A novel immersed interface method is presented which is based on a local Taylor series expansion at irregular grid points whereby numerical stability is enforced through a local stability condition. In the past, various immersed interface/boundary methods were developed by solely considering the order of the local truncation error at the irregular grid points. The numerical stability of these schemes was demonstrated in a global sense by applying either a matrix stability analysis or considering a number of different test cases. None of these schemes used a concrete local stability condition to derive the stencils at irregular grid points for advection-diffusion type equations. This paper will show the derivation of stencil coefficients at irregular grid points for the one dimensional advection-diffusion equation. The advection-diffusion equation may be viewed as a simple model equation for the incompressible Navier-Stokes equations. This paper will demonstrate that the local stability constraints can be justified in a global setting as long as the DFL-number stability limit is not reached. This paper will also present an extension of the one-dimensional immersed interface method to the two-dimensional case. In addition, the novel immersed interface method will be extended with a capability which allows for moving boundaries. Finally, several numerical sample problems as well as numerical matrix stability analysis for advection-diffusion type equations will prove the full operability of this novel immersed interface method.