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

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 thei...

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Bibliographic Details
Published inarXiv.org
Main Authors Zhang, Zhongqiang, Tretyakov, Michael V, Rozovskii, Boris, Karniadakis, George E
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 14.05.2015
Subjects
Advection
Advection-diffusion equation
Algorithms
Collocation methods
Diffusion
Mathematical analysis
Numerical methods
Reaction-diffusion equations
Recursive methods
Time integration
White noise
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Summary: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.
ISSN:2331-8422