A new high-order immersed interface method for solving elliptic equations with imbedded interface of discontinuity

This paper presents a new high-order immersed interface method for elliptic equations with imbedded interface of discontinuity. Compared with the original second-order immersed interface method of [R.J. LeVeque, Z. Li. The immersed interface method for elliptic equations with discontinuous coefficients and singular sources. SIAM J. Numer. Anal. 31 (1994) 1001–25], the new method achieves arbitrarily high-order accuracy for derivatives at an irregular grid point by imposing only two physical jump conditions together with a wider set of grid stencils. The new interface difference formulas are expressed in a general explicit form so that they can be applied to different multi-dimensional problems without any modification. The new interface algorithms of up to O(h) accuracy have been derived and tested on several one and twodimensional elliptic equations with imbedded interface. Compared to the standard second-order immersed interface method, the test results show that the new fourth-order immersed interface method leads to a significant improvement in accuracy of the numerical solutions. The proposed method has potential advantages in the application to two-phase flow because of its high-order accuracy and simplicity in applications. 2007 Elsevier Inc. All rights reserved.