Difference Methods and Their Extrapolations

Marchuk, G. I.; Shaidurov, V. V.

doi:10.1007/978-1-4613-8224-9

Cited by 155 publications

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“…The difference scheme (12), (11) is constructed on the basis of the nonlinear scheme (5), (8) using a Richardson extrapolation technique on two embedded meshes. We call this scheme the Richardson scheme (12), (11).…”

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

confidence: 99%

“…We call this scheme the Richardson scheme (12), (11). We call the function z 0 (12) (x, t), (x, t) ∈ G 0 h , the solution of the Richardson scheme (12), (11); we call the functions z 1 (12) …”

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

confidence: 99%

“…To increase the accuracy of computed approximations of the solutions of classical boundary value problems, defect correction and Richardson extrapolation (see, e.g., [2,11,12] and their bibliographies) are used. 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]).…”

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

confidence: 99%

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

confidence: 99%

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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“…Let a λµ , b λ , p κ , q κ and their derivatives in x up to order l be functions and be bounded by a constant C for all λ, µ ∈ Λ 1 and κ ∈ Λ 0 . Then for every n ≥ 0 17) where N stands for constants depending only on n, d, l, C and |Λ 0 |.…”

Section: Gyöngymentioning

confidence: 99%

“…Since then Richardson extrapolation has been applied to a wide range of numerical approximations. (See, for example, the textbook [17], the monograph [22] and the survey papers [1] and [12]). To show that it is applicable to an approximation method to solve numerically a problem under certain conditions one has to show the existence of a suitable power series expansion, which can be quite difficult in some situations.…”

Section: Introductionmentioning

confidence: 99%

On stochastic finite difference schemes

Gyöngy

2014

Stoch PDE: Anal Comp

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Abstract. Finite difference schemes in the spatial variable for degenerate stochastic parabolic PDEs are investigated. Sharp results on the rate of Lp and almost sure convergence of the finite difference approximations are presented and results on Richardson extrapolation are established for stochastic parabolic schemes under smoothness assumptions. IntroductionWe consider finite difference schemes to stochastic partial differential equations (SPDEs). The stochastic PDEs we are interested in are linear second order stochastic parabolic equations in the whole R d in the spatial variable. They may degenerate and become first order stochastic or deterministic PDEs. The finite difference schemes which we investigate are spatial discretizations of such SPDEs. One can view them as (possibly degenerate) infinite systems of stochastic differential equations, whose components describe the time evolution of approximate values at the grid points of the solutions to SPDEs. Adapting the approach of [13] we view stochastic finite difference schemes, like in [7] and [3], as stochastic equations for random fields on the whole R d not only on grids.Our aim is to investigate the rate of convergence in the supremum norm of the finite difference approximations. We show that under the stochastic parabolicity condition, if the coefficients and the data are sufficiently smooth, then the solutions to the finite difference schemes admit power series expansions in terms of h, the mesh-size of the grid. The coefficients in these power series are random fields, independent of h, and for any p > 0 the p-th moments of the sup norm of the remainder term is estimated by a power of h. This is Theorem 2.2. Hence for h → 0 we get the convergence (and the sharp rate) of the solutions of the finite difference schemes to a random field which is the solution to the corresponding SPDE. Moreover, by Richardson extrapolation we get that the rate of convergence can be accelerated to any high order if one takes appropriate mixtures of approximations corresponding to different grid sizes. In Theorem 2.4 we obtain convergence estimates for any (high) p-th moments of the sup norm of the approximation error. Hence in 2000 Mathematics Subject Classification. 65M15, 35J70, 35K65.

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Bibliography

2021

Numerical Methods in Computational Finance

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Difference Methods and Their Extrapolations

Abstract: Library of Congress Cataloging in Publication Data Marchuk, G. I. (Gurii Ivanovich), 1925-Difference methods and their extrapolations.

Cited by 155 publications

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

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

On stochastic finite difference schemes

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