Локально-Одномерная Разностная Схема Третьей Краевой Задачи Для Параболического Уравнения Общего Вида С Нелокальным Источником

Бештокова, З. В.; Shkhanukov-Lafishev, M. Kh.

doi:10.1134/s0374064118070051

Cited by 3 publications

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“…In the works [13][14][17][18][19][20][21][22][23][24] for the numerical solution of multidimensional parabolic equations, a LOS was constructed Thus, in [13], in an arbitrary domain G, a locally one-dimensional scheme is considered for solving linear and quasilinear parabolic equations. The stability of the difference scheme with respect to the right-hand side, boundary and initial data is proved, as well as convergence with a rate O(h 2 + τ ).…”

Section: Introductionmentioning

confidence: 99%

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К Численным Методам Решения Многомерных Интегро-Дифференциальных Уравнений

Бештокова

2023

Bulletin of the BSU. Mathematics Informatics

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The third boundary value problem for a multidimensional convection-diffusion equation with memory effect and non-local (integral) source is investigated. To solve numerically the multidimensional problem, a locally one-dimensional difference scheme is constructed, the essence of the idea of which is to reduce the transition from layer to layer to sequential solving of a number of onedimensional problems in each of the coordinate directions. Using the method of energy inequalities for the solution of a locally one-dimensional difference scheme, an a priori estimate is obtained. The main research method is the method of energy inequalities. An a priori estimate of the LOS solution is obtained, from which follow uniqueness, stability, and convergence of the solution of the difference problem to the solution of the original differential problem at a rate equal to the approximation error. Numerical experiments were carried out.

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Section: Introductionmentioning

confidence: 99%

“…This work is a continuation of the author's series of works [23][24][25] devoted to the study of local and nonlocal boundary value problems for multidimensional parabolic equations.…”

Section: Introductionmentioning

confidence: 99%

К Численным Методам Решения Многомерных Интегро-Дифференциальных Уравнений

Бештокова

2023

Bulletin of the BSU. Mathematics Informatics

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“…Численным методам решения локальных и нелокальных краевых задач для многомерных дифференциальных уравнений в частных производных параболического типа на основе различных методов расщепления посвящены работы автора [18][19][20].…”

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Finite Difference Method for Solving Parabolic-Type Integro-Differential Equations in Multidimensional Domain with Nonhomogeneous First-Order Boundary Conditions

Beshtokova

2024

CMIT

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Introduction. We investigate a multidimensional (in terms of spatial variables) parabolic-type integro-differential equation with nonhomogeneous first-order boundary conditions. The locally one-dimensional finite difference scheme developed herein can be applied to solve various applied problems leading to multidimensional parabolic-type integro-differential equations. Examples include mathematical modelling of cloud processes, addressing the issue of active intervention in convective clouds to prevent hail and artificially enhance precipitation, as well as describing the droplet mass distribution function due to microphysical processes such as condensation, coagulation, fragmentation, and freezing of droplets in convective clouds.Materials and Methods. In this study, an approximate solution to the initial-boundary value problem is constructed using the locally one-dimensional scheme of A.A. Samarsky with a specified order of approximation О(h2 +τ). The primary research method employed is the method of energy inequalities.Results. A priori estimates have been obtained in the discrete interpretation, from which uniqueness, stability, and convergence of the solution of the locally one-dimensional difference scheme to the solution of the original differential problem follow, with a convergence rate equal to the order of approximation of the difference scheme.Discussion and Conclusions. The research findings can be utilized for further development of boundary value problem theory for parabolic equations with variable coefficients. Additionally, they may find applications in the fields of difference scheme theory, computational mathematics, and numerical modelling.

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Stability and convergence of the locally one-dimensional scheme A. A. Samarskii, approximating the multidimensional integro-differential equation of convection-diffusion with inhomogeneous boundary conditions of the first kind

Бештокова,

Beshtokova

2023

Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki

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Изучена первая начально-краевая задача для многомерного (по пространственным переменным) интегро-дифференциального уравнения конвекции-диффузии. Для приближенного решения поставленной задачи предложена локально-одномерная схема А. А. Самарского с порядком аппроксимации $O(h^2+\tau)$. Исследование единственности и устойчивости решения проводится с помощью метода энергетических неравенств. Получены априорные оценки решения локально-одномерной разностной схемы, откуда следуют единственность решения, непрерывная и равномерная зависимость решения от входных данных, а также сходимость решения схемы к решению исходной дифференциальной задачи со скоростью, равной порядку аппроксимации разностной схемы. Для двумерной задачи построен алгоритм численного решения, проведены численные расчеты тестовых примеров, иллюстрирующие полученные в работе теоретические результаты.

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Локально-Одномерная Разностная Схема Третьей Краевой Задачи Для Параболического Уравнения Общего Вида С Нелокальным Источником

Cited by 3 publications

References 0 publications

К Численным Методам Решения Многомерных Интегро-Дифференциальных Уравнений

К Численным Методам Решения Многомерных Интегро-Дифференциальных Уравнений

Finite Difference Method for Solving Parabolic-Type Integro-Differential Equations in Multidimensional Domain with Nonhomogeneous First-Order Boundary Conditions

Stability and convergence of the locally one-dimensional scheme A. A. Samarskii, approximating the multidimensional integro-differential equation of convection-diffusion with inhomogeneous boundary conditions of the first kind

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