Immersed Interface Methods (IIM) arise as a very effective tool to solve many interface problems encountered in fluid dynamics, mechanics and other related fields of study. Despite their versatility and potential, IIM-inspired techniques impose as constraints different types of jump conditions in order to be mathematically tractable and usable in practice. To cope with this issue, in this paper we introduce a novel Immersed Interface method for solving Poisson equations with discontinuous coefficients on Cartesian grids. Different from most conventional methods which assume some derivative information at the interface to produce a valid approximation, our approach reduces the number of regular constraints when solving the Poisson problem, requiring to be given only the ordinary jumps of the function. We combine Finite Difference schemes, ghost node strategy, correction formulas, and interpolation rules into a unified and stable numerical model. Moreover, the present method is capable of producing high-order solutions from a unique resource of available data. We attest to the accuracy and robustness of our single jump-based method through a variety of numerical experiments comprising Poisson problems with interfaces that can be now solved from a reduced number of jump conditions.

沉浸式界面方法（IIM）作为解决流体动力学，力学和其他相关研究领域中遇到的许多界面问题的一种非常有效的工具而出现。尽管具有通用性和潜力，但受IIM启发的技术仍将不同类型的跳跃条件作为约束条件，以使其在数学上易于处理并在实践中可用。为了解决这个问题，在本文中，我们介绍了一种新颖的浸入式界面方法，用于在笛卡尔网格上求解具有不连续系数的泊松方程。与大多数常规方法不同，大多数常规方法在接口处假定一些导数信息以产生有效的近似值，而在解决泊松问题时，我们的方法减少了常规约束的数量，只需要给出函数的普通跳转即可。我们结合了有限差分方案，重影节点策略，校正公式和插值规则成为统一而稳定的数值模型。而且，本方法能够从可用数据的唯一资源产生高阶解。我们通过各种数值实验证明了我们基于单个跳转的方法的准确性和鲁棒性，这些数值实验包括带有接口的泊松问题，现在可以通过减少跳转条件来解决这些问题。