Solution of the Dirichlet Problem by the Moving Node Method

Abstract

The method of displaced nodes is applied when solving the Dirichlet problem for the Poisson equation in a rectangular domain. By approximating the Laplace operator using moving nodes, we obtain an approximate analytical solution to the Dirichlet problem, i.e., a solution expressed as a combination of analytic functions. Further improvement of accuracy is achieved by applying the method of lines. Within this approach, spatial variables are discretized using the moving nodes method, resulting in the original boundary value problem for the Poisson equation in a two-dimensional domain being transformed into an ordinary differential equation (ODE) - a two-point boundary value problem where the independent variable becomes one of the coordinates (e.g., ????), and for another coordinate (????). Each of the obtained ODEs describes the behavior of the solution along a line parallel to one of the axes and contains boundary conditions given on the opposite sides of the rectangular domain. Thus, the combination of the method of displaced nodes with the method of lines allows us to move from a two-dimensional problem to a one-dimensional one. To verify the effectiveness of the proposed approach, test problems with analytically given solutions were considered. The results confirm that the proposed approach ensures increased solution accuracy.