An ADI scheme for convection diffusion equation with variable coefficients on irregular domain

The alternating direction implicit (ADI) scheme is a type of well-known fast solvable scheme, which is frequently studied for time-dependent problems with constant coefficients on rectangular domains. Due to the complicated algebraic matrix structure, it is rare to study the ADI schemes for time-dependent problems with variable coefficients on irregular domains in the literature. In this paper, we study a novel ADI scheme for the well-known challenging problem: convection-dominated convection diffusion equation with variable coefficients on two-dimensional irregular domain. An upwind difference scheme is proposed for the spatial discretization, which leads to diagonally dominant spatial discretization matrices. The ADI temporal discretization reduces the two-dimension spatial problem into two one-dimension spatial sub-problems, in which the sub-problem along x-direction has a tridiagonal structure while the sub-problem along y-direction is sparse and unstructured due to the complicated domain geometry. To handle the complicated structure of the y-sub-problems, a grid adaptive permutation technique is proposed to convert the y-sub-problems into tridiagonal systems. As a result, all the one-dimension sub-problems arising from the proposed ADI schemes are diagonally dominant tridiagonal systems, which can be fast and directly solved by banded LU factorization with optimal complexity (i.e., linear complexity). It is well-known that the complicated domain geometry and the dominance of the convection terms bring challenge to numerical solution of the equation. Remarkably, both theoretical results and numerical results show that the proposed scheme is unconditionally stable with respect to the dominance of the convection term, time-space grid ratio and is flexible to the domain geometry.