  A novel Douglas alternating direction implicit (ADI) method is proposed in
this work to solve a two-dimensional (2D) heat equation with interfaces. The
ADI scheme is a powerful finite difference method for solving parabolic
equations, due to its unconditional stability and high efficiency. However, it
suffers from a serious accuracy reduction in space for interface problems with
different materials and nonsmooth solutions. If the jumps in a function and its
derivatives are known across the interface, rigorous ADI schemes have been
successfully constructed in the literature based on the immersed interface
method (IIM) so that the spatial accuracy can be restored. Nevertheless, the
development of accurate and stable ADI methods for general parabolic interface
problems with physical interface conditions that describe jumps of a function
and its flux, remains unsolved. To overcome this difficulty, a novel tensor
product decomposition is proposed in this paper to decouple 2D jump conditions
into essentially one-dimensional (1D) ones. These 1D conditions can then be
incorporated into the ADI central difference discretization, using the matched
interface and boundary (MIB) technique. Fast algebraic solvers for perturbed
tridiagonal systems are developed to maintain the computational efficiency.
Stability analysis is conducted through eigenvalue spectrum analysis, which
numerically demonstrates the unconditional stability of the proposed ADI
method. The matched ADI scheme achieves the first order of accuracy in time and
second order of accuracy in space in all tested parabolic interface problems
with complex geometries and spatial-temporal dependent jump conditions.
