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.
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