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

• Main Authors: Zhang, Zhongqiang, Tretyakov, Michael V, Rozovskii, Boris, Karniadakis, George E
• Format: Journal Article
• Language: English
• Published: 14.05.2015
• Subjects: Mathematics - Numerical Analysis; Mathematics - Probability
• DOI: 10.48550/arxiv.1505.03771

SIAM J. Numer. Anal., 53(1): 153-183, 2015 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 multi-stage algorithm for long-time integration. We derive error estimates for both methods and compare their numerical performance. Numerical results confirm that the recursive multi-stage stochastic collocation method is of order $\Delta$ (time step size) in the second-order moments while the recursive multi-stage 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 non-commutative noises, both methods are of order one in the second-order moments.