一种求解有交界面的椭圆型方程的隐式间断条件差分格式

A finite Difference Scheme for Elliptic Interface Problem with Implicit Jump Conditions

本文将文献中的求解二维的有交界面的椭圆型方程的浸入界面方法推广到界面及间断条件都由定义在界面某个邻域的网格函数点上的函数隐式提供的情形，给出了一种间断条件捕捉格式。它特别适合干隐式界面跟踪法如水平集方法。对原浸入界面方法中的界面间断关系，确定不规则点差分格式的系数的代数方程组和修正项都针对新的情形进行了相应的修正。该格式利用标准的二阶拉格朗日插值计算间断函数沿界面的导数，避免了文献中的用样条函数的局部界面重构，易于执行。数值计算验证了该法的关于最大模的二阶收敛性。

In this paper, the immersed interface method in the literature for solving elliptic interface problems in 2D was extended to the situation when the interface and jumps were implicitly given by the data on the grid points in a neighborhood of the interface. It was convenient when the interface was implicitly captured by level-set function. The formulations of the original immersed interface method have been modified for this implicit setting, including the interface jump relations, the algebraic equations to determine the coefficients of the finite difference scheme at irregular grid points and the correction terms. Particularly, the interface quantities and their derivatives along the interface were calculated by using standard quadratic Lagrange interpolation, avoiding the interface reconstruction by means of a spline function, and leading to relatively easy implementation. Second order accuracy was achieved in maximum norm, as demonstrated by numerical examples