An acoustic finite-difference time-domain algorithm with isotropic dispersion

The classic Yee Finite-Difference Time-Domain (FDTD) algorithm employs central differences to achieve second-order accuracy, i.e., if the spatial and temporal step sizes are reduced by a factor of n, the phase error associated with propagation through the grid will be reduced by a factor of n(2). The Yee algorithm is also second-order isotropic meaning the error as a function of the direction of propagation has a leading term which depends on the square of the discretization step sizes. An FDTD algorithm is presented here that has second-order accuracy but fourth-order isotropy. This algorithm permits a temporal step size 50% larger than that of the three-dimensional Yee algorithm. Pressure-release resonators are used to demonstrate the behavior of the algorithm and to compare it with the Yee algorithm. It is demonstrated how the increased isotropy enables post-processing of the simulation spectra to correct much of the dispersion error. The algorithm can also be optimized at a specified frequency, substantially reducing numerical errors at that design frequency. Also considered are simulations of scattering from penetrable spheres ensonified by a pulsed plane wave. Each simulation yields results at multiple frequencies which are compared to the exact solution. In general excellent agreement is obtained.