The peculiarities of error accumulation in solving problems for simple equations of mathematical physics by finite difference methods
Vladimir Pavlovich Zhitnikov1, Nataliya Mikhailovna Sherykhalina1, Roza Ravilevna Muksimova2
1Ufa State Aviation Technical University, K. Marksa str., 12, Ufa, Russia, 450000
2Saint-Petersburg State University of Civil Aviation, Pilotov str., 38, St. Petersburg, Russia, 196210
Keywords: уравнение теплопроводности, волновое уравнение, явная и неявная схемы, число Куранта, модели погрешности, численная фильтрация, heat equation, explicit and implicit schemes, the Courant number, model error, numerical filtration
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.
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