A Multistage Wiener Chaos Expansion Method for Stochastic Advection-Diffusion-Reaction Equations

Abstract

Using Wiener chaos expansion (WCE), we develop numerical algorithms for solving second-order linear parabolic stochastic partial differential equations (SPDEs). We propose a deterministic WCE-based algorithm for computing moments of the SPDE solutions without any use of the Monte Carlo technique. We also compare the proposed deterministic algorithm with two other numerical methods based on the Monte Carlo technique and demonstrate that the new method is more efficient for highly accurate solutions. Numerical tests verify that the scheme is of mean-square order) for diffusion and for diffusion-reaction SPDEs with constant or variable coefficients, where Δ is the time step, and N is the Wiener chaos order. Introduction.In this paper we develop a new numerical method, based on nonlinear filtering ideas and spectral expansions, for advection-diffusion-reaction equations perturbed by random fluctuations, which form a broad class of second-order linear parabolic stochastic differential equations (SPDEs). The standard approach to constructing SPDE solvers starts with a space discretization of an SPDE, for which spectral methods (see, e.g., [4,10,14]), finite element methods (see, e.g., [1,8,32]) or spatial finite differences (see, e.g., [1,11,30,33]) can be used. The result of such a space discretization is a large system of ordinary stochastic differential equations (SDEs) which requires time discretization to complete a numerical algorithm. In [5,6] an SPDE is first discretized in time and then a finite element or finite difference method can be applied to this semidiscretization. Other numerical approaches include those making use of splitting techniques [2,17,12], quantization [9], or an approach based on the averaging-over-characteristic formula [26,27]. In [22,19] numerical algorithms based on the Wiener chaos expansion (WCE) were introduced for solving the nonlinear filtering problem for hidden Markov models. Since then the WCE-based numerical methods have been successfully developed in a number of directions (see, e.g., [13,31]).In computing moments of SPDE solutions, the existing approaches to solving SPDEs are usually complemented by the Monte Carlo technique. Consequently, in these approaches numerical approximations of SPDE moments have two errors: numerical integration error and Monte Carlo (statistical) error. To reach a high accuracy, we have to run a very large number of independent simulations of the SPDE to reduce the Monte Carlo error. Instead, here we exploit WCE numerical methods to construct a deterministic algorithm for computing moments of the SPDE solutions without any

Numerical methods for hyperbolic SPDEs: a Wiener chaos approach

“…N = 1) may lead to very accurate approximation of the random field u. Similar results are also produced by [19] for the stochastic advection-diffusion equation.…”

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

“…Noticing the linear property and Markovian properties of the solution to (2.1), we take the solution at t k−1 as initial condition to Figure 1. Illustration of the idea of recursive multi-stage approach for long-time integration in [37]. · · · 0 ∆ 2∆ · · · · · · i∆ · · · · · · T = K∆ δt solve the solution over (t k−1 , t k ].…”

“…Remark 2.3. The complexity of this algorithm is of order M 4 but can be reduced to the order of M 2 by making full use of the sparsity of the data [37].…”

“…In random space, we will employ functional expansion methods, WCE [4,21] and SCM [38], instead of the Monte Carlo method. These functional expansion methods have no statistical errors as no random number generators are used; they have only errors from truncations of Wiener processes and functional expansions and allow efficient short-time integration of SPDEs [4,5,15,21,22,37,38].…”

“…Some numerical results of WCE for SPDEs have been presented in [37] for linear advection-diffusionreaction equations and in [15] for nonlinear SPDEs including the stochastic Burgers equation and the Navier-Stokes equations. Numerical results for SCM have also been provided in [38] for linear stochastic advection-diffusion-reaction equations and the stochastic Burgers equation.…”