Alternating direction ghost-fluid methods for solving the heat equation with interfaces

This work presents two new alternating direction implicit (ADI) schemes for solving parabolic interface problems in two and three dimensions. First, the ghost fluid method (GFM) is adopted for the first time in the literature to treat interface jumps in the ADI framework, which results in symmetric and tridiagonal finite difference matrices in each ADI step. The proposed GFM-ADI scheme achieves a first order of accuracy in both space and time, as confirmed by numerical experiments involving complex interface shapes and spatial–temporal dependent jumps. The GFM-ADI scheme also maintains the ADI efficiency — the computational complexity for each time step scales as O ( N ) for a total degree of freedom N in higher dimensions. Second, a new matched interface and boundary (MIB) scheme is constructed, which downgrades the quadratic polynomial bases in the existing second order MIB to linear ones. Interestingly, the resulting MIB-ADI or mADI scheme produces the same finite difference matrices as the GFM-ADI scheme in all dimensions. Hence, the mADI scheme can be regarded as an improvement of the GFM, because it calculates tangential jumps which are omitted in the GFM. Consequently, the present mADI scheme is constantly more accurate than the GFM-ADI in all numerical examples, while keeping the same computational efficiency. Nevertheless, the mADI scheme is semi-implicit due to tangential jump approximations, while the GFM-ADI scheme is fully implicit without tangential corrections. Thus, the GFM-ADI scheme could be more stable than the mADI scheme when a huge contrast is presented in diffusion coefficients.