Science and Education Publishing American Journal of Numerical Analysis 2328-7292 2 5 2014 11 17 The Immersed Interface Method for Elliptic and Parabolic Problems with Discontinuous Coefficients 152 166 EN Noufe Aljahdaly Department of Mathematics King Abduall-Aziz University AJNA2014253 10.12691/ajna-2-5-3 2014 08 28 2014 11 12 2014 11 17 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.