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 $O\left(N\right)$ 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.

这项工作提出了两个新的交替方向隐式（ADI）方案，用于解决二维和三维的抛物线界面问题。首先，文献中首次采用幻影流体方法（GFM）来处理ADI框架中的界面跳变，从而在每个ADI步骤中产生对称和三对角有限差分矩阵。所提出的GFM-ADI方案在空间和时间上均达到了第一级精度，这一点已通过涉及复杂界面形状和时空相关跃变的数值实验得到了证实。GFM-ADI方案还保持了ADI的效率—每个时间步长的计算复杂度随$Ø（ñ）$ 完全自由 $ñ$在更高的尺寸。其次，构造了新的匹配接口和边界（MIB）方案，该方案将现有的二阶MIB中的二次多项式基降级为线性基。有趣的是，所得的MIB-ADI或mADI方案在所有维度上都产生与GFM-ADI方案相同的有限差分矩阵。因此，mADI方案可以视为GFM的改进，因为它可以计算出GFM中省略的切向跳跃。因此，在保持相同的计算效率的同时，当前的mADI方案在所有数值示例中始终比GFM-ADI更加准确。但是，由于切向跳跃近似，mADI方案是半隐式的，而GFM-ADI方案是完全隐式的，没有切向校正。从而，