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 Zh. Vychisl. Mat. Mat. Fiz., 2016, Volume 56, Number 7, Pages 1299–1322 (Mi zvmmf10435)

On the perturbation algorithm for the semidiscrete scheme for the evolution equation and estimation of the approximate solution error using semigroups

D. V. Guluaa, D. L. Rogavab

a Department of Computational Mathematics, Georgian Technical University
b Department of Mathematics,Tbilisi State University

Abstract: In a Banach space, for the approximate solution of the Cauchy problem for the evolution equation with an operator generating an analytic semigroup, a purely implicit three-level semidiscrete scheme that can be reduced to two-level schemes is considered. Using these schemes, an approximate solution to the original problem is constructed. Explicit bounds on the approximate solution error are proved using properties of semigroups under minimal assumptions about the smoothness of the data of the problem. An intermediate step in this proof is the derivation of an explicit estimate for the semidiscrete Crank–Nicolson scheme. To demonstrate the generality of the perturbation algorithm as applied to difference schemes, a four-level scheme that is also reduced to two-level schemes is considered.

Key words: evolution equation, semidiscrete scheme, perturbation algorithm, explicit error estimation, method of semigroups.

 Funding Agency Grant Number Shota Rustaveli National Science Foundation D-13/1830/28 IRSES FP7-PEOPLE-2012-JRSES, ¹ 317721

DOI: https://doi.org/10.7868/S0044466916070085

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English version:
Computational Mathematics and Mathematical Physics, 2016, 56:7, 1269–1292

Bibliographic databases:

UDC: 519.622.2
Revised: 01.06.2015

Citation: D. V. Gulua, D. L. Rogava, “On the perturbation algorithm for the semidiscrete scheme for the evolution equation and estimation of the approximate solution error using semigroups”, Zh. Vychisl. Mat. Mat. Fiz., 56:7 (2016), 1299–1322; Comput. Math. Math. Phys., 56:7 (2016), 1269–1292

Citation in format AMSBIB
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