A spatially second order alternating direction implicit (ADI) method for solving three dimensional parabolic interface problems

A new matched alternating direction implicit (ADI) method is proposed in this paper for solving three-dimensional (3D) parabolic interface problems with discontinuous jumps and complex interfaces. This scheme inherits the merits of its ancestor for two-dimensional problems, while possesses several novel features, such as a non-orthogonal local coordinate system for decoupling the jump conditions, two-side estimation of tangential derivatives at an interface point, and a new Douglas-Rachford ADI formulation that minimizes the number of perturbation terms, to attack more challenging 3D problems. In time discretization, this new ADI method is found to be of first order and stable in various experiments. In space discretization, the matched ADI method achieves the second order accuracy based on simple Cartesian grids for various irregularly-shaped surfaces and spatial-temporal dependent jumps. Computationally, the matched ADI method is as efficient as the fastest implicit scheme based on the geometrical multigrid for solving 3D parabolic equations, in the sense that its complexity in each time step scales linearly with respect to the spatial degree of freedom N, i.e., O(N). Furthermore, unlike iterative methods, the ADI method is an exact or non-iterative algebraic solver which guarantees to stop after a certain number of computations for a fixed N. Therefore, the proposed matched ADI method provides a very promising tool for solving 3D parabolic interface problems.