Spectral Methods for Uncertainty Quantification: With Applications to Computational Fluid Dynamics
This book deals with the application of spectral methods to problems of uncertainty propagation and quanti?cation in model-based computations. It speci?cally focuses on computational and algorithmic features of these methods which are most useful in dealing with models based on partial differential equations, with special att- tion to models arising in simulations of ?uid ?ows. Implementations are illustrated through applications to elementary problems, as well as more elaborate examples selected from the authors’ interests in incompressible vortex-dominated ?ows and compressible ?ows 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 - ludes 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 ?uid dynamics. The book assumes familiarity with elementary methods for the numerical solution of time-dependent, partial differential equations; prior experience with spectral me- ods is naturally helpful though not essential. Full appreciation of elaborate examples in computational ?uid 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 suf?cient to address and reconstruct all but those highly elaborate examples.
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Adapted from 171 algorithms approach approximation space boundary conditions Boussinesq Burgers equation coefficients cold wall computational consider construction convection convergence rate corresponding cubature decomposition defined denote density function deterministic deterministic problem diffusion dimension distribution domain efficiency expansion order flow formulation Galerkin methods Galerkin projection Gaussian grid level increases integration iterations L2 norm linear mean methods multi-index Navier-Stokes equations Newton NISP nodes nonlinear norm obtained orthogonal parameters particle PC expansion plotted in Fig polynomial chaos preconditioner predictions probability density function probability space quadrature random variables refinement representation residual sampling scheme second-order Sect simulation solved solver sparse grid spatial spectral problem standard deviation steady stochastic approximation stochastic basis stochastic discretization stochastic modes stochastic problem stochastic solution strategy temperature field tensor tion truncation uncertainty values variance velocity field viscosity vorticity wavelet