Consistent modeling of boundaries in acoustic finite-difference time-domain simulations

The finite-difference time-domain method is one of the most popular for wave propagation in the time domain. One of its advantages is the use of a structured staggered grid, which makes it simple and efficient on modern computer architectures. A drawback, however, is the difficulty in approximating oblique boundaries, having to resort to staircase approximations. In many scattering problems this means that the grid resolution required to obtain an accurate solution is much higher than what is dictated by propagation in a homogeneous material. In this paper zero boundary data are considered, first for the velocity and then the pressure. These two forms of boundary conditions model perfectly rigid and pressure-release boundaries, respectively. A simple and efficient method to consistently model curved rigid boundaries in two dimensions was developed in Tornberg and Engquist [J. Comput. Phys. 227, 6922-6943 (2008)]. Here this treatment is generalized to three dimensions. Based on the approach of this method, a technique to model pressure-release surfaces with second order accuracy and without additional restriction on the timestep is also introduced. The structure of the standard method is preserved, making it easy to use in existing solvers. The effectiveness is demonstrated in several numerical tests.