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(h4) accuracy have been derived and tested on several one and two-dimensional 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.