Approximate method for solving a loaded multidimensional diffusion equation with inhomogeneous boundary conditions of the first kind

Z. V. Beshtokova^a, V. A. Vodakhova^b, F. M. Nakhusheva^b, M. Z. Khudalov<sup>c

a</sup> Institute of Applied Mathematics and Automation KBNC RAS 89a Shortanova St., Nalchik, 360000, Russian Federation
^b Institute of Physics and Mathematics of the Kabardino-Balkarian State University named after KH. M. Berbekov 173 Chernyshevsky St., Nalchik, 360004, Russian Federation
^c North Ossetian State University named after K. L. Khetagurov 44-46 Vatutina str., Vladikavkaz, 362025, Russian Federation

Abstract: In this paper, a loaded multidimensional diffusion equation with non-homogeneous boundary conditions of the first kind is considered. For an approximate solution of the initial-boundary value problem, a locally one-dimensional scheme proposed by A. A. Samarskiy with an approximation order of $O(h^2+\tau)$ is developed. Using the method of energy inequalities, a priori estimates in difference form are obtained, which allows us to establish the uniqueness, stability and convergence of the solution of the locally one-dimensional scheme to the solution of the original differential problem with a convergence rate corresponding to the approximation order of the scheme. For the two-dimensional problem, an algorithm for a numerical solution was developed, and numerical experiments were carried out, which confirm the theoretical results obtained in the work.

Keywords: multidimensional problem, first initial-boundary value problem, loaded equation, diffusion equation, locally one-dimensional difference scheme, a priori estimate, stability, convergence.

UDC: 519.63

MSC: 65M06, 65N12

Language: English