This paper presents a fourth-order compact immersed interface method to solve two-dimensional Poisson equations with discontinuous solutions on arbitrary domains divided by an interface. The compact scheme only employs a nine-point stencil for each grid point on the computational domain. The new approach is based on an implicit formulation obtained from generalized Taylor series expansions, and it is constructed from a few modifications to the central finite difference near the interface. The discretization results in a linear system in which the matrix coefficients are the same as the ones for smooth solutions, and the right-hand side system is modified by adding terms known as jump contributions. These contributions are only calculated at those points where the nine-point stencil cuts the interface. However, the contribution formulas require the knowledge of Cartesian jumps up to fourth-order. In this paper, we derived them using only the principal jump conditions and the jumps coming from the known right-hand function of the Poisson equation. We present numerical experiments in two dimensions to verify the feasibility and accuracy of the proposed method. Thus, the implicit immersed interface method results in an attractive fourth-order compact scheme that is easy to be implemented and applied to arbitrary interface shapes.

本文提出了一种四阶紧凑浸入式界面方法，用于求解由界面划分的任意域上具有不连续解的二维 Poisson 方程。紧凑方案仅对计算域上的每个网格点使用九点模板。新方法基于从广义泰勒级数展开获得的隐式公式，并通过对界面附近的中心有限差分的一些修改构建而成。离散化产生一个线性系统，其中矩阵系数与平滑解的矩阵系数相同，并且通过添加称为跳跃贡献的项来修改右侧系统。这些贡献仅在九点模板切割界面的那些点处计算。然而，贡献公式需要笛卡尔跳到四阶的知识。在本文中，我们仅使用主要跳跃条件和来自泊松方程已知右手函数的跳跃来推导它们。我们提出了二维数值实验，以验证所提出方法的可行性和准确性。因此，隐式浸入式界面方法产生了一种有吸引力的四阶紧凑方案，该方案易于实现并应用于任意界面形状。