Numerical errors and accuracy-efficiency tradeoff in frequency and time-domain integral equation solvers
This thesis presents a detailed study of the numerical errors and the associated accuracy-efficiency tradeoffs encountered in the solution of frequency- and time-domain integral equations. For frequency-domain integral equations, the potential integrals contain singular Green’s function kernels and the resulting singular and near-singular integrals must be carefully evaluated, using singularity extraction or cancellation techniques, to ensure the accuracy of the method-of-moments impedance matrix elements. This thesis presents a practical approach based on the progressive Gauss-Patterson quadrature rules for implementing the radial-angular-transform singularity-cancellation method such that all singular and near-singular integrals are evaluated to an arbitrary pre-specified accuracy. Numerical results for various scattering problems in the high- and low-frequency regimes are presented to quantify the efficiency of the method and contrast it to the singularity extraction method. For time-domain integral equations, the singular Green’s function kernels are functions of space and time and sub-domain temporal basis functions rather than entire-domain sinusoidal/Fourier basis functions are used to represent the time variation of currents/fields. This thesis also investigates the accuracy-efficiency tradeoff encountered when choosing sub-domain temporal basis functions by contrasting two prototypical ones: The causal piecewise polynomial interpolatory functions, sometimes called shifted Lagrange interpolants, and the band-limited interpolatory functions based on approximate prolate spheroidal wave functions. It is observed that the former is more efficient for low to moderate accuracy levels and the latter achieves higher, but extrapolation-limited, accuracy levels.