Backward difference formulae and spectral Galerkin methods for the Riesz space fractional diffusion equation

Abstract

Approximating Riesz space fractional diffusion equation in time by $k$-step backward difference formula and in space by spectral Galerkin method, we establish a fully discrete scheme with high order both in time and in space. For $k\le 5$, we prove the stability of full discretization and obtain the error estimate with order $O({\tau }^k+N^{\frac{\alpha }{2}-m})$, which depends only on the regularity of initial value and right-hand function. Moreover, we extend the proposed method to two dimensional case and derive similar results. Finally, we illustrate the theoretical estimates by numerical examples.

Introduction

Consider the Riesz space fractional diffusion equation (RSFDE)

$$\frac{\partial }{\partial t}u(x,t)-\frac{{\partial }^{\alpha }}{\partial {\left|x\right|}^{\alpha }}u(x,t)=f(x,t),t\in (0,T],x\in (-1,1),$$

with the initial condition $u(x,0)=u_0\left(x\right)$ for $x\in (-1,1)$ and the boundary condition $u(1,t)=u(-1,t)=0$ for $t\in (0,T]$. Here, $\alpha \in (1,2)$ and $\frac{{\partial }^{\alpha }}{\partial {\left|x\right|}^{\alpha }}$ is the Riesz fractional derivative operator. RSFDE has important applications in physics, finance and engineering, such as the description of particle transport with non-integer space–time scaling [1] and the theory of tick-by-tick dynamics in financial markets [15].

To give the existence, uniqueness and regularity results for the solution of RSFDE (1.1), we define ${\mathcal{H}}^m\left(\Omega \right)$ to be Sobolev space with semi-norm ${\left|\cdot \right|}_{{\mathcal{H}}^m\left(\Omega \right)}$ and norm ${\Vert \cdot \Vert }_{{\mathcal{H}}^m\left(\Omega \right)}$, where $\Omega =(-1,1)$ and $m\in \mathbb{N}$. When $m=0$, it reduces to ${\mathcal{L}}^2$-space with inner product $〈\cdot ,\cdot 〉$ and norm ${\Vert \cdot \Vert }_{{\mathcal{L}}^2\left(\Omega \right)}$. For $\mu >0$, we introduce the fractional derivative space ${\mathcal{H}}_0^{\mu }\left(\Omega \right)$ defined in [7], which is equipped with semi-norm ${\left|\cdot \right|}_{{\mathcal{H}}_0^{\mu }\left(\Omega \right)}$ and norm ${\Vert \cdot \Vert }_{{\mathcal{H}}_0^{\mu }\left(\Omega \right)}={\left({\Vert \cdot \Vert }_{{\mathcal{L}}^2\left(\Omega \right)}^2+{\left|\cdot \right|}_{{\mathcal{H}}_0^{\mu }\left(\Omega \right)}^2\right)}^{1∕2}$. Similar to Theorem 2.2 in [10], we have the following conclusion.

Lemma 1.1

Suppose that $m\in \mathbb{N}\cup \left\{0\right\}$ , $f(x,t)\in {\mathcal{L}}^2(0,T;{\mathcal{H}}^m\left(\Omega \right))$ , $u_0\left(x\right)\in {\mathcal{H}}_0^{m+\frac{\alpha }{2}}\left(\Omega \right)$ , then there exists a unique solution of RSFDE (1.1) , which satisfies $u(x,t)\in {\mathcal{L}}^2(0,T;{\mathcal{H}}_0^{m+\alpha }\left(\Omega \right))$ and $\frac{\partial }{\partial t}u(x,t)\in {\mathcal{L}}^2(0,T;{\mathcal{H}}^m\left(\Omega \right))$ . Furthermore, the solution $u(x,t)$ fulfils the estimates

$${\int }_0^T{\Vert {(-\Delta )}^{\frac{\alpha }{2}}u(x,t)\Vert }_{{\mathcal{H}}^m\left(\Omega \right)}^2\mathrm{d}t\le C_0\left[{\Vert u_0\left(x\right)\Vert }_{{\mathcal{H}}_0^{m+\frac{\alpha }{2}}\left(\Omega \right)}^2+{\int }_0^T{\Vert f(x,t)\Vert }_{{\mathcal{H}}^m\left(\Omega \right)}^2\mathrm{d}t\right],$$$${\int }_0^T{\Vert \frac{\partial }{\partial t}u(x,t)\Vert }_{{\mathcal{H}}^m\left(\Omega \right)}^2\mathrm{d}t\le C_0\left[{\Vert u_0\left(x\right)\Vert }_{{\mathcal{H}}_0^{m+\frac{\alpha }{2}}\left(\Omega \right)}^2+{\int }_0^T{\Vert f(x,t)\Vert }_{{\mathcal{H}}^m\left(\Omega \right)}^2\mathrm{d}t\right],$$

where $-{(-\Delta )}^{\frac{\alpha }{2}}$ is the fractional Laplacian and $C_0$ is a positive constant independent of $x$ and $T$ .

In the last decades, a lot of numerical methods have been proposed to solve RSFDE, including finite difference methods based on fractional centred difference [18] and Fourier transform technique [3], [6], finite element methods [2], [11], [22], spectral methods built upon Jacobi polynomials [16], [21] and generalized Jacobi functions [4], [20]. Most of them were devoted to constructing high order schemes to approximate the spatial term of RSFDE. However, they discretized the temporal term of RSFDE by the methods with convergence order less than or equal to second-order, which does not match with the high order in space. So, it is necessary to develop a high order method with matching accuracy in time and space for RSFDE.

As we know, the $k$-step backward difference formula (BDF) not only has low computation cost, but also has order of $k$. Recently, [2], [22] use $1$-step BDF method and [12], [17] apply $2$-step BDF method to discretize the temporal term of RSFDE. They obtain convergence order of $1$ and $2$ in time, respectively. In this paper, we approximate the temporal term of RSFDE (1.1) by $k$-step BDF method with $k\le 5$. Since the fractional derivative operator is nonlocal, we discretize the spatial term by spectral Galerkin method, which is a global method. This paper has two features: (i) we prove the stability by using the method based on the G-stability theory [5] and the multiplier technique [14], which is first proposed by Lubich in [13], and (ii) we give the rigorous error analysis of the proposed method, which shows that the order in time is $k$th order and the order in space depends on the regularity of $u_0\left(x\right)$ and $f(x,t)$.

The contents of this paper is as follows. In Section 2, we derive the full discretization of RSFDE (1.1) by using the spectral Galerkin method and the $k$-step BDF method. In Section 3, we give the stability and error analysis of the full discretization. In Section 4, we use the proposed method to solve the two dimensional RSFDE. In Section 5, several numerical examples are provided to show the theoretical estimates.