A matched Peaceman-Rachford ADI method for solving parabolic interface problems

A new Peaceman-Rachford alternating direction implicit (PR-ADI) method is proposed in this work for solving two-dimensional (2D) parabolic interface problems with discontinuous solutions. The classical ADI schemes are known to be inaccurate for handling interfaces. This motivates the development of a matched Douglas ADI (D-ADI) method in the literature, in which the finite difference is locally corrected according to the jump conditions. However, the unconditional stability of the matched ADI method cannot be maintained if the D-ADI is simply replaced by the PR-ADI. To stabilize the computation, the tangential derivative approximations in the jump conditions decomposition are substantially improved in this paper. Moreover, a new temporal discretization is adopted for formulating the PR-ADI method, which involves less perturbation terms. Stability analysis is conducted through eigenvalue spectrum analysis, which demonstrates the unconditional stability of the proposed method. The matched PR-ADI method achieves second order of accuracy in space in all tested problems with complex geometries and jumps, while maintaining the efficiency of the ADI. The proposed PR-ADI method is found to be more accurate than the D-ADI method in time integration, even though its formal temporal order is limited in the matched ADI framework.