Let Γ be a smooth curve inside a two-dimensional rectangular region Ω. In this paper, we consider the Poisson interface problem $-{\mathrm{\nabla }}^{2}u=f$ in $\mathrm{\Omega }\setminus \mathrm{\Gamma }$ with Dirichlet boundary condition such that f is smooth in $\mathrm{\Omega }\setminus \mathrm{\Gamma }$ and the jump functions $[u]$ and $[\mathrm{\nabla }u\cdot \overrightarrow{n}]$ across Γ are smooth along Γ. This Poisson interface problem includes the weak solution of $-{\mathrm{\nabla }}^{2}u=f+g{\delta }_{\mathrm{\Gamma }}$ in Ω as a special case. Because the source term f is possibly discontinuous across the interface curve Γ and contains a delta function singularity along the curve Γ, both the solution u of the Poisson interface problem and its flux $\mathrm{\nabla }u\cdot \overrightarrow{n}$ are often discontinuous across the interface. To solve the Poisson interface problem with singular sources, in this paper we propose a sixth order compact finite difference scheme on uniform Cartesian grids. Our proposed compact finite difference scheme with explicitly given stencils extends the immersed interface method (IIM) to the highest possible accuracy order six for compact finite difference schemes on uniform Cartesian grids, but without the need to change coordinates into the local coordinates as in most papers on IIM in the literature. Also in contrast with most published papers on IIM, we explicitly provide the formulas for all involved stencils and therefore, our proposed scheme can be easily implemented and is of interest to practitioners dealing with Poisson interface problems. Note that the curve Γ splits Ω into two disjoint subregions ${\mathrm{\Omega }}^{+}$ and ${\mathrm{\Omega }}^{-}$. The coefficient matrix A in the resulting linear system $Ax=b$, following from the proposed scheme, is independent of any source term f, jump condition $g{\delta }_{\mathrm{\Gamma }}$, interface curve Γ and Dirichlet boundary conditions, while only b depends on these factors and is explicitly given, according to the configuration of the nine stencil points in ${\mathrm{\Omega }}^{+}$ or ${\mathrm{\Omega }}^{-}$. The constant coefficient matrix A facilitates the parallel implementation of the algorithm in case of a large size matrix and only requires the update of the right hand side vector b for different Poisson interface problems. Due to the flexibility and explicitness of the proposed scheme, it can be generalized to obtain the highest order compact finite difference scheme for non-uniform grids as well. We prove the order 6 convergence for the proposed scheme using the discrete maximum principle. Our numerical experiments confirm the sixth accuracy order of the proposed compact finite difference scheme on uniform meshes for the Poisson interface problems with various singular sources.

设Γ是二维矩形区域Ω内的平滑曲线。在本文中，我们考虑泊松接口问题$-{\mathrm{\nabla }}^2你=F$ 在 $\mathrm{\Omega }\setminus \mathrm{\Gamma }$具有狄利克雷边界条件，使得f在$\mathrm{\Omega }\setminus \mathrm{\Gamma }$ 和跳转功能 $[你]$ 和 $[\mathrm{\nabla }你\cdot \overrightarrow{n}]$Γ 上的 Γ 是平滑的。这个泊松接口问题包括弱解$-{\mathrm{\nabla }}^2你=F+G{\delta }_{\mathrm{\Gamma }}$在Ω中作为特例。因为源项f可能在界面曲线 Γ 上不连续，并且包含沿曲线 Γ 的 delta 函数奇点，泊松界面问题的解u及其通量$\mathrm{\nabla }你\cdot \overrightarrow{n}$在整个界面上通常是不连续的。为了解决奇异源的泊松接口问题，本文提出了均匀笛卡尔网格上的六阶紧致有限差分格式。我们提出的具有明确给定模板的紧凑有限差分方案将浸入式界面方法 (IIM) 扩展到最高可能的精度六阶，用于均匀笛卡尔网格上的紧凑有限差分方案，但无需像大多数论文中那样将坐标更改为局部坐标关于文献中的 IIM。与大多数已发表的 IIM 论文相比，我们明确提供了所有涉及的模板的公式，因此，我们提出的方案可以轻松实现，并且对处理泊松接口问题的从业者感兴趣。${\mathrm{\Omega }}^{+}$ 和 ${\mathrm{\Omega }}^{-}$. 所得线性系统中的系数矩阵A$\mathrm{一种}X=乙$，根据提议的方案，独立于任何源项f，跳跃条件$G{\delta }_{\mathrm{\Gamma }}$, 界面曲线 Γ 和 Dirichlet 边界条件，而只有b取决于这些因素并明确给出，根据 9 个模板点的配置${\mathrm{\Omega }}^{+}$ 要么 ${\mathrm{\Omega }}^{-}$. 在大矩阵的情况下，常系数矩阵A 有利于算法的并行实现，并且对于不同的泊松接口问题只需要更新右侧向量b。由于所提出方案的灵活性和明确性，它也可以推广到非均匀网格的最高阶紧致有限差分方案。我们使用离散最大值原理证明了所提出方案的 6 阶收敛性。我们的数值实验证实了针对具有各种奇异源的泊松接口问题的均匀网格上所提出的紧凑有限差分方案的六阶精度。