eng Science and Education Publishing American Journal of Numerical Analysis 2328-7292 2014-11-17 2 5 152 166 10.12691/ajna-2-5-3 AJNA2014253 article Noufe Aljahdaly nhaljahdaly@kau.edu.sa 1 Department of Mathematics King Abduall-Aziz University In this paper we consider numerical methods for solving elliptic as well as time dependent advection- diffusion-reaction (ADR) equations in one spatial dimension. We consider the case in which the difference diffusion coefficients as well as advection coefficients and reaction coefficients are discontinuous across a fixed interface. Using the immersed interface method (IIM) for finite difference approximations, we demonstrate how to modify numerical methods constructed for the constant coefficient case around interfaces of discontinuity of the diffusion, advection, and reaction coefficient. http://pubs.sciepub.com/ajna/2/5/3/ajna-2-5-3.pdf advection diffusion reaction immersed interface