A Multilevel Cartesian Non-uniform Grid Time Domain (ML-CNGTD) Method

A novel algorithm that accelerates marching on in time-based time domain integral equation solvers by rapidly evaluating transient fields due to spatially finite and temporally bandlimited (historical and surface-bound) source constellations is described. The multi-level Cartesian non-uniform grid time domain (ML-CNGTD) algorithm accomplishes the rapid evaluation by restoring delay- and amplitude-compensated fields following their interpolation from highly sparse and non-uniform grids. In doing so, it exploits two key properties of radial and angular bandwidths of the compensated field due to a spatially finite and temporally bandlimited source constellation: (i) both the radial and angular bandwidths are directly proportional to the maximum linear dimension of the source constellation and (ii) the radial bandwidth is inversely proportional to the observer distance to the constellation center. The ML-CNGTD scheme extends the two-level spherical non-uniform grid time domain algorithm (SNGTD) into a multilevel framework. Compared to a direct multilevel interpretation of the SNGTD, the extension described here has two features of note. First, it samples delayand amplitude-compensated fields on Cartesian as opposed to spherical grids; this feature simplifies the scheme's implementation without (measurably) increasing its computational complexity. Second, it uses a modified tree construct to compute fields due to sources clustered within a fraction of the minimum wavelength; this feature extends the scheme's reach to dense/non-uniform source constellations encountered in low- or mixed-frequency applications. Irrespective of the nature of the source constellation, the ML-CNGTD permits the evaluation of instantaneous fields produced by surface-bound source constellations comprising N
_s
sources in O(N
_s
log
²
N
_s
) operations. Compared to previously developed multilevel schemes for rapidly computing transient fields produced by historical source constellations, notably the plane wave time domain algorithm, the proposed scheme is easy to implement