The use of the line-by-line recurrent method for solving systems of difference elliptic equations with nine-diagonal matrices

The applying of the line-by-line recurrent method for solving systems of difference elliptic equations with nine-diagonal matrices is the subject of the article. Such matrices take place in the case of difference approximation of 2D differential problems of a higher order of accuracy on a regular grid covering the area under consideration. The technology of the so-called compensatory transform which allows replacing the initial nine-diagonal matrix of the system with the five-diagonal one is offered in the article, due to the fact that originally the line-by-line recurrent method was designed for solving systems of difference equations with a five-diagonal matrix. The efficiency of this technology is analyzed by comparing the solutions of the test boundary value problem in a unit square. The solutions are found both with the help of different implementations of the compensatory transform technology and by other modern highly efficient iterative methods for solving the systems of difference equations. The problem is solved on the sequence of grids from coarse (501$\times$501) to fine (4001$\times$4001) nodes. The accuracy of the solution convergence is determined by the relative norm of the residual, which is equal to $10^{-12}$ in the present work. It is shown that the line-by-line recurrent method retains its high efficiency over the entire range of the grids under consideration despite the use of the intermediate technology of the compensatory transform.