一类求解一维带有不连续系数和奇异源项椭圆型方程的高精度有限差分方法

High order finite difference scheme for solving one-dimensional elliptic equations with discontinuous coefficient and singular sources

针对一维带有不连续系数和奇异源项的椭圆型方程,采用匹配界面和边界(MIB,matched interface and boundary)方法进行求解.该方法对微分方程和跳跃条件的离散是分别进行的,通过在界面附近构造虚拟点达到提高差分格式整体精度的目的,文中对Neumann边界也给出了处理办法.通过数值算例对文中构造的差分方法进行了验证,并与文献中的浸入界面方法进行了对比,数值结果证明了方法的有效性和可行性.

A matched interface and boundary method is proposed for solving onedimensional elliptic equations with a discontinuous coefficient and a singular source term. The method discretizes the differential equations and the jump conditions separately, and uses some ghost points near the interface to improve the accuracy of the scheme. The paper also discusses how to deal with the Neumann boundary conditions. Numerical experiments are presented to confirm the efficiency and accuracy of the proposed method and compared with the immersed interface method （IIM）.