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Differential Equations and Mathematical Physics
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
Z. V. Beshtokova Institute of Applied Mathematics and Automation,
Kabardino-Balkarian Scientific Center of RAS,
Nalchik, 360000, Russian Federation
(published under the terms of the Creative Commons Attribution 4.0 International License)
Abstract:
The first initial-boundary value problem for a multidimensional (in space variables) integro-differential equation of convection-diffusion is studied. For an approximate solution of the problem a locally one-dimensional scheme by A. A. Samarskii with order of approximation $O(h^2+\tau)$ is proposed. The study of the uniqueness and stability of the solution is carried out using the method of energy inequalities. A priori estimates for the solution of a locally one-dimensional difference scheme are obtained, which imply the uniqueness of the solution, the continuous and uniform dependence of the solution on the input data, and the convergence of the solution of the scheme to the solution of the original differential problem at a rate equal to the order of approximation of the difference scheme. For a two-dimensional problem, a numerical solution algorithm is constructed, numerical calculations of test cases are carried out, illustrating the theoretical results obtained in the study.
Keywords:
convection-diffusion equation, first initial boundary value problem, nonlocal source, multidimensional problem, difference schemes, a priori estimate, stability and convergence.
Received: April 26, 2023 Revised: August 23, 2023 Accepted: September 19, 2023 First online: September 28, 2023
Citation:
Z. V. Beshtokova, “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”, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 27:3 (2023), 407–426
Linking options:
https://www.mathnet.ru/eng/vsgtu2014 https://www.mathnet.ru/eng/vsgtu/v227/i3/p407
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