Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki
RUS  ENG    JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB  
General information
Latest issue
Archive
Impact factor

Search papers
Search references

RSS
Latest issue
Current issues
Archive issues
What is RSS



Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki:
Year:
Volume:
Issue:
Page:
Find






Personal entry:
Login:
Password:
Save password
Enter
Forgotten password?
Register


Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2021, Volume 163, Book 3-4, Pages 261–275
DOI: https://doi.org/10.26907/2541-7746.2021.3-4.261-275
(Mi uzku1595)
 

Analysis of formulas for numerical differentiation of functions with large gradients on a Bakhvalov mesh

N. A. Zadorin

Sobolev Institute of Mathematics, Siberian Branch, Russian Academy of Sciences, Novosibirsk, 630090 Russia
References:
Abstract: The article gives an estimate of the error of the classical formulas for the numerical differentiation of a function of one variable with large gradients in the exponential boundary layer. It is assumed that the function is decomposed in the form of the sum of the regular and singular components, which is valid for the solution of a boundary value problem for the ordinary second-order differential equation with a small parameter $\varepsilon $ affecting the highest derivative. It is known that the application of the classical polynomial formulas of numerical differentiation to such a function in the case of a uniform mesh can lead to unacceptable errors. The article estimates the error of the formulas for numerical differentiation on the Bakhvalov mesh, which is condensed in the boundary layer region. Bakhvalov's mesh is widely used to construct uniformly converging difference schemes; therefore, the error estimation of the numerical differentiation formulas on this mesh is of interest. The estimates of the error on the Bakhvalov mesh are obtained taking into account the uniformity in the small parameter for the classical difference formulas widely used to calculate the first, second, and third derivatives. The results of numerical experiments are presented, which agree with the obtained error estimates. A numerical comparison of the obtained errors on the Bakhvalov and Shishkin meshes and on a uniform mesh is carried out.
Keywords: function of one variable, boundary layer, large gradients, Bakhvalov mesh, formulas for numerical differentiation, error estimation.
Funding agency Grant number
Russian Foundation for Basic Research 19-31-60009
The study was supported by the Russian Foundation for Basic Research (project no. 19-31-60009).
Received: 03.02.2021
Bibliographic databases:
Document Type: Article
UDC: 519.653
Language: Russian
Citation: N. A. Zadorin, “Analysis of formulas for numerical differentiation of functions with large gradients on a Bakhvalov mesh”, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 163, no. 3-4, Kazan University, Kazan, 2021, 261–275
Citation in format AMSBIB
\Bibitem{Zad21}
\by N.~A.~Zadorin
\paper Analysis of formulas for numerical differentiation of functions with large gradients on a Bakhvalov mesh
\serial Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki
\yr 2021
\vol 163
\issue 3-4
\pages 261--275
\publ Kazan University
\publaddr Kazan
\mathnet{http://mi.mathnet.ru/uzku1595}
\crossref{https://doi.org/10.26907/2541-7746.2021.3-4.261-275}
Linking options:
  • https://www.mathnet.ru/eng/uzku1595
  • https://www.mathnet.ru/eng/uzku/v163/i3/p261
  • Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki
    Statistics & downloads:
    Abstract page:207
    Full-text PDF :124
    References:17
     
      Contact us:
     Terms of Use  Registration to the website  Logotypes © Steklov Mathematical Institute RAS, 2024