Loading [MathJax]/jax/output/SVG/config.js
Vestnik Tomskogo Gosudarstvennogo Universiteta. Matematika i Mekhanika
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



Vestn. Tomsk. Gos. Univ. Mat. Mekh.:
Year:
Volume:
Issue:
Page:
Find






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


Vestnik Tomskogo Gosudarstvennogo Universiteta. Matematika i Mekhanika, 2019, Number 57, Pages 5–25
DOI: https://doi.org/10.17223/19988621/57/1
(Mi vtgu686)
 

This article is cited in 10 scientific papers (total in 10 papers)

MATHEMATICS

A refinement of the boundary element collocation method near the boundary of domain in the case of two-dimensional problems of non-stationary heat conduction with boundary conditions of the second and third kind

Ivanov D.Yu.

Moscow State University of Railway Engeneering (MIIT), Moscow, Russian Federation
References:
Abstract: In this paper, we consider initial-boundary value problems (IBVPs) for the equation $\partial_tu=a^2\Delta_2u-pu$ with constants $a,p>0$ in an open two-dimensional spatial domain $\Omega$ with boundary conditions of the second and third kind at a zero initial condition. A fully justified collocation boundary element method is proposed, which makes it possible to obtain uniformly convergent in the space-time domain $\Omega\times[0,T]$ approximate solutions of the abovementioned IBVPs. The solutions are found in the form of the single-layer potential with unknown density functions determined from boundary integral equations of the second kind.
To ensure the uniform convergence, integration on arc-length $s$ when calculating the potential operator is carried out in two ways. If the distance $r$ from the point $x\in\Omega$ at which the potential is calculated to the integration point $x'\in\partial\Omega$ does not exceed approximately one-third of the radius of the Lyapunov circle $R_{\text{Л}}$, then we use exact integration with respect to a certain component $\rho$ of the distance $r:\,\rho\equiv(r^2-d^2)^{\frac12}$ ($d$ is the distance from the point $x\in\Omega$ to the boundary $\partial\Omega$). This exact integration is practically feasible for any analytically defined curve $\partial\Omega$. In this integration, functions of the variable $\rho$ are taken as the weighting functions and the rest of the integrand is approximated by quadratic interpolation on $\rho$. The functions of $\rho$ are generated by the fundamental solution of the heat equation. The integrals with respect to $s$ for $r>R_{\text{Л}}/3$ are calculated using Gaussian quadrature with $\gamma$ points.
Under the condition $\partial\Omega\in C^5\cap C^{2\gamma}$ ($\gamma\geqslant2$), it is proved that the approximate solutions converge to an exact one with a cubic velocity uniformly in the domain $\Omega\times [0, T]$. It is also proved that the approximate solutions are stable to perturbations of the boundary function uniformly in the domain $\Omega\times [0, T]$. The results of computational experiments on the solution of the IBVPs in a circular spatial domain are presented. These results show that the use of the exact integration with respect to $\rho$ can substantially reduce the decrease in the accuracy of numerical solutions near the boundary $\partial\Omega$, in comparison with the use of exclusively Gauss quadratures in calculating the potential.
Keywords: non-stationary heat conduction, boundary integral equation, single-layer heat potential, singular boundary element, collocation, operator, uniform convergence.
Received: 31.08.2018
Bibliographic databases:
Document Type: Article
UDC: 519.642.4
MSC: 80М15, 65Е05
Language: Russian
Citation: Ivanov D.Yu., “A refinement of the boundary element collocation method near the boundary of domain in the case of two-dimensional problems of non-stationary heat conduction with boundary conditions of the second and third kind”, Vestn. Tomsk. Gos. Univ. Mat. Mekh., 2019, no. 57, 5–25
Citation in format AMSBIB
\Bibitem{Iva19}
\by Ivanov~D.Yu.
\paper A refinement of the boundary element collocation method near the boundary of domain in the case of two-dimensional problems of non-stationary heat conduction with boundary conditions of the second and third kind
\jour Vestn. Tomsk. Gos. Univ. Mat. Mekh.
\yr 2019
\issue 57
\pages 5--25
\mathnet{http://mi.mathnet.ru/vtgu686}
\crossref{https://doi.org/10.17223/19988621/57/1}
\elib{https://elibrary.ru/item.asp?id=37113830}
Linking options:
  • https://www.mathnet.ru/eng/vtgu686
  • https://www.mathnet.ru/eng/vtgu/y2019/i57/p5
  • This publication is cited in the following 10 articles:
    1. D. Yu. Ivanov, “On the Uniform Convergence of Approximations to the Tangential and Normal Derivatives of the Single-Layer Potential Near the Boundary of a Two-Dimensional Domain”, Comput. Math. and Math. Phys., 64:7 (2024), 1504  crossref
    2. Ivanov D.Yu., “Poluanaliticheskaya approksimatsiya normalnoi proizvodnoi teplovogo potentsiala prostogo sloya vblizi granitsy dvumernoi oblasti”, Izv. Sarat. un-ta. Nov. ser. Ser.: Matematika. Mekhanika. Informatika, 24:4 (2024), 476–487  mathnet  crossref
    3. Ivanov D.Yu., “Ob odnoi poluanaliticheskoi approksimatsii normalnoi proizvodnoi potentsiala prostogo sloya vblizi granitsy dvumernoi oblasti”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 33:3 (2023), 434–451  mathnet  crossref
    4. D. Yu. Ivanov, “On uniform convergence of semi-analytic solution of Dirichlet problem for dissipative Helmholtz equation in vicinity of boundary of two-dimensional domain”, Ufa Math. J., 15:4 (2023), 76–99  mathnet  crossref
    5. Ivanov D.Yu., “Ob approksimatsii normalnoi proizvodnoi teplovogo potentsiala prostogo sloya vblizi granitsy dvumernoi oblasti”, Vestn. Tomsk. gos. un-ta. Matem. i mekh., 2023, no. 83, 31–51  mathnet  crossref
    6. D. Yu. Ivanov, O. A. Platonova, MODERN APPROACHES IN ENGINEERING AND NATURAL SCIENCES: MAENS-2021, 2526, MODERN APPROACHES IN ENGINEERING AND NATURAL SCIENCES: MAENS-2021, 2023, 020007  crossref
    7. Ivanov D.Yu., “O ravnomernoi skhodimosti approksimatsii potentsiala dvoinogo sloya vblizi granitsy dvumernoi oblasti”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 32:1 (2022), 26–43  mathnet  crossref  mathscinet
    8. A. I. Kanareykin, “COOLING OF AN INFINITE RECTANGULAR PLATE UNDER BOUNDARY CONDITIONS OF THE SECOND AND THIRD KIND”, CJ, 7:9(71) (2022), 36  crossref
    9. Ivanov D.Yu., “O sovmestnom primenenii kollokatsionnogo metoda granichnykh elementov i metoda Fure dlya resheniya zadach teploprovodnosti v konechnykh tsilindrakh s gladkimi napravlyayuschimi”, Vestn. Tomsk. gos. un-ta. Matem. i mekh., 2021, no. 72, 15–38  mathnet  crossref
    10. Ivanov D.Yu., “Utochnenie kollokatsionnogo metoda granichnykh elementov vblizi granitsy dvumernoi oblasti s pomoschyu poluanaliticheskoi approksimatsii teplovogo potentsiala dvoinogo sloya”, Vestn. Tomsk. gos. un-ta. Matem. i mekh., 2020, no. 65, 30–52  mathnet  crossref
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Вестник Томского государственного университета. Математика и механика
    Statistics & downloads:
    Abstract page:228
    Full-text PDF :55
    References:36
     
      Contact us:
    math-net2025_03@mi-ras.ru
     Terms of Use  Registration to the website  Logotypes © Steklov Mathematical Institute RAS, 2025