The Immersed Interface Technique for Parabolic Problems with Mixed Boundary Conditions

A finite difference scheme is presented for a parabolic problem with mixed boundary conditions. We use an immersed interface technique to discretize the Neumann condition, and we use the Shortley-Weller approximation for the Dirichlet condition. The proof of a discrete maximum principle is given as well as the proof of convergence of the scheme. This convergence is also validated on numerical examples.