Stability and convergence of finite difference method for two-sided space-fractional diffusion equations

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Abstract

In this paper, we study and analyse Crank–Nicolson (CN) temporal discretization with certain spatial difference schemes for one- and two-dimensional two-sided space-fractional diffusion equations (TSFDEs) with variable diffusion coefficients. The stability and convergence of the resulting discretization linear systems for TSFDEs with variable diffusion coefficients are proven by a new technique. That is, under mild assumption, the scheme is unconditionally stable and convergent with O(τ2+hl)(l1), where τ and h denote the temporal and spatial mesh steps, respectively. Further, we show that several numerical schemes with lth order accuracy from the literature satisfy the required assumption. Numerical examples are implemented to illustrate our theoretical analyses.

Introduction

Fractional partial differential equations (FPDEs) have attracted considerable attentions during the past several decades, mainly due to its wide applications in fields including turbulent flow [1], groundwater contaminant transport [2], [3], finance [4], biological systems [5], signal and image processing [6], etc. Because of the nonlocal properties of the fractional derivatives, it can model many complicated phenomena more accurately than integer-order derivatives do. Especially, space fractional diffusion equations (SFDEs) have been widely and successfully used in nonlocal dynamics and image processing [3], [6], [7]. However, it is very difficult to obtain analytical solutions for most SFDEs. Therefore, establishing an effective discretization to obtain a high accurate numerical solution is a very popular and necessary way.

As we all know, there are several kinds of numerical methods for SFDEs, among them, finite element method [8], [9], [10], finite difference method [11], [12], [13] and spectral method [14]. In recent years, these numerical treatments have been successfully used by many scholars to solve various fractional order models. Moreover, finite difference method is a common and effective numerical technique to approximate fractional derivatives of Grünwald–Letnikov type. Here is a brief literature review on it.

Firstly, Meershaert and Tadjeran proposed the shifted Grünwald–Letnikov formula to approximate the FPDEs with two-sided fractional derivatives and variable coefficients; see [11]. Nevertheless, the authors did not give the convergence analysis. Whereafter, several second order numerical schemes were proposed for solving TSFDEs with constant and variable coefficients, while only provide the convergence analysis for the constant coefficients case, see [12], [13], [15], [16] for more details. Numerical schemes with spatial third or fourth order convergence for TSFDEs were developed by applying the technique of compact operator which is only available for constant coefficient case; see [17], [18]. Based on Lubich’s operator [19], Chen and Deng [20] derived a class of fourth order approximations for space fractional derivatives. Nevertheless, no convergence analysis for variable coefficients was provided in [20]. In [21], the stability of a second order numerical scheme for TSFDEs with variable coefficients has been discussed, the theorem of convergence has been mentioned but no proof was given. Lin and Liu [15] introduced a series of second order approximations for TSFDEs with variable coefficients. Again, the stability analysis was presented but no convergence analysis was given. In [22], several numerical schemes for the Riesz SFDEs (RSFDEs) with variable coefficients have been proven to be stable and convergent, but the proof methods used in [22] cannot be applied to the case of TSFDEs. Under two strict conditions, Vong and Lyu [23] introduced a second order scheme for TSFDEs with variable coefficients and proved that this scheme was stable and convergent. Very recently, the stability and convergence analyses of a second order numerical scheme were proposed for one-sided SFDEs with variable coefficients, while the proof technique cannot be applied to TSFDEs; see [24]. Usually, the high order schemes have the same algebraic structures comparing with the low order schemes, they are both in form of, (IC)Un+1=(I+C)Un+bn+1, where I is identity matrix, and C is Toeplitz-like matrix. As usual, after the finite difference discretizations, the matrix C can be rewritten as the sum of diagonal matrices times Toeplitz matrices, i.e., D1W+D2WT, where Di(i=1,2) are positive diagonal matrices, W and its transpose WT are Toeplitz matrices.

In this paper, we present a method combining CN temporal discretization with certain spatial difference schemes for one- and two-dimensional TSFDEs with variable diffusion coefficients. A new technique is proposed to implement the stability and convergence analyses. In order to show the convergence, almost all the difference schemes in the mentioned above require a condition, that is, C is negative definite, see [20] for instance. Actually, this is a strong condition, and C is indefinite in most of the cases. In our discussions, we only need the mild assumption that W is negative definite instead of the negative definiteness of C. This is a weakening condition.

The rest of the paper is organized as follows. In Section 2, we briefly introduce the CN-type scheme of one-dimensional TSFDEs and analyse the stability and convergence for the corresponding scheme. Also, we extend our results to two-dimensional case in Section 3, and derive the stability and convergence analyses for the cases of the separable diffusion coefficients and non-separable, respectively. In Section 4, we show that several numerical schemes from the literature satisfying the required assumption in our analyses. In Section 5, some numerical results are presented to verify the theoretical results. Finally, some concluding remarks are given in Section 6.

Section snippets

The CN method of one-dimensional TSFDE

In this subsection, we consider the initial–boundary value problem for a one-dimensional two-sided space fractional diffusion equation (TSFDE) [11], [20]: u(x,t)t=d(x)k1xLDxαu(x,t)+k2xDxRαu(x,t)+f(x,t),(x,t)(xL,xR)×(0,T],u(xL,t)=u(xR,t)=0,t(0,T],u(x,0)=u0(x),x[xL,xR],where α(1,2), d(x) is a bounded positive-valued function, k1,k20 and k1+k2>0, f(x,t) is the source term. xLDxαu(x,t) and xDxRαu(x,t) are the left- and right-sided Riemann–Liouville fractional derivatives, respectively,

The CN method of two-dimensional TSFDE

In this subsection, we consider the initial–boundary value problem for a two-dimensional TSFDE [20]: u(x,y,t)t=d(x,y)k1xLDxαu(x,y,t)+k2xDxRαu(x,y,t)+v(x,y)k3yLDyβu(x,y,t)+k4yDyRβu(x,y,t)+f(x,y,t),(x,y,t)Ω×(0,T],u(x,y,0)=u0(x,y),(x,y)Ω,u(x,y,t)=0,(x,y,t)Ω×(0,T],where α,β(1,2), Ω=(xL,xR)×(yL,yR), Ω is the boundary of Ω, d(x,y) and v(x,y) are two bounded and positive-valued functions. The coefficients satisfy ki0, i=1,2,3,4, and i=14ki>0. xLDxαu(x,y,t), xDxRαu(x,y,t) and yLDyβu(x,y,t)

Numerical schemes satisfying Wα0

In the previous section, we see that our theoretical analyses depend on one assumption of the differentiation matrix, i.e., Wα is negative definite (denote by Wα0) for α(1,2).

In this section, we introduce several numerical schemes that satisfy this assumption. For simplicity, we only discuss one-dimensional case, and two-dimensional case can be obtained similarly. Firstly, we introduce some notations.

For θR, let Lθ(R){w:RRxDθw(x)L1(R),F[xDθw(x)]L1(R),Dxθw(x)L1(R),F[Dxθw(x)]L1(R)},

Numerical results

In this section, we give several examples to verify the theoretical results derived in Section 4. Several numerical schemes are tested, namely, CN-SG, CN-WSGD(γ1), CN-WSGD(γ2), CN-WSGD(γ3), CN-WAFD, CN-ND, CN-WSLD2, CN-WSLD3 and CN-WSLD4, where γ1=α2, γ2=α+24 and γ3=α+14. For CN-WSLD2, CN-WSLD3 and CN-WSLD4 schemes, the parameters are q=2, (q,r)=(2,2) and (q,r)=(3,3), respectively. All the numerical schemes use the same parameters in the two-dimensional case, e.g., the parameters of CN-WSGD(γ1

Conclusion

In this paper, we study a series of effective numerical methods to solve the TSFDEs by using the CN scheme with several spatial discretization. A new technique is proposed to implement the stability and convergence analyses when these schemes are used. Moreover, several numerical experiments are given to verify our theoretical analyses. In our future work, we will further consider the stability and convergence of the numerical schemes for TSFDEs with non-smooth solutions.

Acknowledgements

We thank the referees for providing valuable comments and suggestions, which are very helpful for us to improve our paper.

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    This research was supported by Natural Science Foundation of Guangdong Provincial Department of Education, China No. 2018KQNCX156; 2020KZDZX1147.

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