A numerical algorithm for solving Laplace and Poisson type partial differential equations on a uniform rectangular mesh with Dirichlet and Neumann boundary conditions

摘要

In this paper a numerical algorithm is described for solving the boundary value problem associated with Laplace and Poisson type equations with Dirichlet, Neumann and Robin boundary conditions. An analytic treatment of the governing partial differential equation is undertaken to illustrate the unsuitability of attempting a solution using the technique of separation of variables. Exact solutions are set up and the algorithm is compared against this exact solution determined on a unit square. The technique used was found to be in good agreement with the exact solutions that were considered. The governing partial differential equation arises naturally when considering flow problems when the physical plane is mapped onto an infinite rectangle with the mutually orthogonal coordinates being the stream function Ψ(x, y) and the velocity potential function Φ(x, y) as described in Pavlika.