On the stability of the spline-collocation difference scheme for a semilinear differential-algebraic index system (1,0)

In the paper, a semi-linear differential-algebraic system of partial differential equations of index $(1,0)$ with a rectangular domain of definition and compatible initial-boundary conditions is considered. It is assumed that the matrix pencil constructed from the coefficients of a differential-algebraic system is smoothly similar to the special canonical form. A uniform grid, in the rectangular domain of definition, for a numerical solving of the system, is constructed. On the grid, a rectangular elementary sub-region is allocated with a fixed number of nodes in each direction. The solution of the system, in each such sub-domain, is sought in the form of the Newton polynomial. The values of polynomial on the joint lines of the elementary sub-regions must coincide. A differential-algebraic system is written in the inner nodes of an elementary sub-region. Derivatives entering the system at each node of the elementary sub-region are approximated by the corresponding derivatives of the Newton polynomial. As a result, a nonlinear spline-collocation difference scheme the order of approximation of which coincides with the order of the spline for each independent variable is written out. Using the transformation of the matrix pencil of the system and the properties of the interpolation spline, the spline-collocation difference scheme is transformed to a matrix-difference equation. It is shown, in the paper, that the matrix-difference equation can be written in normal form. This form of writing of the difference scheme makes it possible to apply the method of simple iterations to it. Using the simple iteration method, an iterative process is written and it is proved that the corresponding transition operator is a compression operator and maps the grid space into itself. Incidentally, it is proved that the difference scheme has a unique solution and is stable in the grid space. To justify the last statement, the results of the author's previous work are used. As a result, in the work, the existence and stability of a unique solution of a spline-collocation difference scheme with an arbitrary order of approximation are justified. The stability of the difference scheme in the present work is understood in the sense of the definition by A.A. Samarskii. The results of a numerical solving of a semi-linear differential-algebraic system of partial differential equations are demonstrated in the test example.