A coupling interface method for elliptic interface problems
A coupling interface method for elliptic interface problems
复制标题
DOI：
10.1016/j.jcp.2007.03.012
复制
发表时间：
2007-08-10
影响因子：
4.1
通讯作者：
Shu, Yu-Chen
中科院分区：
文献类型：
--
作者：
Chern, I-Liang;Shu, Yu-Chen
本文提出了一种笛卡尔网格下的耦合界面法（CIM），用于求解任意维的椭圆型复界面问题，其中系数、源项和解在界面上可以是间断的或奇异的。它由一阶形式（CIM 1）和二阶形式（CIM 2）组成，在一维中，CIM 1由界面两侧的线性近似导出。该方法通过逐维方法扩展到高维。为了连接来自每个维度的信息，通过每个坐标方向上的跳跃条件导出一阶导数的耦合方程。生成的模具在二维中使用标准的5个网格点，在三维中使用7个网格点。类似地，CIM 2是从每个维度的二次近似导出的。在高维情况下，通过每个坐标方向上的跳跃条件，导出了主二阶导数u（xkxk）的耦合方程。交叉导数由单侧插值近似。这种方法减少了单侧插值所需的网格点数量。所得到的模板在二维中包含8个网格点，在三维中包含12-14个网格点，并对一维中所得到的CIM 2线性系统的条件数进行了数值研究.它示出的条件数具有相同的行为的离散拉普拉斯算子，独立的界面在网格单元中的相对位置。进一步，我们还证明了当界面曲率K满足kappa h时，耦合方程的可解性
We propose a coupling interface method (CIM) under Cartesian grid for solving elliptic complex interface problems in arbitrary dimensions, where the coefficients, the source terms, and the solutions may be discontinuous or singular across the interfaces. It consists of a first-order version (CIM1) and a second-order version (CIM2).In one dimension, the CIM1 is derived from a linear approximation on both sides of the interface. The method is extended to high dimensions through a dimension-by-dimension approach. To connect information from each dimension, a coupled equation for the first-order derivatives is derived through the jump conditions in each coordinate direction. The resulting stencil uses the standard 5 grid points in two dimensions and 7 grid points in three dimensions. Similarly, the CIM2 is derived from a quadratic approximation in each dimension. In high dimensions, a coupled equation for the principal second-order derivatives u(xkxk) is derived through the jump conditions in each coordinate direction. The cross derivatives are approximated by one-side interpolation. This approach reduces the number of grid points needed for one-side interpolation. The resulting stencil involves 8 grid points in two dimensions and 12-14 grid points in three dimensions.A numerical study for the condition number of the resulting linear system of the CIM2 in one dimension has been performed. It is shown that the condition number has the same behavior as that of the discrete Laplacian, independent of the relative location of the interface in a grid cell. Further, we also give a proof of the solvability of the coupling equations, provided the curvature K of the interface satisfies kappa h