Abstract
The paper deals with the scheme of the second and fourth order approximation error for solving convection-diffusion problems. To model initial boundary value problem in the case when the functions of the right and the initial condition can be represented by finite sums of Fourier series in the trigonometric basis, we investigated the accuracy of difference schemes. It was found that the accuracy of the numerical solution depends on the number of units attributable to half the wavelength corresponding to the most high frequency harmonics in the final sum of the Fourier series, necessary to describe the behavior of calculated objects. The dependence of the diffusion approximation error terms difference schemes of second and fourth order of accuracy of the number of nodes. The comparison of the calculation results of two-dimensional convectiondiffusion problems and tasks of the Poisson-based schemes of the second and fourth order accuracy. In the expediency of transition to a scheme of high accuracy for solving applied problems of the estimates and is easy to obtain the numerical values of the gain in computation time by using schemes of higher order accuracy.
Highlights
To model initial boundary value problem in the case when the functions of the right and the initial condition can be represented by finite sums of Fourier series in the trigonometric basis, we investigated the accuracy of difference schemes
Bulletin of the South Ural State University
Summary
При выполнении оценок (12) аппроксимации задачи (1) – (2) разностными схемами (16) для каждой гармоники функции решения u скорость диффузионного обмена k меньше реальных значений и отличаются на величину \alpha 1 = 1 - 2 (1 - cos (\omega mh)) / (\omega mh). Описывающая зависимость погрешности аппроксимации диффузионных операторов разностными схемами четвертого порядка точности от количества узлов на структуру примет вид \alpha 2 (r) = 1 - (15 - 16 cos (\pi /r) + cos (2\pi /r)) /6 (\pi /r). 2. График зависимости погрешности аппроксимации операторов диффузионного переноса разностными схемами от количества узлов, используемых для описания объекта: 1 — схемы четвертого порядка точности, 2 — второго. 3. График зависимости погрешности аппроксимации операторов конвективного переноса разностными схемами от количества узлов, используемых для описания объекта: 1 — схемы четвертого порядка точности, 2 — второго. В табл. 2 приведены значения погрешностей расчета задачи диффузии-конвекции на различных расчетных сетках схемами второго порядка точности, На рис. 6 приведено поле, описывающее погрешность вычислений, полученное как разность между аналитическим и численным решением для задачи диффузии-конвекции
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