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

TL;DR

This paper compares Wiener chaos and stochastic collocation methods for solving linear advection-reaction-diffusion equations with multiplicative white noise, analyzing their error estimates and numerical performance for different noise types.

Contribution

It provides a detailed comparison of Wiener chaos and stochastic collocation methods, including error analysis and performance for commutative and non-commutative noise cases.

Findings

Recursive stochastic collocation achieves order Δ in second moments.

Wiener chaos method achieves order Δ^N + Δ^2 for commutative noise.

Both methods are order one for non-commutative noise.

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 $\Delta$ (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.