Spectral collocation method for stochastic partial differential equations with fractional Brownian motion

Abstract

In this paper, we consider the numerical approximation of stochastic partial differential equations driven by infinite dimensional fractional Brownian motion with Hurst index $H>\frac{1}{2}$. A Fourier spectral collocation approximation is used in space and semi-implicit Euler method is applied for the temporal approximation. Our aim is to investigate the convergence of the proposed method. Optimal strong convergence error estimates in mean-square sense are derived and numerical experiments are presented and confirm theoretical results.

Introduction

Nowadays, stochastic partial differential equations (SPDEs) are used to model various different phenomena in physics, engineering, economics, biology and fluid mechanics. Hence, the extensive applications of random effects in describing practical sciences show the importance of the theory of SPDEs. Since most of SPDEs do not have the exact analytical solution, numerical analysis of these equations is currently an active area of research and due to their application in describing uncertainties, there are a large number of literatures on numerical methods for SPDEs driven by standard Wiener process, see, for instance [1], [2], [3], [4], [5], [6]. On the other hand, since fractional Brownian motion provides better models than Markovian ones in some applicable sciences, it has been an important subject of interest of many researchers, see, [7]. Very recently, SPDEs driven by infinite dimensional fractional Brownian motion is also a growing research area in probability theory, see, for example [8], [9]. In contrast to the extensive studies on fractional Brownian motions, numerical methods for SPDEs driven by fractional Brownian motion do not appear so often in the literature, see [10], [11]. Indexed by Hurst index $H\in (0,1)$, there are two principle difficulties in investigating SPDEs driven by fractional Brownian motion with $H\ne \frac{1}{2}$ in comparison with the standard Brownian setting, the lack of martingale properties and the correlation between increments. From this point of view, the fundamental stochastic calculus and the classical stochastic integration are invalid in numerical analysis of equations driven by these processes.

In [12], collocation method was already considered for the numerical approximation of SPDEs with nonlinear multiplicative trace class noise. The main idea of this article is to propose a spectral collocation method for solving numerical solution of SPDEs driven by infinite dimensional fractional Brownian motions.

Given a real separable Hilbert space $(\mathcal{H},〈\cdot ,\cdot 〉,\Vert \cdot \Vert )$ and let $A:\mathcal{D}\left(A\right)\subset \mathcal{H}\to \mathcal{H}$ be a positive self-adjoint, linear unbounded operator with a compact inverse, densely defined in $\mathcal{D}\left(A\right)\subset \mathcal{H}$. Let $(\Omega ,\mathcal{F},{\left\{{\mathcal{F}}_t\right\}}_{t\ge o},\mathbb{P})$ be a filtered probability space and ${\left\{W^H\left(t\right)\right\}}_{t\in [0,t]}$ be a cylindrical Brownian motion with Hurst index $H>\frac{1}{2}$. With these settings, we will then consider the following equation

$$du\left(t\right)+\left(Au\left(t\right)+F\left(u\left(t\right)\right)\right)dt=\Psi d{W}^H\left(t\right),t\in [0,T],$$$$u\left(0\right)=\xi ,$$

where $F:\mathcal{H}\to \mathcal{H}$ and $\Psi :\mathcal{H}\to \mathcal{H}$ are deterministic mappings. The advantage of using the collocation methods lies in the fact that since no integration is required, the construction of the final system of equations is very efficient. Moreover, in contrast to other methods, the functions must evaluate just at the collocation nodes. It is worth nothing that by applying collocation method, the computational cost of calculating nonlinear terms and incorporating general boundary conditions (Dirichlet, Neumann, and mixed) is reasonably low with high numerical accuracy.

The rest of the paper is organized as follows: In Section 2, we recall some definitions and results about fractional calculus and stochastic integration with respect to infinite dimensional fractional Brownian motion. In Section 3, we introduce the spectral collocation operator. Section 4 is devoted to the study of spectral collocation method for SPDEs with infinite fractional Brownian motion and our main results about the optimal convergence rate are stated and proved. In Section 5, the numerical results are presented and the confirmation of the theoretical rate of convergence is illustrated.