Abstract
The stochastic collocation method for solving differential equations with random inputs has gained lots of popularity in many applications, since such a scheme exhibits exponential convergence with smooth solutions in the random space. However, in some circumstance the solutions do not fulfill the smoothness requirement; thus a direct application of the method will cause poor performance and slow convergence rate due to the well known Gibbs phenomenon. To address the issue, we propose an adaptive high-order multi-element stochastic collocation scheme by incorporating a WENO (Weighted Essentially non-oscillatory) interpolation procedure and an adaptive mesh refinement (AMR) strategy. The proposed multi-element stochastic collocation scheme requires only repetitive runs of an existing deterministic solver at each interpolation point, similar to the Monte Carlo method. Furthermore, the scheme takes advantage of robustness and the high-order nature of the WENO interpolation procedure, and efficacy and efficiency of the AMR strategy. When the proposed scheme is applied to stochastic problems with non-smooth solutions, the Gibbs phenomenon is mitigated by the WENO methodology in the random space, and the errors around discontinuities in the stochastic space are significantly reduced by the AMR strategy. The numerical experiments for some benchmark stochastic problems, such as the Kraichnan-Orszag problem and Burgers’ equation with random initial conditions, demonstrate the reliability, efficiency and efficacy of the proposed scheme.
Highlights
Problems subject to uncertainty arise in many engineering [1,2], environmental and biological applications
One of the popular stochastic methodologies is generalized polynomial chaos [3], which is an extension of the standard polynomial chaos method [4]
We propose an adaptive high-order multi-element stochastic collocation method in conjunction with Weighted Essentially non-oscillatory (WENO) reconstruction methodology and adaptive mesh refinement (AMR) strategy to address the challenging issue regarding standard multi-element approaches for solving stochastic differential equations with discontinuities in the random space
Summary
Problems subject to uncertainty arise in many engineering [1,2], environmental and biological applications. Such uncertainty is mainly due to a lack of knowledge about the true value of model parameters or the random nature of the quantity of interest being studied. Uncertainty quantification (UQ) for practical problems has been drawing growing interest in recent years, in particular in developing numerical methods for stochastic computations. Advanced gPC algorithms for high-dimensional stochastic problems have been developed [5,6,7,8,9,10,11,12,13,14,15]
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.