Decimation in Time and Space of Finite-Difference Time-Domain Schemes: Standard Isotropic Lossless Model
Finite-difference time-domain (FDTD) schemes permit changes in the grid density on selected regions of the wave propagation domain, which can reduce the computational load of the simulations. One possible alternative to varying the spatial density is to change the simulation temporal rate. This idea looks attractive when the wave signals exhibit pronounced bandwidth fluctuations across time. This is particularly true in sound synthesis, where a physically based acoustic resonator can be conveniently modeled using such schemes. To overcome the computational constraints that must be met by real-time distributed resonator models, this paper deals with the decimation in time and space of isotropic lossless finite-difference time-domain schemes holding conventional Nyquist-Shannon limits on the bandwidth of the wave signals. Formulas for the reconstruction of these signals at runtime over the interpolated grid are provided for both the 1D and 2D orthogonal case, depending on the ideal boundary conditions (either Neumann or Dirichlet) holding at each side of the grid in connection with the domain side lengths (either even or odd). Together, the boundaries and size determine the type of Discrete Cosine Transform used in the corresponding interpolation formula. Numerical artifacts arising as a consequence of decimating in space in 2D are discussed in terms of dispersion error and aliasing. Considerations concerning the temporal reconstruction of components lying at the decimated Nyquist frequency are addressed in the conclusion.