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.Mathematics subject classification: 65N06.

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In this article, we propose simplified immersed interface methods for elliptic partial/ordinary differential equations with discontinuous coefficients across interfaces that are few isolated points in 1D, and straight lines in 2D. For one-dimensional problems or two-dimensional problems with circular interfaces, we propose a conservative second-order finite difference scheme whose coefficient matrix is symmetric and definite. For two-dimensional problems with straight interfaces, we first propose a conservative first-order finite difference scheme, then use the Richardson extrapolation technique to get a second-order method. In both cases, the finite difference coefficients are almost the same as those for regular problems. Error analysis is given along with numerical example.

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In this paper, the ill-posed problem of the two-dimensional modified Helmholtz equation is investigated in a strip domain. For obtaining a stable numerical approximation solution, a mollification regularization method with the de la Vallée Poussin kernel is proposed. An error estimate between the exact solution and approximation solution is given under suitable choices of the regularization parameter. Two numerical experiments show that our procedure is effective and stable with respect to perturbations in the data.