A matched alternating direction implicit (ADI) method for solving the   heat equation with interfaces
  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. 
Shan Zhao
math.NA