A numerical scheme for multi-point special boundary-value problems and application to fluid flow through porous layers

In this paper we propose a numerical scheme based on finite differences for the numerical solution of nonlinear multi-point boundary-value problems over adjacent domains. In each subdomain the solution is governed by a different equation. The solutions are required to be smooth across the interface nodes. The approach is based on using finite difference approximation of the derivatives at the interface nodes. Smoothness across the interface nodes is imposed to produce an algebraic system of nonlinear equations. A modified multi-dimensional Newton's method is proposed for solving the nonlinear system. The accuracy of the proposed scheme is validated by examples whose exact solutions are known. The proposed scheme is applied to solve for the velocity profile of fluid flow through multilayer porous media.