Criteria of Solvability for Asymmetric Difference Schemes at High-Accuracy Approximation of Boundary Conditions

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