Long-term behavior of polynomial chaos in stochastic flow simulations

Wan, Xiaoliang; Karniadakis, George Em

doi:10.1016/j.cma.2005.10.016

Cited by 149 publications

(111 citation statements)

References 21 publications

(46 reference statements)

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“…There are essentially two different techniques to quantify the resulting uncertainty in the solution. Non-intrusive methods [1,2,3,4,5,6,7] use multiple runs of existing deterministic codes for a particular statistical input. Standard quadrature techniques, often in combination with sparse grid techniques [8] can be used to obtain the statistics of interest.…”

Section: Introductionmentioning

confidence: 99%

Variance reduction through robust design of boundary conditions for stochastic hyperbolic systems of equations

Nordström

Wahlsten

2015

Journal of Computational Physics

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We consider a hyperbolic system in one space dimension with uncertainty in the boundary and initial data. Our aim is to show that different boundary conditions gives different convergence rates of the variance of the solution. This means that we can with the same knowledge of data get a more or less accurate description of the uncertainty in the solution. A variety of boundary conditions are compared and both analytical and numerical estimates of the variance of the solution is presented. As an application, we study the effect of this technique on a subsonic outflow boundary for the Euler equations

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Section: Introductionmentioning

confidence: 99%

Variance reduction through robust design of boundary conditions for stochastic hyperbolic systems of equations

Nordström

Wahlsten

2015

Journal of Computational Physics

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“…An incomplete list of references on gPC methods for uncertainty quantification in hyperbolic conservation laws includes [3,8,27,42,34,44] and other references therein. Although these deterministic methods show some promise, they suffer from the disadvantage that they are highly intrusive: existing codes for computing deterministic solutions of balance (conservation) laws need to be completely reconfigured for implementation of the gPC based stochastic Galerkin methods.…”

Section: Uncertainty Quantification (Uq)mentioning

confidence: 99%

Multi-level Monte Carlo Finite Volume Methods for Uncertainty Quantification in Nonlinear Systems of Balance Laws

Mishra

Schwab

Šukys

2013

Lecture Notes in Computational Science and Engineering

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“…For the approximation of the second derivative, we can either use the first derivative operator twice, or use u xx ≈ P −1 (−M +BD)u, where M + M T ≥ 0,B is given by (40), and D is a first-derivative approximation at the boundaries, i.e., 2, 3, . .…”

Section: The Initial Boundary Value Problemmentioning

confidence: 99%

“…al. investigated the advection-diffusion equation in two dimensions with random transport velocity [41], and the effect of long-term time integration of flow problems with gPC methods [40]. Xiu and Karniadakis studied the Navier-Stokes equations with various stochastic boundary conditions [47], as well as steady-state problems with random diffusivity [45].…”

Section: Introductionmentioning

confidence: 99%

On stability and monotonicity requirements of finite difference approximations of stochastic conservation laws with random viscosity

Pettersson¹,

Doostan²,

Nordström³

2013

Computer Methods in Applied Mechanics and Engineering

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The stochastic Galerkin and collocation methods are used to solve an advection-diffusion equation with uncertain and spatially varying viscosity. We investigate well-posedness, monotonicity and stability for the extended system resulting from the Galerkin projection of the advection-diffusion equation onto the stochastic basis functions. High-order summationby-parts operators and weak imposition of boundary conditions are used to prove stability of the semi-discrete system.It is essential that the eigenvalues of the resulting viscosity matrix of the stochastic Galerkin system are positive and we investigate conditions for this to hold. When the viscosity matrix is diagonalizable, stochastic Galerkin and stochastic collocation are similar in terms of computational cost, and for some cases the accuracy is higher for stochastic Galerkin provided that monotonicity requirements are met. We also investigate the total spatial operator of the semi-discretized system and its impact on the convergence to steadystate.

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Long-term behavior of polynomial chaos in stochastic flow simulations

Cited by 149 publications

References 21 publications

Variance reduction through robust design of boundary conditions for stochastic hyperbolic systems of equations

Variance reduction through robust design of boundary conditions for stochastic hyperbolic systems of equations

Multi-level Monte Carlo Finite Volume Methods for Uncertainty Quantification in Nonlinear Systems of Balance Laws

On stability and monotonicity requirements of finite difference approximations of stochastic conservation laws with random viscosity

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