Solving the Helmholtz equation for general smooth geometry using simple grids

•Method of difference potentials applied to general geometry and multiple scattering.•High order accurate numerical method for boundary value problems of wave motion.•Includes transmission and scattering of acoustic and electromagnetic waves.•Highly effective finite differences for complicated geometries and boundary condition.•Curvilinear interfaces, variable coefficients using regular structured grid.

The method of difference potentials was originally proposed by Ryaben’kii, and is a generalized discrete version of the method of Calderon’s operators. It handles non-conforming curvilinear boundaries, variable coefficients, and non-standard boundary conditions while keeping the complexity of the solver at the level of a finite-difference scheme on a regular structured grid. Compact finite difference schemes enable high order accuracy on small stencils and so require no additional boundary conditions beyond those needed for the differential equation itself. Previously, we have used difference potentials combined with compact schemes for solving transmission/scattering problems in regions of a simple shape. In this paper, we generalize our previous work to incorporate smooth general shaped boundaries and interfaces, including a formulation that involves multiple scattering.