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Numerical Analysis and Applications

2021 year, number 2

The errors investigation in problems for solving simple equations of mathematical physics by iterative methods

V.P. Zhitnikov1, N.M. Sherykhalina1, R.R. Muksimova2
1Ufa State Aviation Technical University, Ufa, Bashkortostan, Russia
2Saint Petersburg State University of Civil Aviation, Saint Petersburg, Russia
Keywords: heat equation, implicit scheme, Laplace equation, biharmonic equation, iteration method, numerical filtration


The error caused by the inaccuracy of the equation system solution by iterative methods has been investigated. The upper error estimate for the axially symmetric heat equation is found in the accumulation process in several time steps. The upper estimate shows the linear dependence of the error on the threshold value of the limiting criterion for the iterations number, the quadratic error growth from the range partitions number, and its independence of the time partitions number. The computing experiment shows a good correspondence of the obtained estimate to real errors with boundary and initial conditions of various types. The quadratic error growth for the Laplace equation, caused by the accuracy limitation for applying the iteration method, on the number of range partitions n, is empirically found. A similar error growth for the biharmonic equation is found in proportion to n4.