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arXiv · 1505.03771

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

Zhongqiang Zhang·Michael V. Tretyakov·Boris Rozovskii·George E. Karniadakis
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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 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 $Δ$ (time step size) in the second-order moments while the recursive multi-stage Wiener chaos method is of order $Δ^{\mathsf{N}}+Δ^2$ ($\mathsf{N}$ 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.

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Zhongqiang Zhang, Michael V. Tretyakov, Boris Rozovskii, George E. Karniadakis. 2015-05-14. Wiener chaos vs stochastic collocation methods for linear advection-diffusion equations with multiplicative white noise. https://arxiv.org/abs/1505.03771

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