Finite difference discretizations with the boundary embedded in a Cartesian grid provide an alternative to body-fitted overset-grid methods, finite-element and finite-volume methods for handling complex geometries. Embedded boundary methods, being based on structured Cartesian grids, are highly efficient both in terms of operations and required memory per degree of freedom. They are also well suited for massively parallel computers as they are easy to load balance and scale to thousands of cores. Another advantage is that the geometry can be represented in a lower dimension and only local information such as the location and the normal of the boundary are required to enforce the boundary conditions, no costly (parallel) grid generation is needed.

In this talk we describe the development of a fourth-order-accurate embedded boundary method for the wave equation with Dirichlet or Neumann boundary conditions. The method uses a compact finite- difference approximation for spatial derivatives and a modified equation approach to achieve fourth-order-accuracy in time. We describe two approaches, related to the second-order-accurate methods of Kreiss and Petersson, to impose the boundary conditions. However, unlike previously used techniques, where values are assigned by extrapolation to ghost points outside the boundary, we impose the boundary conditions by assigning values via interpolation to "ghost points" just inside the boundary. Our new approach is more accurate and removes the small cell stiffness problem as the boundary is always well separated from the point inside the ghost point.