Symmetrized Versions of the Seidel and Successive OverRelaxation Methods for Solving Two-Dimensional Difference Problems of Elliptic Type

Abstract

Introduction. This article is devoted to the consideration of options for symmetrization of two-layer implicit iterative methods for solving grid equations that arise when approximating boundary value problems for two-dimensional elliptic equations. These equations are included in the formulation of many problems of hydrodynamics, hydrobiology of aquatic systems, etc. Grid equations for these problems are characterized by a large number of unknowns — from 10⁶ to 10¹⁰, which leads to poor conditionality of the corresponding system of algebraic equations and, as a consequence, to a significant increase in the number of iterations, necessary to achieve the specified accuracy. The article discusses a method for reducing the number of iterations for relatively simple methods for solving grid equations, based on the procedure of symmetrized traversal of the grid region.

Materials and Methods. The methods for solving grid equations discussed in the article are based on the procedure of symmetrized traversal along the rows (or columns) of the grid area.

Results. Numerical experiments have been performed for a model problem — the Dirichlet difference problem for the Poisson equation, which demonstrate a reduction in the number of iterations compared to the basic algorithms of these methods.

Discussion and Conclusions. This work has practical significance. The developed software allows it to be used to solve specific physical problems, including as an element of a software package.