Abstract:
A mixed problem for a one-dimensional heat equation with several versions of initial and boundary conditions is considered. Explicit and implicit schemes are applied for the solution. The sweep method and the iteration methods are used for the implicit scheme for solving the implicit system of equations. The numerical filtration of a finite sequence of results obtained for different grids with an increasing number of nodal points is used to analyze errors of the method and rounding. In addition, to investigate the rounding errors, the results obtained with several lengths of the machine word mantissa are compared. The numerical solution of the mixed problem for the wave equation is studied by similar methods.
The occurrence of deterministic dependencies of the error in the numerical method and the rounding on spatial coordinates, time and the number of nodes is revealed. The source models to describe the behavior of errors in terms of time are based on the analysis of the results of numerical experiments for different versions of conditions of problems. In accord with such models, which were verified by the experiment, the errors can increase, decrease or stabilize depending on conditions over time similar to changing the energy or mass.
Key words:
heat equation, explicit and implicit schemes, the Courant number, model error, numerical filtration.
Citation:
V. P. Zhitnikov, N. M. Sherykhalina, R. R. Muksimova, “The peculiarities of error accumulation in solving problems for simple equations of mathematical physics by finite difference methods”, Sib. Zh. Vychisl. Mat., 19:2 (2016), 139–152; Num. Anal. Appl., 9:2 (2016), 107–117
\Bibitem{ZhiSheMuk16}
\by V.~P.~Zhitnikov, N.~M.~Sherykhalina, R.~R.~Muksimova
\paper The peculiarities of error accumulation in solving problems for simple equations of mathematical physics by finite difference methods
\jour Sib. Zh. Vychisl. Mat.
\yr 2016
\vol 19
\issue 2
\pages 139--152
\mathnet{http://mi.mathnet.ru/sjvm608}
\crossref{https://doi.org/10.15372/SJNM20160202}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=3509198}
\elib{https://elibrary.ru/item.asp?id=25984438}
\transl
\jour Num. Anal. Appl.
\yr 2016
\vol 9
\issue 2
\pages 107--117
\crossref{https://doi.org/10.1134/S1995423916020026}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=000377110400002}
\elib{https://elibrary.ru/item.asp?id=27140173}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84975856220}
Linking options:
https://www.mathnet.ru/eng/sjvm608
https://www.mathnet.ru/eng/sjvm/v19/i2/p139
This publication is cited in the following 2 articles:
V. P. Zhitnikov, N. M. Sherykhalina, R. R. Muksimova, “The errors investigation in problems for solving simple equations of mathematical physics by iterative methods”, Num. Anal. Appl., 14:2 (2021), 115–125
V P Zhitnikov, R R Muksimova, N M Sherykhalina, N I Zhitnikova, “Influence of different components of data error on the result of solving identification and approximation problems”, IOP Conf. Ser.: Mater. Sci. Eng., 919:5 (2020), 052016
Citation in format AMSBIB
Contact us: