Criteria of solvability of asymmetric difference schemes at high-precision approximation of boundary conditions

In this paper we study the technology of calculating difference problems with internal boundary conditions of flow balance constructed by means of one-sided multipoint difference analogs of first derivatives of arbitrary order of accuracy. The proposed technology is equally suitable for any type of differential equations to be solved and admits a uniform realization at any order of accuracy. It, unlike approximations based on the continued system of equations, does not lead to complications in splitting multidimensional problems into one-dimensional ones. Sufficient conditions of solvability and stability of the realization of algorithms by the run method for boundary conditions of arbitrary order of accuracy are formulated. The proof is based on the reduction of multipoint boundary conditions to a form that does not violate the tridiagonal structure of matrices, and the establishment of the conditions of diagonal dominance in the transformed matrix rows corresponding to the external and internal boundary conditions.