Crank–Nicolson method for the fractional diffusion equation with the Riesz fractional derivative

doi:10.1016/j.jcp.2011.11.008

Journal of Computational Physics

Volume 231, Issue 4, 20 February 2012, Pages 1743-1750

https://doi.org/10.1016/j.jcp.2011.11.008 Get rights and content

Abstract

We examine a numerical method to approximate to a fractional diffusion equation with the Riesz fractional derivative in a finite domain, which has second order accuracy in time and space level. In order to approximate the Riesz fractional derivative, we use the “fractional centered derivative” approach. We determine the error of the Riesz fractional derivative to the fractional centered difference. We apply the Crank–Nicolson method to a fractional diffusion equation which has the Riesz fractional derivative, and obtain that the method is unconditionally stable and convergent. Numerical results are given to demonstrate the accuracy of the Crank–Nicolson method for the fractional diffusion equation with using fractional centered difference approach.

Introduction

Fractional differential equations appear in many applications of the applied sciences, such as the fractional diffusion and wave equations [1], [2], subdiffusion and superdiffusion equations [3], [4], electrical systems [5], viscoelasticity theory [5], control systems [5], bioengineering [6], and finance [7].

Let $\frac{\partial^{α}}{\partial | x |^{α}}$ be the Riesz fractional derivative operator for 1 < α ⩽ 2 defined by [8], [9], [10] $\frac{\partial^{α} u (x, t)}{\partial | x |^{α}} = - \frac{1}{2 \cos (α π / 2) Γ (2 - α)} \frac{d^{2}}{{dx}^{2}} \int_{- \infty}^{\infty} | x - ξ |^{1 - α} u (ξ, t) d ξ,$ where $Γ (z) = \int_{0}^{\infty} t^{z - 1} e^{- t} dt$ .

We consider the following equation in a finite domain associated with initial and Dirichlet boundary conditions $\frac{\partial u (x, t)}{\partial t} = D \frac{\partial^{α} u (x, t)}{\partial | x |^{α}} + f (x, t), a < x < b, 0 < t ⩽ T,$ $\begin{matrix} u (a, t) = φ (t), 0 < t ⩽ T, \\ u (b, t) = Φ (t), 0 < t ⩽ T, \\ u (0, t) = ψ (x), a ⩽ x ⩽ b, \end{matrix}$ where D > 0 is diffusion coefficient, and f(x, t), φ(t), Φ(t) and ψ(x) are sufficiently smooth functions.

In order to approximate to the Riesz fractional derivative, it was generally used the Grünwald–Letnikov derivative approximation scheme of order O(h) [11], [12], [13], [14], [15], [16], [17], [18], [19], [20]. Meerschaert and Tadjeran [13] and Tadjeran et al. [14] applied the Crank–Nicolson method with Grünwald–Letnikov derivative approximation to a linear fractional diffusion equation which has one-sided fractional derivative, and showed that the used method is unconditionally stable for given problems. Shen et al. [15] applied implicit and explicit finite difference methods with Grünwald–Letnikov derivative approximation to a linear Riesz fractional diffusion equation, and showed that the explicit method is conditionally and the implicit method is unconditionally stable for given problem. The Richardson extrapolation method was used in [17], [13], [15], [14] to improve the convergence order of used approximations. Shen et al. [16] investigated the Green function and a discrete random walk model for Riesz fractional advection–dispersion equation on infinite domain with an initial condition, and also presented implicit and explicit finite differences to this problem on a finite domain. Yang et al. [18] presented the standard and shifted Grünwald–Letnikov derivative approximations, the method of lines, the matrix transform method, the L1/L2 approximation method and a spectral representation method for Riesz fractional advection–dispersion equation on a finite domain. Yang et al. [19] implemented finite difference with matrix transform method and Laplace transform to a homogeneous fractional diffusion equation with Neumann and Dirichlet boundary conditions. Zhang and Liu [20] applied an implicit finite difference method with Grünwald–Letnikov derivative approximation to a non-linear Riesz fractional diffusion equation and showed that the used method is stable for small time. Zhang et al. [21] used Galerkin finite element approximation and a backward difference technique to Riesz fractional advection–dispersion equation, and gave the stability and convergence analysis.

As a new approach, Ortigueira [22] defined the “fractional centered derivative” and proved that the Riesz fractional derivative of an analytic function can be represented by the fractional centered derivative.

In Section 2, we show that the fractional centered difference approximates to the Riesz fractional derivative for 1 < α ⩽ 2 with O(h²) accurate. In Section 3, we apply the Crank–Nicolson method for the problem (2), (3) with using the fractional centered difference discretization. In Section 4, we give the stability and the convergence properties of the Crank–Nicolson method for the problem (2), (3). Finally, the last section illustrates the stability and convergence of the Crank–Nicolson method trough the numerical solution of an example.

Section snippets

Approximation by fractional centered difference

In [22], for α > −1 the fractional centered difference is defined by $Δ_{h}^{α} θ (x) = \sum_{k = - \infty}^{\infty} \frac{(- 1)^{k} Γ (α + 1)}{Γ (\frac{α}{2} - k + 1) Γ (\frac{α}{2} + k + 1)} θ (x - kh)$ and it is shown that $\lim_{h \to 0} \frac{Δ_{h}^{α} θ (x)}{h^{α}} = \lim_{h \to 0} \frac{1}{h^{α}} \sum_{k = - \infty}^{\infty} \frac{(- 1)^{k} Γ (α + 1)}{Γ (\frac{α}{2} - k + 1) Γ (\frac{α}{2} + k + 1)} θ (x - kh),$ represents the Riesz fractional derivative (1) for the case of 1 < α ⩽ 2.

We use Eq. (5) as a discretization to the Riesz fractional derivative. Therefore, we prove the following property and lemma.

Property 2.1

Let $g_{k} = \frac{(- 1)^{k} Γ (α + 1)}{Γ (α / 2 - k + 1) Γ (α / 2 + k + 1)}$ be the coefficients of the centered finite difference approximation (5)

The Crank–Nicolson discretization for the fractional diffusion equation

The Crank–Nicolson discretization for the problem (2), (3) is $\frac{u_{i}^{j + 1} - u_{i}^{j}}{τ} = - \frac{1}{2} {Dh}^{- α} \sum_{k = - m + i}^{i} g_{k} u_{i - k}^{j + 1} - \frac{1}{2} {Dh}^{- α} \sum_{k = - m + i}^{i} g_{k} u_{i - k}^{j} + f_{i}^{j + \frac{1}{2}}$ or $u_{i}^{j + 1} + \frac{D τ}{2 h^{α}} \sum_{k = - m + i}^{i} g_{k} u_{i - k}^{j + 1} = u_{i}^{j} - \frac{D τ}{2 h^{α}} \sum_{k = - m + i}^{i} g_{k} u_{i - k}^{j} + τ f_{i}^{j + \frac{1}{2}}$ for $i = 1, 2, \dots, m - 1, j = 0, 1, \dots, N - 1, h = \frac{b - a}{m}, x_{i} = a + ih, τ = \frac{T}{N}, t_{j} = j τ$ , and $u_{i}^{j} = u (x_{i}, t_{j})$ . We can write the system (22) in matrix–vector form as $(I + A) U^{j + 1} = (I - A) U^{j} + τ F^{j + \frac{1}{2}},$ where $U^{j} = {(u_{1}^{j}, u_{2}^{j}, \dots, u_{m - 1}^{j})}^{T}, F^{j + \frac{1}{2}} = {(f_{1}^{j + \frac{1}{2}}, f_{2}^{j + \frac{1}{2}}, \dots, f_{m - 1}^{j + \frac{1}{2}})}^{T}, f_{i}^{j + \frac{1}{2}} = f (x_{i}, t_{j + \frac{1}{2}}), t_{j + \frac{1}{2}} = j τ + \frac{τ}{2}$ , and A is an (m − 1) × (m − 1) symmetric matrix which has the entries a = τD/(2 h^α

Stability and convergence of the Crank–Nicolson difference approximation scheme

Theorem 4.1

The discrete Eq. (22) for the problem (2), (3) is unconditionally stable.

Proof

Let λ be the eigenvalue of the matrix A. Then by the Gerschgorin’s circle theorem [23] and by (13), we have $| λ - {ag}_{0} | ⩽ r_{i} = a \sum_{\binom{k = - m + i}{k \neq 0}}^{i - 1} | g_{k} | < {ag}_{0},$ where $\sum_{\binom{k = - \infty}{k \neq 0}}^{\infty} | g_{k} | = g_{0}$ , that is, we have $0 < λ < 2 {ag}_{0} .$ Hence the eigenvalues of the matrix (I + A)⁻¹(I − A) satisfy $|\frac{1 - λ}{1 + λ}| < 1 .$ Therefore, the spectral radius of the matrix (I + A)⁻¹(I − A) is less than one. Thus, the discrete Eq. (22) is unconditionally stable. □

Theorem 4.2

Let $\bar{u}$ be the exact solution of the

Numerical example

One can reduce a non-homogeneous Dirichlet boundary conditions to a homogeneous Dirichlet boundary conditions using the following transformation $u (x, t) = W (x, t) + V (x, t)$ where $V = φ (t) + (Φ (t) - φ (t)) \frac{x - a}{b - a}$ and W unknown function satisfying W(a, t) = 0, W(b, t) = 0.

As a numerical test example, we consider $\frac{\partial u (x, t)}{\partial t} = \frac{\partial^{α} u (x, t)}{\partial | x |^{α}} + f (x, t), 0 < x < 1, 0 < t ⩽ T$ associated with the initial and boundary conditions $\begin{matrix} u (x, 0) = x^{2} (1 - x)^{2}, 0 < x < 1 \\ u (0, t) = 0, 0 < t ⩽ T \\ u (1, t) = 0, 0 < t ⩽ T \end{matrix}$ and the non-homogeneous part is $f (x, t) = (1 + t)^{- 1 + α} (- 1 + x)^{2} x^{2} α + \frac{1}{Γ (5 - α)} x^{- α} ((1 +))$

Conclusion

In this work, fractional centered difference approximation to the Riesz fractional derivative is used. The Crank–Nicolson method is applied to a linear fractional diffusion Eq. (2) subject to the conditions (3), and it is proved that the method is unconditionally stable and convergent. A numerical simulation is given. The experimental and theoretical results show that the accuracy is of order h² + τ². The order of approximation to the derivative can be improved by using the Richardson

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Crank–Nicolson method for the fractional diffusion equation with the Riesz fractional derivative

Abstract

Introduction

Section snippets

Approximation by fractional centered difference

The Crank–Nicolson discretization for the fractional diffusion equation

Stability and convergence of the Crank–Nicolson difference approximation scheme

Numerical example

Conclusion

Physics Reports

Physica A: Statistical Mechanics and its Applications

Applied Numerical Mathematics

Journal of Computational Physics

Applied Mathematical Modelling

Applied Mathematics and Computation

Fractional diffusion and wave equations

Journal of Mathematical Physics

The fractional diffusion equation

Journal of Mathematical Physics

The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics

Journal of Physics A: Mathematical and General

Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering

Fractional Calculus in Bioengineering