We present the new method for computation of a gravitational potential for isolated systems in cylindrical coordinates. This method solves the main difficulty arising when treating isolated systems: in order to correctly state the Dirichlet problem for the Poisson equation at the boundary of a finite computational domain, one must provide the boundary values of a gravitational potential which are unknown.
To develop this method, we adapt the ideas of the convolution method and the James algorithm to the Cartesian coordinates and rectangular computational domains. To solve the Dirichlet problem for the Poisson equation, we use a finite difference 7-point stencil. System of linear equations obtained after a difference approximation is solved by means of Fast Fourier transform applied to the az-imuthal coordinate, Fast sine transform for the vertical coordinate and 3-diagonal elimination to determine the radial component of potential.