  Numerical solution of the Poisson equation in metallic enclosures, open at
one or more ends, is important in many practical situations such as High Power
Microwave (HPM) or photo-cathode devices. It requires imposition of a suitable
boundary condition at the open end. In this paper, methods for solving the
Poisson equation are investigated for various charge densities and aspect
ratios of the open ends. It is found that a mixture of second order and third
order local asymptotic boundary condition (ABC) is best suited for large aspect
ratios while a proposed non-local matching method, based on the solution of the
Laplace equation, scores well when the aspect ratio is near unity for all
charge density variations, including ones where the centre of charge is close
to an open end or the charge density is non-localized. The two methods
complement each other and can be used in electrostatic calculations where the
computational domain needs to be terminated at the open boundaries of the
metallic enclosure.
