on the parameter ε, here T = O(ln (min[N 1 , N 2 ])) and T = O(ln ln (min[N 1 , N 2 ])) in the case of the truncated linearized scheme and the truncated Newton scheme, respectively.

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Section: A Priori Estimates Of Solutions and Derivativesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Iterative Newton solution method for the Richardson scheme for a semilinear singular perturbed elliptic convection–diffusion equation

Шишкин

Shishkina

2011

Russian Journal of Numerical Analysis and Mathematical Modelling

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“…In addition, other difficulties appear in the construction of difference schemes of a higher order of accuracy. Thus, it was shown in [20] and [25] that we cannot construct Richardson schemes on piecewise-uniform grids for a singularly perturbed reaction-diffusion problem so that its solutions converge ε-uniformly with the accuracy order exceeding three or with the order exceeding two for a convection-diffusion problem.…”

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confidence: 99%

Data perturbation stability of difference schemes on uniform grids for a singularly perturbed convection–diffusion equation

Шишкин

2013

Russian Journal of Numerical Analysis and Mathematical Modelling

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An approach to the study of the conditioning of difference schemes and their stability to data perturbations is developed for the Dirichlet problem for a singularly perturbed convection-diffusion ordinary differential equation with the perturbation parameter ε, ε ∈ (0, 1]. We consider a standard difference scheme, which is a monotone scheme on a uniform grid, and a special scheme of the grid solution decomposition method. Constructing the special scheme, we use a decomposition of the grid solution into a regular and a singular components that are the solutions to grid subproblems considered on uniform grids.The following facts are proved for the standard scheme: (a) the scheme converges only for N −1 = o(ε) at the rate O N −1 (ε + N −1 ) −1 , where N + 1 is the number of grid nodes, (b) the scheme is not ε-uniformly well conditioned and stable to perturbations, (c) the condition number of the scheme has the order O ε −1 δ −2 , where δ is the accuracy of the grid solution, δ = δ st ≈ N −1 (ε + N −1 ) −1 . In the case of convergence of the standard scheme proved theoretically, the actual accuracy of the computed solution decreases with the decrease in the parameter ε under the presence of perturbations and may completely vanish for a sufficiently small ε (namely, under the condition t = O ln ε −1 + ln δ −2 st , where t is the number of digits in the machine word). At the same time, the special scheme of the solution decomposition method converges ε-uniformly in the uniform norm at the rate O N −1 ln N ; in the variables ε and δ , the special scheme is ε-uniformly well conditioned and stable to grid problem data perturbations; the condition number of the scheme of the decomposition method is of the order O δ −2 ln δ −1 , δ = δ dec ≈ N −1 ln N.Methods for the construction of ε-uniformly convergent difference schemes for singularly perturbed partial differential equations have been successfully developed on the base of the method of refining grids and the method of adjustment (see, e.g., [2,6,7,11,12,16,24] and the references therein).An essential drawback of the approach based on standard methods on uniform grids is that the constructed schemes converge formally, i.e., under the condition that all calculations are performed exactly. In reality, if the parameter ε is sufficiently small, the errors appearing in actual calculations Brought to you by | Florida International University Libraries Authenticated Download Date | 5/30/15 10:12 PM Data perturbation stability 383 cause errors in the numerical solution exceeding the accuracy of the solution obtained without computation errors by several orders. The cause is that the standard numerical methods are not, generally speaking, ε-uniformly stable to grid problem data perturbations appearing in calculations even in the case of their convergence (see, e.g., the assertions of Theorems 6.2, 6.3 and Remark 6.1 for a convection-diffusion problem).Thus, in the case of singularly perturbed problems, there is an interest to develop special ε-uniformly convergent numerical method...

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Second-order parameter-uniform finite difference scheme for singularly perturbed parabolic problem with a boundary turning point

Sahoo

Gupta

2021

Journal of Difference Equations and Applications

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A higher order richardson scheme for a singularly perturbed semilinear elliptic convection-diffusion equation

Cited by 8 publications

References 8 publications

Iterative Newton solution method for the Richardson scheme for a semilinear singular perturbed elliptic convection–diffusion equation

Iterative Newton solution method for the Richardson scheme for a semilinear singular perturbed elliptic convection–diffusion equation

Data perturbation stability of difference schemes on uniform grids for a singularly perturbed convection–diffusion equation

Second-order parameter-uniform finite difference scheme for singularly perturbed parabolic problem with a boundary turning point

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