High order compact finite difference schemes for the Helmholtz equation with discontinuous coefficients

In this paper, third- and fourth-order compact finite difference schemes are proposed for solving Helmholtz equations with discontinuous media along straight interfaces in two space dimensions. To keep the compactness of the finite difference schemes and get global high order schemes, even at the interface where the wave number is discontinuous, the idea of the immersed interface method is employed. Numerical experiments are included to confirm the efficiency and accuracy of the proposed methods.