A Higher-Order Richardson Method for a Quasilinear Singularly Perturbed Elliptic Reaction-Diffusion Equation

Shishkin, G. I.; Shishkina, L. P.

doi:10.1007/s10625-005-0245-8

Cited by 25 publications

(13 citation statements)

References 6 publications

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“…2. To justify the convergence of the Richardson scheme (12), (11), we apply a technique similar to one used in [7,19,21]. It is convenient to consider the expansions of the functions …”

Section: Richardson Extrapolation For (2) (1)mentioning

confidence: 99%

“…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%

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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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“…2. To justify the convergence of the Richardson scheme (12), (11), we apply a technique similar to one used in [7,19,21]. It is convenient to consider the expansions of the functions …”

Section: Richardson Extrapolation For (2) (1)mentioning

confidence: 99%

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 computed solution  U is the numerical approximation of u with improved rate of convergence. To prove this we apply a technique similar to [18]. We write…”

Section: Richardson Extrapolation Schemementioning

confidence: 99%

High order parameter-uniform discretization for singularly perturbed parabolic partial differential equations with time delay

Kumar¹,

Kumar²

2014

Computers & Mathematics with Applications

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“…• the approach to the construction of difference schemes of higher orders of accuracy based on the Richardson technique on nested piecewise-uniform grids [4,20,21];…”

Section: 2mentioning

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%

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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A Higher-Order Richardson Method for a Quasilinear Singularly Perturbed Elliptic Reaction-Diffusion Equation

Cited by 25 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

High order parameter-uniform discretization for singularly perturbed parabolic partial differential equations with time delay

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

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