Bystryts'kyj M. Finite-difference schemes on non-ortho-go-nal patterns for hyperbolic partial differential equations.

The author introduced and analyzed the new finite-difference approximations - discrete Laplace operators on nonorthogonal patterns of the re-ctangular grid. The main difference operators have been constructed on non-orthogonal patterns. Two particular cases - discrete Laplace operator on nonorthogonal seven-point pattern on the plane, and nonothogonal thirteen-point pattern in three-dimensional space - have been analyzed in detail. A comparison is also drawn between the pro-perties of both nonorthogonal and classical schemes. It has been pro-ved in the dissertation that dispersional properties of the finitedifference schemes on nonorthogonal patterns are better then the corresponding properties of the classical schemes. Evaluations of the convergence rate for the difference elliptic and hyperbolic problems have been obtained for sche-mes on nonorthogonal patterns. It has also been pro-ved that the four order of approximation by choosing the right side in the scheme both for elliptic and for hyperbolic equations can be obtained for a sche-me on the sevenpoint pattern. Some direct methods for computation schemes on nonorthogonal patterns have been proposed. It has been shown that going onto the rectangular grid (two-dimensional case) allows to increase effectiveness of algorithms by twice in comparison with the triangular grid without loosing dispersional properties. The dissertation consists of the Introduction, three Chapters, the Con-clusion, and the List of the References. The total volume of the ma-nu-script is 137 pages. In the Introduction there is grounded the importance of the work, its goals, the main issues of the research, and the main results are presented with their theoretical and practical value, as well as it includes the comprise content of the work. In Chapter I a new method for building difference approximations of the Laplace operator has been developed that is based upon the reduction of derivatives on variables to derivatives on artificially introduced directions. As a result the main difference operators have been constructed on nonorthogonal patterns both for the plane and for a three-dimensional space. The vicinity of patterns to a circumference (in two-dimensional case) and to a sphere (in three-dimensional case) causes better dispersional characterstics of such schemes, comparing to sche-mes on usual approximations using orthogonal patterns. In this chapter only nonorthogonal patterns are considered, i.e., the ones in which at least one of axis directions is absent. Dispersional and approximation pro-perties of Laplace operators on nonorthogonal seven-point and non-or-thogonal thirteen-points patterns have been analyzed in detail. In Chapter II the convergence rates has been obtained for finite-dif-ference schemes with solutions from the Sobolev class. Elliptic and hy-perbolic equations on nonorthogonal patterns have been analyzed. We used two-dimensional discrete Fourier transforms, discrete analogue to Marcinkiewicz Theorem on multiplier, and Bramble-Hilbert Lemma to find a priori e stimates and error bounds. Error bounds for hyperbolic equation for both the grid method and the method of lines have been obtained. In Chapter III direct methods of computing elliptic and hyperbolic sche-mes on nonorthogonal seven-point pattern on the plane were con-struc-ted. Consecutively the technique of separation of variables (Fourier method), the technique of the incomplete reduction for Poisson equa-tion were used. The result has been adopted for hyperbolic equations. A simple factorization has been proposed for implicit scheme of the hyperbolic equation. Additionally, in the Chapter the stability of this factorization has been analyzed. It is proved that this factorization does not make worse dispersional properties of the finite-difference scheme on the nonorthogonal pattern. Thus, we have the following results: 1. The new class of finite-difference approxmations of the Laplace operator on rectangular grid has been constructed. 2. Dispersional and approximational properties of main schemes - on nono rthogonal seven-point pattern on the plane and on thir-teen-point pattern on 3d space on the rectangular grid have been analyzed. 3. For elliptic and hyperbolic equations the error estimates for schemes with solutions from the Sobolev space have been obtained. 4. Some computational algorithms for schemes on nonorthogonal patterns have been proposed.