KELLER Colloquium in Computing & Mathematical Sciences

We present fast spectral solvers that address some of the main difficulties associated with the simulation of realistic scientific and engineering problems in the frequency- and time-domain. Based on fast high-order methods for evaluation of integral operators and new approaches for accurate Fourier expansion of non-periodic functions, these algorithms can solve, with high-order accuracy and without numerical dispersion, problems of electromagnetic and acoustic propagation and scattering for large and complex three-dimensional geometries such as aircraft, submarines, and ships; they can be applied to challenging fluid-dynamic and seismic problems; and they can be used to design and optimize optical and photonic devices of large electrical size. A variety of applications will be presented which demonstrate the significant capabilities inherent in the new methods, as well as the improvements these algorithms can provide, over other methods, in generality, accuracy, and speed.