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Abstract

Many physical processes are described by the Helmholtz equation. By adding the Dirichlet, Neumann, or Robin boundary conditions, the corresponding problems are obtained. An eective method for solving such boundary problems is the nite dierence method. In this paper, we consider a Dirichlet problem for the Helmholtz equation. A nite dierence approximate scheme of the higher-order was constructed. For rst- and second-order derivatives the fourth-order nite dierence approximations are used. By adding boundary conditions at the grid nodes, we obtain a system of dierence equations (a system of linear algebraic equations) with a symmetric matrix that has a diagonal advantage. Therefore, it is advisable to solve this system by iterative methods. In the paper, the simple iterations method and Seidel method are used. They give an approximate solution with a given accuracy by a small number of iterations. A test problem with a constant wave number and a known exact solution is considered. The dierence equation and the iterative formulas of the simple iterations method and Seidel method for this problem are given. The results of numerical experiments, which conrm the eciency of the method and the theoretical order of convergence, are presented. The calculations are performed under dierent values of the wave number. Iterative processes are compared by the number of iterations and the values of absolute errors.