Wiener chaos versus stochastic collocation methods for linear advection-diffusion-reaction equations with multiplicative white noise

Zhang, Zhongqiang; Tretyakov, Mikhail; Rozovskii, Boris; Karniadakis, George E.

Abstract

We compare Wiener chaos and stochastic collocation methods for linear advection-reaction-diffusion equations with multiplicative white noise. Both methods are constructed based on a recursive multistage algorithm for long-time integration. We derive error estimates for both methods and compare their numerical performance. Numerical results confirm that the recursive multistage stochastic collocation method is of order $\Delta$ (time step size) in the second-order moments while the recursive multistage Wiener chaos method is of order $\Delta^{\mathsf{N}}+\Delta^2$ ($\mathsf{N}$ is the order of Wiener chaos) for advection-diffusion-reaction equations with commutative noises, in agreement with the theoretical error estimates. However, for noncommutative noises, both methods are of order one in the second-order moments.

Citation

Zhang, Z., Tretyakov, M., Rozovskii, B., & Karniadakis, G. E. (2015). Wiener chaos versus stochastic collocation methods for linear advection-diffusion-reaction equations with multiplicative white noise. SIAM Journal on Numerical Analysis, 53(1), doi:10.1137/130932156

• Journal Article Type: Article
• Acceptance Date: Nov 5, 2014
• Online Publication Date: Jan 8, 2015
• Publication Date: Jan 8, 2015
• Deposit Date: Mar 8, 2018
• Print ISSN: 0036-1429
• Publisher: Society for Industrial and Applied Mathematics
• Peer Reviewed: Peer Reviewed
• Volume: 53
• Issue: 1
• Article Number: 153-183
• DOI: https://doi.org/10.1137/130932156
• Public URL: http://dx.doi.org/10.1137/130932156
• Publisher URL: https://epubs.siam.org/doi/abs/10.1137/130932156