THE REES DISTINGUISHED LECTURER SERIES
Thursday November 18, 2010
3:30 – 4:30 pm
KRB 006
TITLE: Numerical simulations in science and engineering: accuracy and Efficiency
Professor of Applied & Computational Mathematics
California Institute of Technology
ABSTRACT
Numerical simulations play exceedingly important roles in industry,
science and engineering, as they impact upon areas such as fluid
dynamics, materials science, electrodynamics, geophysics, biology and
medicine amongst many others: together with theory and experiment, the
field is indeed one of the three pillars of science. Fundamental
advances on numerical analysis and computational science over the last
century have given rise to a solid and reliable set of numerical
techniques, which have enabled much progress in the field of numerical
simulation, and, consequently, in science and engineering as a
whole. Yet, it is no exaggeration to state that we have merely
scratched the surface of this important field. In this lecture we will
discuss some of the fundamental challenges arising in the area of
numerical analysis and computational science, and we will suggest a
few possible avenues of approach, which, on the basis of recent
successes, hold promise to contribute towards a highly desirable set
of capabilities, enabling accurate simulations for general, fully
realistic scientific and engineering configurations.
Friday November 19, 2010
3:30 – 4:30 pm
KRB 006
TITLE Spectral FFT-based integral and differential PDE solvers general domains
Professor of Applied & Computational Mathematics
California Institute of Technology
ABSTRACT
We present new methodologies for the numerical solution of Partial
Differential Equations (PDE) in general spatial domains. Based on a
novel Fourier-Continuation (FC) method for the resolution of the Gibbs
phenomenon, associated surface-representation methods and fast
high-order methods for evaluation of integral operators, these
methodologies give rise to fast and highly accurate frequency- and
time-domain solvers for PDEs on general three-dimensional spatial
domains. Our fast integral algorithms can solve, with high-order
accuracy, problems of electromagnetic and acoustic scattering for
complex three-dimensional geometries - including, possibly, singular
elements such as wires, corners, edges and open screens. Our
differential solvers for time-dependent PDEs, in turn, which are based
on the FC method in conjunction with both, explicit and alternating
direction time-stepping strategies, give rise to essentially spectral
behavior, free of pollution or dispersion errors. Unlike previous
alternating direction methods of order higher than one, which can only
deliver unconditional stability for rectangular domains, the present
spectral Fourier-Continuation Alternating-Direction (FC-AD) algorithm
possesses the desirable property of unconditional stability for
general domains; the computational time required by the algorithm to
advance a solution by one time-step, in turn, grows in a linear manner
with the number of spatial discretization points used. A variety of
applications to linear and nonlinear PDE problems demonstrate the very
significant improvements the new algorithms provide over the accuracy
and speed resulting from other approaches.