The solvability of difference schemes with universal approximation of ѕows in boundary conditions

Исследуется способ аппроксимации граничных условий с произвольным порядком точности, основанный на односторонних многоточечных разностных аналогах потоков максимально возможной точности на данном шаблоне. Рассматриваемая технология универсальна в смысле однообразия алгоритма при любых порядках точности схемы и в смысле независимости граничного условия от вида решаемого уравнения. Исследована проблема реализации и разрешимости разностной задачи путем приведения ҡдлинныхә граничных условий к эквивалентным двухточечным условиям. Сформулированы условия диагонального преобладания в строках матрицы приведенной системы, соответствующих граничным соотношениям. Показано, что обсуждаемая универсальная технология расчета, в отличие от традиционных способов, не порождает проблем при расщеплении многомерных задач на одномерные. 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 ѕows. 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 onedimensional ones at the intermediate steps. However, the number of nodes in the boundary relation stencil, i. e. the ҡlengthә 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 ҡlongә 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 justiҥcation. 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 ҡlongә 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 ѕow approximation. For several difference schemes up to the fourth order of accuracy, it is found that the criterion is satisҥed unconditionally or under not burdensome restrictions on the ratio of grid steps.