A spectral method for stochastic fractional PDEs using dynamically-orthogonal/bi-orthogonal decomposition
Zhao, Yue1,2,3; Mao, Zhiping4; Guo, Ling5; Tang, Yifa1,2; Karniadakis, George Em3
AbstractModeling uncertainty propagation in anomalous transport applications leads to formulating stochastic fractional partial differential equations (SFPDEs), which require special algorithms for obtaining satisfactory accuracy at reasonable computational complexity. Here, we consider a stochastic fractional diffusion-reaction equation and combine a Galerkin spectral method based on poly-fractonomials with the modal decomposition of the stochastic fields to formulate effective numerical methods for SFPDEs. Specifically, we employ a generalized Karhunen-Loeve (KL) expansion and proper dynamically-orthogonal/bi-orthogonal (DO/BO) constraints to derive new Galerkin formulations for the mean solution, the time-dependent spatial basis, and the stochastic time-dependent coefficients. In addition, we employ a hybrid approach to tackle the singular limit of DO and BO equations for the case of deterministic initial conditions. The DO and BO formulations are mathematically equivalent but they exhibit computationally complementary properties. In demonstration examples, we investigate the interplay between randomness and non-locality and quantify this interaction for first time. In particular, we apply generalized polynomial chaos (gPC), DO, BO, and hybrid methods (i.e., combining gPC with DO or BO) to linear problems and (combining Quasi Monte Carlo (QMC) method with DO or BO) to nonlinear problems, and we compare our results to reference solutions obtained by well-resolved QMC simulations. We find that the DO and BO methods are both accurate approaches suitable for SFPDEs, with the BO method seemingly more accurate overall. The fractional order alpha affects strongly the accuracy of the solution and hence the required number of modes for multi-dimensional problems with parametric uncertainty. Both the DO and BO methods converge fast with respect to the number of modes, and they are especially effective for nonlinear problems and long-time integration for a modest number of stochastic dimensions. With regards to temporal discretization, the mean of the solution using the backward differentiation formula (BDF3) exhibits very high accuracy. (C)& nbsp;2022 Elsevier Inc. All rights reserved.
KeywordUncertainty quantification Anomalous transport Quasi Monte Carlo simulation Generalized polynomial chaos Long-time integration Poly-fractonomials
Indexed BySCI
Funding ProjectCSC
WOS Research AreaComputer Science ; Physics
WOS SubjectComputer Science, Interdisciplinary Applications ; Physics, Mathematical
WOS IDWOS:000793700800001
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Document Type期刊论文
Corresponding AuthorMao, Zhiping; Guo, Ling
Affiliation1.Chinese Acad Sci, Acad Math & Syst Sci, ICMSEC, LSEC, Beijing 100190, Peoples R China
2.Univ China Acad Sci, Sch Math Sci, Beijing 100049, Peoples R China
3.Brown Univ, Div Appl Math, Providence, RI 02912 USA
4.Xiamen Univ, Sch Math Sci, Xiamen 361005, Peoples R China
5.Shanghai Normal Univ, Dept Math, Shanghai, Peoples R China
Recommended Citation
GB/T 7714
Zhao, Yue,Mao, Zhiping,Guo, Ling,et al. A spectral method for stochastic fractional PDEs using dynamically-orthogonal/bi-orthogonal decomposition[J]. JOURNAL OF COMPUTATIONAL PHYSICS,2022,461:17.
APA Zhao, Yue,Mao, Zhiping,Guo, Ling,Tang, Yifa,&Karniadakis, George Em.(2022).A spectral method for stochastic fractional PDEs using dynamically-orthogonal/bi-orthogonal decomposition.JOURNAL OF COMPUTATIONAL PHYSICS,461,17.
MLA Zhao, Yue,et al."A spectral method for stochastic fractional PDEs using dynamically-orthogonal/bi-orthogonal decomposition".JOURNAL OF COMPUTATIONAL PHYSICS 461(2022):17.
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