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Peculiarities of error accumulation in solving problems for simple equations of mathematical physics by finite difference methods

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Abstract

A mixed problem for the one-dimensional heat conduction equation with several versions of initial and boundary conditions is considered. To solve this problem, explicit and implicit schemes are used. Sweep and iterative methods are used for the implicit scheme when solving the system of equations. Numerical filtering of a finite sequence of results obtained for various grids with an increasing number of node points is used to analyze the method and rounding errors. To investigate rounding errors, the results obtained for various machine word mantissa lengths are compared. The numerical solution of a mixed problem for the wave equation is studied by similar methods. Some deterministic dependencies of the numerical method and rounding errors on the spatial coordinates, time, and the number of nodes are found. Some models of sources to describe the behavior of the errors in time are constructed. They are based on the results of computational experiments under various conditions. According to these models, which have been experimentally verified, the errors increase, decrease, or stabilize in time under the conditions, similarly to energy or mass.

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References

  1. Zhitnikov, V.P. and Sherykhalina, N.M., Modelirovanie techenii vesomoi zhidkosti s primeneniem metodov mnogokomponentnogo analiza (Modeling of Heavy Fluid Flows with Methods of Multi-Component Analysis), Ufa: Gilem, 2009.

    Google Scholar 

  2. Zhitnikov, V.P. and Sherykhalina, N.M., Mnogokomponentnyi analiz chislennykh rezul’tatov (Multi-Component Analysis of Numerical Results), Saarbrucken, Germany: LAP LAMBERT Academic Publishing, 2012.

    Google Scholar 

  3. Zhitnikov, V.P. and Sherykhalina, N.M., Multiple Filtering in Numerical Problem Solving by Methods of the Theory of Functions of Complex Variables, Vych. Tekhnol., 2013, vol. 18, no. 1, pp. 15–24.

    Google Scholar 

  4. Zhitnikov, V.P. and Muksimova, R.R., Solving a Problem of Nonstationary Electrochemical Treatment by a Plane Electrode Tool with Limited Roughness, Vest. UGATU, 2011, vol. 15, no. 1, pp. 113–118.

    Google Scholar 

  5. Borshchuk, O.S. and Zhitnikov, V.P., Nonlinear Filtering in Low-PermeabilityReservoirs:Numerical Scheme, Analysis of Stability and Convergence, Nauchno-Tekhn. Vest. OAO “NK “Rosneft,” 2013, no. 2, pp. 13–16.

    Google Scholar 

  6. Volkov, E.A., Chislennye metody (NumericalMethods), Moscow: Nauka, 1988.

    MATH  Google Scholar 

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Authors and Affiliations

  1. Ufa State Aviation Technical University, ul. K. Marksa 12, Ufa, 450000, Russia

    V. P. Zhitnikov & N. M. Sherykhalina

  2. St. Petersburg State University of Civil Aviation, ul. Pilotov 38, St. Petersburg, 196210, Russia

    R. R. Muksimova

Authors
  1. V. P. Zhitnikov
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  2. N. M. Sherykhalina
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  3. R. R. Muksimova
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Corresponding author

Correspondence to V. P. Zhitnikov.

Additional information

Original Russian Text © V.P. Zhitnikov, N.M. Sherykhalina, R.R. Muksimova, 2016, published in Sibirskii Zhurnal Vychislitel’noi Matematiki, 2016, Vol. 19, No. 2, pp. 139–152.

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Zhitnikov, V.P., Sherykhalina, N.M. & Muksimova, R.R. Peculiarities of error accumulation in solving problems for simple equations of mathematical physics by finite difference methods. Numer. Analys. Appl. 9, 107–117 (2016). https://doi.org/10.1134/S1995423916020026

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  • DOI: https://doi.org/10.1134/S1995423916020026

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