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We compare Wiener chaos and stochastic collocation methods for linearadvection-reaction-diffusion equations with multiplicative white noise. Bothmethods are constructed based on a recursive multi-stage algorithm forlong-time integration. We derive error estimates for both methods and comparetheir numerical performance. Numerical results confirm that the recursivemulti-stage stochastic collocation method is of order $\Delta$ (time step size)in the second-order moments while the recursive multi-stage Wiener chaos methodis of order $\Delta^{\mathsf{N}}+\Delta^2$ ($\mathsf{N}$ is the order of Wienerchaos) for advection-diffusion-reaction equations with commutative noises, inagreement with the theoretical error estimates. However, for non-commutativenoises, both methods are of order one in the second-order moments.

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