The explicit-jump immersed interface method: Finite difference methods forPDEs with piecewise smooth solutions

Many boundary value problems (BVPs) or initial BVPs have nonsmooth solution s, with jumps along lower-dimensional interfaces. The explicit-jump immerse d interface method (EJIIM) was developed following Li's fast iterative imme rsed interface method (FIIIM), recognizing that the foundation for the effi cient solution of many such problems is a good solver for elliptic BVPs. EJ IIM generalizes the class of problems for which FIIIM is applicable. It han dles interfaces between constant and variable coefficients and extends the immersed interface method (IIM) to BVPs on irregular domains with Neumann a nd Dirichlet boundary conditions. Proofs of second order convergence for a one-dimensional (1D) problem with piecewise constant coefficients and for t wo-dimensional (2D) problems with singular sources are given. Other problem s are reduced to the singular sources case, with additional equations deter mining the source strengths. The advantages of EJIIM are high quality of so lutions even on coarse grids and easy adaptation to many problems with comp licated geometries, while still maintaining the efficiency of the FIIIM.