TY - JOUR

T1 - The solvability of difference schemes with universal approximation of flows in boundary conditions

AU - Paasonen, V. I.

N1 - Acknowledgements. The research results presented in Sect. 1–3 (related to the development of the idea) were obtained within the framework of the state assignment of the Ministry of Education and Science of Russia for the FRC ICT. The research results presented in Sect. 4 (related to the derivation of the general criterion of diagonal predominance for boundary conditions of arbitrary order of accuracy) were obtained by a grant from the Russian Science Foundation (project No. 20-11-20040, https://rscf.ru/project/20-11-20040/).

PY - 2024

Y1 - 2024

N2 - One of the ways of setting the difference boundary conditions with high order of accuracy is based on the direct multi-point one-sided approximation of the flows. Such boundary relations, unlike traditional ones, are universal in the sense of uniformity of their structure at different orders of accuracy, as well as in the sense of their independence from the differential equation being solved. In addition, this technology does not create any obstacles in splitting multidimensional problems into one-dimensional ones, since the boundary conditions turn out to be the same universal one-dimensional ones at the intermediate steps. However, the number of nodes in the boundary relation stencil, i. e. the “length” of the boundary condition, increases as the order of accuracy of the scheme increases. This leads to a violation of the traditional tridiagonal structure of the matrices to be reversed, and a related violation of the diagonal predominance in the rows corresponding to “long” boundary conditions. Although extensive experience of applying universal boundary conditions in numerical simulations of various types of boundary value problems has not revealed violations of computational stability, this technique required a theoretical justification. This paper addresses the question of the solvability of such problems and the stability of calculations when they are implemented by the proposed method. For this purpose, matrix rows with “long” boundary conditions are reduced by means of local Gaussian procedures to equivalent short two-point rows, and the solvability and stability conditions for solutions of the transformed systems are established based on the requirement of a diagonal predominance in the transformed rows corresponding to the boundary conditions. A general criterion for diagonal predominance in a transformed string is formulated for an arbitrary order of flow approximation. For several difference schemes up to the fourth order of accuracy, it is found that the criterion is satisfied unconditionally or under not burdensome restrictions on the ratio of grid steps.

AB - One of the ways of setting the difference boundary conditions with high order of accuracy is based on the direct multi-point one-sided approximation of the flows. Such boundary relations, unlike traditional ones, are universal in the sense of uniformity of their structure at different orders of accuracy, as well as in the sense of their independence from the differential equation being solved. In addition, this technology does not create any obstacles in splitting multidimensional problems into one-dimensional ones, since the boundary conditions turn out to be the same universal one-dimensional ones at the intermediate steps. However, the number of nodes in the boundary relation stencil, i. e. the “length” of the boundary condition, increases as the order of accuracy of the scheme increases. This leads to a violation of the traditional tridiagonal structure of the matrices to be reversed, and a related violation of the diagonal predominance in the rows corresponding to “long” boundary conditions. Although extensive experience of applying universal boundary conditions in numerical simulations of various types of boundary value problems has not revealed violations of computational stability, this technique required a theoretical justification. This paper addresses the question of the solvability of such problems and the stability of calculations when they are implemented by the proposed method. For this purpose, matrix rows with “long” boundary conditions are reduced by means of local Gaussian procedures to equivalent short two-point rows, and the solvability and stability conditions for solutions of the transformed systems are established based on the requirement of a diagonal predominance in the transformed rows corresponding to the boundary conditions. A general criterion for diagonal predominance in a transformed string is formulated for an arbitrary order of flow approximation. For several difference schemes up to the fourth order of accuracy, it is found that the criterion is satisfied unconditionally or under not burdensome restrictions on the ratio of grid steps.

KW - compact difference scheme

KW - diagonal domination

KW - high-accuracy boundary conditions

KW - multipoint flow approximation

UR - https://www.scopus.com/record/display.uri?eid=2-s2.0-85185813466&origin=inward&txGid=daa344f479640b0493ed64d026661023

UR - https://www.elibrary.ru/item.asp?id=62415623

UR - https://www.mendeley.com/catalogue/c58e58b7-77bf-3fa5-a5d5-c1e0f9dd0ba5/

U2 - 10.25743/ICT.2024.29.1.003

DO - 10.25743/ICT.2024.29.1.003

M3 - Article

VL - 29

SP - 18

EP - 31

JO - Вычислительные технологии

JF - Вычислительные технологии

SN - 1560-7534

IS - 1

ER -