具有分段常数系数和界面跃变的泊松问题的高阶解法
High order solution of Poisson problems with piecewise constant coefficients and interface jumps
AI总结：
本文提出了一种快速准确的算法，利用正则笛卡尔网格解决复杂几何中的泊松问题，结合校正函数法和边界积分法处理界面和边界条件，通过高阶精度有限差分与快速傅里叶变换求解器实现高阶精度解。
摘要：
我们提出了一种快速且准确的算法，用于在复杂几何中求解泊松问题，使用正则笛卡尔网格。我们考虑了多种配置，包括具有界面的泊松问题，其中解在界面处不连续（如多流体流动中出现的类型）。该算法基于校正函数法（CFM）和边界积分法（BIM）的结合。界面和边界条件可通过边界积分方程和BIM快速准确地处理。然而，当需要在整个网格中求解时，BIM可能成本较高，例如流体流动问题。我们使用CFM来解决这个问题。BIM的解用于将问题重写为一系列在矩形域中的泊松问题，这需要BIM在界面/边界处的解。这些泊松问题涉及界面处的不连续性，而CFM可以处理此类不连续性。因此，我们使用CFM（通过有限差分和基于快速傅里叶变换的快速泊松求解器）来求解它们，以高阶精度。我们展示了该算法应用于包含复杂几何的泊松问题的2D示例，包括解不连续的情况。我们证明该算法生成的解在3阶或4阶精度下收敛，取决于边界条件和解的不连续性类型。
英文摘要：
We present a fast and accurate algorithm to solve Poisson problems in complex geometries, using regular Cartesian grids. We consider a variety of configurations, including Poisson problems with interfaces across which the solution is discontinuous (of the type arising in multi-fluid flows). The algorithm is based on a combination of the Correction Function Method (CFM) and Boundary Integral Methods (BIM). Interface and boundary conditions can be treated in a fast and accurate manner using boundary integral equations, and the associated BIM. Unfortunately, BIM can be costly when the solution is needed everywhere in a grid, e.g. fluid flow problems. We use the CFM to circumvent this issue. The solution from the BIM is used to rewrite the problem as a series of Poisson problems in rectangular domains - which requires the BIM solution at interfaces/boundaries only. These Poisson problems involve discontinuities at interfaces, of the type that the CFM can handle. Hence we use the CFM to solve them (to high order of accuracy) with finite differences and a Fast Fourier Transform based fast Poisson solver. We present 2-D examples of the algorithm applied to Poisson problems involving complex geometries, including cases in which the solution is discontinuous. We show that the algorithm produces solutions that converge with either 3rd or 4th order of accuracy, depending on the type of boundary condition and solution discontinuity.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
