This book deals with the application of spectral methods to problems of uncertainty

propagation and quantification in model-based computations. It specifically focuses

on computational and algorithmic features of these methods which are most useful

in dealing with models based on partial differential equations, with special attention

to models arising in simulations of fluid flows. Implementations are illustrated

through applications to elementary problems, as well as more elaborate examples

selected from the authors’ interests in incompressible vortex-dominated flows and

compressible flows at low Mach numbers.

Spectral stochastic methods are probabilistic in nature, and are consequently

rooted in the rich mathematical foundation associated with probability and measure

spaces. Despite the authors’ fascination with this foundation, the discussion only alludes

to those theoretical aspects needed to set the stage for subsequent applications.

The book is authored by practitioners, and is primarily intended for researchers or

graduate students in computational mathematics, physics, or fluid dynamics. The

book assumes familiarity with elementary methods for the numerical solution of

time-dependent, partial differential equations; prior experience with spectral methods

is naturally helpful though not essential. Full appreciation of elaborate examples

in computational fluid dynamics (CFD) would require familiarity with key, and in

some cases delicate, features of the associated numerical methods. Besides these

shortcomings, our aim is to treat algorithmic and computational aspects of spectral

stochastic methods with details sufficient to address and reconstruct all but those

highly elaborate examples.