Wiener Chaos Versus Stochastic Collocation Methods for Linear Advection-Diffusion-Reaction Equations with Multiplicative White Noise

Abstract

We compare Wiener chaos and stochastic collocation methods for linear advection-reactiondiffusion 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 ∆ (time step size) in the second-order moments while the recursive multi-stage Wiener chaos method is of order ∆ N + ∆ 2 (N is the order of Wiener chaos) for advectiondiffusion-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. Notation. q: number of Brownian motions (noises). N: highest order of Hermite polynomial chaos. n: number of basis modes in approximating the Brownian motion. L: level of Smolyak sparse grid collocation. M: number of Fourier collocation nodes in physical space. ∆: element size (in time) for multi-element spectral approximation of Brownian motion. K: number of elements in time, which is T /∆ with T the final integration time. δt: time step size for time discretization in the time interval (0, ∆]. η(L, nq): number of sparse grid points at level L with dimension nq. † To cite this paper, use Z. Zhang, M. V. Tretyakov, B. Rozovskii, and G. E. Karniadakis. Wiener chaos vs stochastic collocation methods for linear advection-diffusion equations with multiplicative white noise. SIAM J. Numer. Anal., 53(1): 153-183, 2015.

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