Research Papers

Mitigating Gibbs Phenomena in Uncertainty Quantification With a Stochastic Spectral Method

[+] Author and Article Information
Piyush M. Tagade

Computer Simulation Group,
Samsung Advanced Institute of Technology,
Samsung R&D Institute India,
Bangalore 560 037, India
e-mail: piyush.tagade@gmail.com

Han-Lim Choi

Department of Aerospace Engineering,
291 Daehak-ro, N7-4303,
Daejeon 34141, South Korea
e-mail: hanlimc@kaist.ac.kr

1Corresponding author.

Manuscript received April 5, 2016; final manuscript received January 19, 2017; published online February 22, 2017. Editor: Ashley F. Emery.

J. Verif. Valid. Uncert 2(1), 011003 (Feb 22, 2017) (12 pages) Paper No: VVUQ-16-1011; doi: 10.1115/1.4035900 History: Received April 05, 2016; Revised January 19, 2017

The use of spectral projection-based methods for simulation of a stochastic system with discontinuous solution exhibits the Gibbs phenomenon, which is characterized by oscillations near discontinuities. This paper investigates a dynamic bi-orthogonality-based approach with appropriate postprocessing for mitigating the effects of the Gibbs phenomenon. The proposed approach uses spectral decomposition of the spatial and stochastic fields in appropriate orthogonal bases, while the dynamic orthogonality (DO) condition is used to derive the resultant closed-form evolution equations. The orthogonal decomposition of the spatial field is exploited to propose a Gegenbauer reprojection-based postprocessing approach, where the orthogonal bases in spatial dimension are reprojected on the Gegenbauer polynomials in the domain of analyticity. The resultant spectral expansion in Gegenbauer series is shown to mitigate the Gibbs phenomenon. Efficacy of the proposed method is demonstrated for simulation of a one-dimensional stochastic Burgers equation and stochastic quasi-one-dimensional flow through a convergent-divergent nozzle.

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Grahic Jump Location
Fig. 1

Time evolution of solution for mean initial condition. Solution evolves from left to right for u > 0, while from right to left for u < 0.

Grahic Jump Location
Fig. 3

Comparison of the Monte Carlo and the DBFE for solution of stochastic Burgers equation. Top row shows results without postprocessing while bottom row shows results with postprocessing. (a) and (c) show 90% confidence bound. (b) and (d) show L1(Ω) error for DBFE method.

Grahic Jump Location
Fig. 4

Comparison of (a) mean and (b) variance for Monte Carlo and the DBFE Method

Grahic Jump Location
Fig. 2

Comparison of numerical solution for the first eigenfunction, obtained by implementing the proposed DBFE method using the central difference scheme, first-order KT scheme in mean and full first-order central KT scheme. The results are obtained using the first three eigenmodes (N = 3) in bi-orthogonal expansion.

Grahic Jump Location
Fig. 5

Comparison of computational time for simulation. 10,000 samples are used for the Monte Carlo method. Third-order Hermite polynomials are used as basis for DBFE and gPC.

Grahic Jump Location
Fig. 8

Comparison of mean obtained using the Monte Carlo and the DBFE method: (a) comparison of mean velocity and pressure and (b) relative L1-error in mean

Grahic Jump Location
Fig. 6

Deterministic solution of quasi-one-dimensional nozzle flow: (a) mean nozzle area profile and (b) deterministic solution for nondimensional flow variables

Grahic Jump Location
Fig. 7

Eigenfunctions at t = 0 and t=3 s. Top row shows eigenfunctions for u1(x,t;ω), middle row shows eigenfunctions for u2(x,t;ω), while the bottom row shows the eigenfunctions for u2(x,t;ω). Left column shows first eigenfunction, middle column shows second eigenfunction, while right column shows third eigenfunction.

Grahic Jump Location
Fig. 9

Comparison of variance obtained using the Monte Carlo and the DBFE method without and with postprocessing: (a) comparison of variance in velocity and (b) comparison of variance in pressure



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