Fast iterative method with a second order implicit difference scheme ...

arXiv:1603.00279v1 [math.NA] 1 Mar 2016

A second order implicit difference scheme for time-space fractional convection-diffusion equations Xian-Ming Gu1,2∗, Ting-Zhu Huang1†, Cui-Cui Ji3‡ Bruno Carpentieri4§, Anatoly A. Alikhanov5¶ 1. School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, 611731, P.R. China 2. Institute of Mathematics and Computing Science, University of Groningen, Nijenborgh 9, P.O. Box 407, 9700 AK Groningen, The Netherlands 3. Department of Mathematics, Southeast University, Nanjing 210096, P.R. China 4. School of Science and Technology, Nottingham Trent University, Clifton Campus, Nottingham, NG11 8NS, UK 5. Kabardino-Balkarian State University, ul. Chernyshevskogo 173, Nalchik, 360004, Russia

Abstract In this paper we want to propose practical efficient numerical methods to solve a class of initial-boundary problem of time-space fractional convection-diffusion equations (TSFCDEs). To start with, an implicit method based on two-sided weighted shifted Gr¨ unwald formulae is proposed with a discussion of the stability and consistency. The established implicit difference scheme can be verified that it converges with second order accuracy in both time and space even for time variable coefficients. Extensive numerical example are reported to illustrate the effectiveness of the proposed implicit difference methods. Keywords: Fractional convection-diffusion equation; Shifted Gr¨ unwald discretization; Toeplitz matrix; Fast Fourier transform. AMSC (2010): 65F15, 65H18, 15A51.

1

Introduction

In recent years there has been a growing interest in the field of fractional calculus. Podlubny [1], Samko et al. [2] and Kilbas et al. [3] provide the history and a comprehensive ∗

E-mail: [email protected], [email protected]. Corresponding author. E-mail: [email protected]. Tel.: +86 28 61831608. ‡ E-mail: [email protected]. § E-mail: [email protected]. ¶ E-mail: [email protected]. †

treatment of this subject. Many phenomena in engineering, physics, chemistry and other sciences can be described very successfully by models using mathematical tools from fractional calculus, i.e., the theory of derivatives and integrals of fractional order. Differential equations with fractional order have recently proved to be valuable tools for the modelling of many physical phenomena [1, 3]. Diffusion with an additional velocity field and diffusion under the influence of a constant external force field are, in the Brownian case, both modelled by the convection-dispersion equation. In the case of anomalous diffusion this is no longer true, i.e., the fractional generalisation may be different for the advection case and the transport an external force field in [4]. In the study, we are very interested in the fast solver for solving the initial-boundary value problem of the time-space fractional convection-diffusion equation (TSFCDE) [5, 6]:  ∂u(x,t) β α β  ∂0,t u(x, t) = γ(t) ∂x + d+ (t)a Dx u(x, t) + d− (t)x Db u(x, t) + f (x, t), (1.1) u(x, 0) = φ(x), a ≤ x ≤ b,   u(a, t) = u(b, t) = 0, 0 < t ≤ T,

where α ∈ (0, 1], β ∈ (1, 2], a < x < b, and 0 < t ≤ T . Here, the parameters α and β are the order of the TSFCDE, f (x, t) is the source term, and diffusion coefficient functions d± (t) are non-negative under the assumption that the flow is from left to right. Moreover, the variable coefficients γ(t) are real. The TSFCDE can be regarded as generalizations of classical convection-diffusion equations with the first-order time derivative replaced by the Caputo fractional derivative of order α ∈ (0, 1], and the second-order space derivatives replaced by the two-sided Riemman-Liouville fractional derivative of order β ∈ (1, 2]. Namely, the time fractional derivative in (1.1) is the Caputo fractional derivative of order α [1] denoted by Z t ∂u(x, ξ) dξ 1 α , (1.2) ∂0,t u(x, t) = Γ(1 − α) 0 ∂ξ (t − ξ)α

and the left-handed (a Dxβ ) and the right-handed (x Dbβ ) space fractional derivatives in (1.1) are the Riemann-Liouville fractional derivatives of order β [2, 3, 7] which are defined by Z x u(η, t)dη ∂m 1 β (1.3) a Dx u(x, t) = Γ(m − β) ∂xm a (x − η)β−m+1 and β x Db u(x, t)

(−1)m ∂ m = Γ(m − β) ∂xm

Z

x

b

u(η, t)dη , (η − x)β−m+1

(1.4)

where Γ(·) denotes the Gamma function. Truly, when α = 1 and β = 2, the above equation reduces to the classical convection-diffusion equation (CDE). The fractional CDE has been recently investigated by a number of authors. It is presented as a useful approach for the description of transport dynamics in complex systems which are governed by anomalous diffusion and non-exponential relaxation patterns [4,5]. The fractional CDE is also used in groundwater hydrology research to model the transport of passive tracers carried by fluid flow in a porous medium [8]. Though analytic approaches, such as the Fourier transform method, the Laplace transform methods, and

the Mellin transform method, have been proposed to seek closed-form solutions [1, 6, 9], there are very few FDEs whose analytical closed-form solutions are available. Therefore, the research on numerical approximation and techniques for the solution of FDEs has attracted intensive interest. Most early established numerical methods are developed for handling the space factional CDE or the time fractional CDE. For space fractional CDEs, many researchers exploited the conventional shifted Gr¨ unwald discretization [14] and the implicit Euler (or Crank-Nicolson) time-step discretization for two-sided Riemman-Liouville fractional derivatives and the first order time derivative, respectively. Then they constructed many numerical approaches for solving the space fractional CDE, refer to [14, 27, 28, 36, 41, 42, 44, 50, 52, 53, 55, 57] and references therein for details. Later, Chen and Deng combined the second-order spatial discretization with the Crank-Nicolson temporal discretization for producing the novel numerical methods, which archive the second order accuracy in both time and space for space fractional CDE [12,13]. Even Chen & Deng and Qu et al. separately designed the fast computational techniques, which can also reduce the required algorithmic storage, for implementing the above mentioned second order numerical methods; see [13, 17] for details. Additionally, there are also some other interesting numerical methods for space fractional CDE, refer, e.g., to [29, 32, 48, 56–58] for a discussion of these issues. On the other hand, for time fractional CDE, many early implicit numerical methods are derived by the combination of the L1 approximate formula [22] for Caputo fractional derivative and the first/second order spatial discretization. These numerical methods are unconditionally convergent with the O(τ 2−α + h) or O(τ 2−α + h2 ) accuracy, where τ and h are time-step size and spatial grid size, respectively. In order to improve the spatial accuracy, Cui [30, 34] and Mohebbi & Abbaszadeh [51] proposed, respectively, two compact exponential methods and a compact finite difference method for a time fractional convection-subdiffusion equation so that the spatial accuracy is improved to the fourthorder. However, the methods and analyses in [30, 51] are only for the equations with constant coefficients. In particular, the discussions in [51] are limited to a special time fractional convection-subdiffusion equation where the diffusion and convection coefficients are assumed to be one. In addition, some other related numerical methods are already for handling the time fractional CDE, see e.g. [31, 33, 43, 46, 47, 49] for details. On contrast, although the numerical methods for the space (or time) fractional CDE are extensively treated in the past research, the work about numerically handling the TSFCDE is not too much. Firstly, Zhang [20,26], Shao & Ma [21], Qin & Zhang [45], and Liu et al. [53] have worked out a series of studies about constructing the implicit difference schemes for TSFCDE, however all these numerical schemes can archive the convergence with first order accuracy in both space and time from the both theoretical and numerical perspectives. Moreover, Liu et al. [18, 53], Zhao et al. [19], and Shen et al. [37] have considered to solve the more general form of TSFCDE, in which the first-order space derivative is replaced by the two-sided Riemman-Liouville fractional derivative of order ν ∈ (0, 1). Again, their numerical methods cannot enjoy the convergence with second order accuracy in both space and time. In addition, some other efficient approaches are also developed for dealing with TSFCDE numerically. Moreover, most of these numerical methods have no complete theoretical analyses for the convergence and stability; refer

to [35, 38–40, 54] for details. In this paper, we firstly derive an implicit difference scheme for solving the TSFCDE (1.1) and then verify that the proposed scheme can archive the stability and convergence with second order accuracy in both space and time, i.e. O(τ 2 + h2 ) from the both theoretical and numerical perspectives. As far as we know, the scheme is the first one who can have the convergence with O(τ 2 + h2 ). The reminder of the paper is organized as follows. In Section 2, we establish an implicit difference scheme for (1.1) and prove that this scheme is unconditionally stable and convergent with O(τ 2 + h2 ) order accuracy. Finally, we present numerical experiments to illustrate the effectiveness of our numerical approaches in Section 3 and provide concluding remarks in Section 4.

2

Implicit difference scheme

In this section, we present an implicit difference method for solving (1.1) by discretizing the TSFCDE defined by (1.1). Unlike former numerical approaches with the first order accuracy in both time and space [18–21, 26], we exploit henceforth two-sided fractional derivatives to approximate the Riemann-Liouville derivatives in (1.3) and (1.4). We can show that, by two-sided fractional derivatives, this proposed method is also unconditionally stable and convergent under second order accuracy in time and space.

2.1

Numerical discretization of the TSFCDE

In order to derive the proposed scheme, we first introduce some notations. In the ¯ T = {(x, t) : a ≤ x ≤ b, 0 ≤ t ≤ T } we introduce the mesh ̟hτ = ̟h × rectangle Q ̟τ , where ̟h = {xi = ih, i = 0, 1, . . . , N; hN = b − a}, and ̟τ = {tj = jτ, j = 0, 1, . . . , M; τ = T /M}. Besides, v = {vi | 0 ≤ i ≤ N} be any grid function. Then, the following lemma introduced in [10] gives a description on the time discretization. Lemma 2.1 Suppose 0 < α < 1, σ = 1 − α2 , u(t) ∈ C 3 [0, T ], and tj+σ = (j + σ)τ . Then

α ∂0,t u(t) − ∆α0,tj+σ u(t) = O(τ 3−α ), (2.1) j+σ Pj (α,σ,j) (α,σ,0) (α,σ) τ −α = a0 for j = 0, where ∆α0,tj+σ u(t) = Γ(2−α) s=0 cj−s [u(ts+1 ) − u(ts )], and c0  (α,σ) (α,σ)  + b1 , m = 0, a0 (α,σ) (α,σ) (α,σ) (α,σ,j) cm = am + bm+1 − bm , 1 ≤ m ≤ j − 1, (2.2)   (α,σ) (α,σ) aj − bj , m = j, (α,σ)

(α,σ)

for j ≥ 1, in which a0 = σ 1−α , aℓ = (ℓ + σ)1−α − (ℓ − 1 + σ)1−α , for ℓ ≥ 1; and (α,σ) 1 bℓ = 2−α [(ℓ + σ)2−α − (ℓ − 1 + σ)2−α ] − 21 [(ℓ + σ)1−α + (ℓ − 1 + σ)1−α ]. R +∞ Denote Ln+β (R) = {v|v ∈ L1 (R) and −∞ (1 + |k|)n+β |ˆ v (k)|dk < ∞}, where vˆ(k) = R +∞ ιkx √ e v(x)dx is the Fourier transformation of v(x), and we use ι = −1 to denote the −∞ imaginary unit. For the discretization on space, we introduce the following Lemma:

Lemma 2.2 ( [11, 15]) Suppose that v ∈ Ln+β (R), and let ] [ x−a h

β δx,+ v(x) =

1 X (β) ω v(x − (k − 1)h), hβ k=0 k

(2.3)

b−x

[ h ] 1 X (β) β δx,− v(x) = β ω v(x + (k − 1)h), h k=0 k

(2.4)

then for a fixed h, we have

where

(

β a Dx v(x)

β = δx,+ v(x) + O(h2 ),

(2.5)

β x Db v(x)

β = δx,− v(x) + O(h2 ),

(2.6)

(β)

(β)

(β)

(β)

(β)

ω 0 = λ1 g 0 , ω 1 = λ1 g 1 + λ0 g 0 , (β) (β) (β) (β) ωk = λ1 gk + λ0 gk−1 + λ−1 gk−2 ,

with β 2 + 3β + 2 λ1 = , 12

4 − β2 λ0 = , 6

β 2 − 3β + 2 = , 12

λ−1

(2.7)

k ≥ 2,

and

(β) gk

β . (2.8) = (−1) k k

4,3 Let u(x, t) ∈ Cx,t be a solution of the problem (1.1). Let us consider Eq. (1.1) for ¯ (x, t) = (xi , tj+σ ) ∈ QT , i = 1, 2, . . . , N − 1, j = 0, 1, . . . , M − 1, σ = 1 − α2 : α ∂0,t u(x, t) = γ(tj+σ ) j+σ

∂u(x, t)

d− (tj+σ )

∂x

(xi ,tj+σ )

β x Db u(x, t)

+ d+ (tj+σ )

(xi ,tj+σ )

β a Dx u(x, t)

+ f (xi , tj+σ ).

(xi ,tj+σ )

+ (2.9)

For simplicity, let us introduce some notations (σ)

ui

(j+σ)

= σuj+1 + (1 − σ)uji , γ (j+σ) = γ(tj+σ ), D± i

= d± (tj+σ ), fij+σ = f (xi , tj+σ ) (2.10)

and (σ)

δhβ ui

= γ (j+σ)

i+1 −i+1 (σ) (σ) (j+σ) NX (j+σ) X ui+1 − ui−1 D+ D− (β) (σ) (β) (σ) ω u + ωk ui+k−1 . + k i−k+1 β β 2h h h k=0

(2.11)

k=0

Then with regard to Lemma 2.1 we derive the implicit difference scheme with the approximation order O(h2 + τ 2 ):  β (σ) j+σ α  ∆0,tj+σ ui = δh ui + fi , 1 ≤ i ≤ N − 1, 0 ≤ j ≤ M − 1, (2.12) u0i = φ(xi ), 1 ≤ i ≤ N − 1,   j j u0 = uN = 0, 0 ≤ j ≤ M.

It is interesting to note that for α → 1 we obtain the Crank-Nicolson difference scheme.

2.2

Analysis of the implicit difference scheme

In this section, we analyze the stability and convergence for the implicit scheme (2.12). Let Vh = {v | v = {vi } is a grid function on ̟h and vi = 0 if i = 0, N} (2.13) For ∀u, v ∈ Vh , we define the discrete inner product and the corresponding discrete L2 norm as follows, N −1 X p (u, v) = h ui vi , and kuk = (u, u). (2.14) i=1

Now, some lemmas are provided for proving the stability and convergence of implicit difference scheme (2.12). (β)

Lemma 2.3 ( [11, 14]) Let 1 < α < 2 and gk be defined in Lemma 2.2. Then we have  (β) (β) (β) (β)  g1 = −β, g2 > g3 > · · · > 0,  0 = 1, gP PN (β) (β) ∞ N > 1, (2.15) k=0 gk < k=0 gk = 0, 0,  (β) (β) (β)  β+1 gk = O(k −(β+1) ), gk = 1 − k gk−1 , k = 1, 2, . . . . (β)

Lemma 2.4 ( [11, 15]) Let 1 < α < 2 and gk be defined in Lemma 2.2. Then we have  (β) (β) (β)  = 1, ω < 0, ωk > 0, k ≥ 3, ω PN P0 ∞ (β) 1 (β) (2.16) N > 1, k=0 ωk < 0, k=0 ωk = 0,   (β) (β) ω0 + ω2 ≥ 0. Lemma 2.5 ( [15]) For 1 < β < 2, and any v ∈ Vh , it holds that β (δx,+ v, v)

=

β (δx,− v, v)

−1 1 N X (β) ≤ ωk kvk2 . β h k=0

(2.17)

Lemma 2.6 For 1 < β < 2, N ≥ 5, and any v ∈ Vh , there exists a positive constant c1 , such that β β (−δx,+ v, v) = (−δx,− v, v) > c1 ln 2kvk2 . (2.18)

Proof. Since 2N +2 X k=N

(β)

ωk =

2N X

k=N

(β)

(β)

(β)

gk + (λ1 + λ0 )g2N +1 + λ1 g2N +2 + ζ(β),

(2.19)

where ζ(β) = (λ0 +

(β) λ−1 )gN −1

+

(β) λ−1 gN −2

h i N −2−β (β) = (λ0 + λ−1 ) + λ−1 gN −2 N −1 (12 − 6β)N + β 3 + 4β 2 − β − 22 (β) gN −2 , = 12(N − 1) ϑ(β) (β) , gN −2 12(N − 1)

(2.20)

then ζ(2) = 0, ϑ(2) = 0 and ϑ′ (β) = −6N + 3β 2 + 8β − 1 ≤ 27 − 6N, which implies ζ(β) is a decreasing function for β ∈ [1, 2], if N ≥ 5 and ϑ′ (β) < 0. Hence ζ(β) > 0 when N ≥ 5. Then, by Lemma 2.3, there exist positive constants c˜1 and c1 , such that ∞ 2N +2 2N 2N X 1 X (β) 1 X (β) 1 X (β) ωk > β ωk > β gk ≥ c˜1 k −(β+1) N β β h k=N h k=N h k=N k=N Z 2N 2N 2N X β c˜1 X 1 1 −(β+1) k k > c˜1 ≥ c1 dx = c1 ln 2, = β 2 2 k x N k=N

k=N

(2.21) N ≥ 5.

Using Lemmas 2.4 and 2.5, we then obtain ∞ 1 X (β) β β ωk kvk2 > c1 ln 2kvk2 , (−δx,+ v, v) = (−δx,− v, v) > β h k=N

(2.22)

which proves the lemma. ✷ Based on the above lemmas, we can obtain the following theorem, which is essential for analyzing the stability of the proposed implicit difference scheme. Theorem 2.1 For any v ∈ Vh , it holds that

(δhβ v, v) ≤ −c ln 2kvk2 ,

(2.23)

where c is a positive constant independent of the spatial step size h. Proof. The concrete expression of (δhβ v, v) can be written by (δhβ v, v)

=γ

(j+σ)

h

N −1 X i=1

0

N

vi+1 − vi−1 (j+σ) β (j+σ) β vi + D+ (δx,+ v, v) + D− (δx,− v, v). 2h

(2.24)

It notes that v = v = 0, we have γ

(j+1)

h

N −1 X i=1

vi+1 − vi−1 vi = 0. 2h

(2.25)

Moreover, according to Lemma 2.6, there exists a positive constant c1 independent of the spatial step size h, such that for any non-vanishing vector u ∈ Vh , we obtain (j+σ) β (j+σ) β (j+σ) (j+σ) D+ (δx,+ v, v) + D− (δx,− v, v) ≤ −c1 ln 2 D+ + D− kvk2 (2.26) (j+σ) (j+σ) . To insert (2.25) and (2.26) into (2.24), then Theorem 2.1 Let c = c1 D+ + D− holds. ✷

Lemma 2.7 ( [10]) Let Vτ = {u|u = (u0, u1 , . . . , uM )} For any u ∈ Vτ ; one has the following inequality 1 [σuj+1 + (1 − σ)uj ]∆α0,tj+σ u ≥ ∆α0,tj+σ (u)2 . 2

(2.27)

Now we conclude the stability and convergence of the scheme (2.12). For simplicity (α,σ,j) P cj−s of presentation, in our proof, we denote aj+1 = . Then ∆α0,tj+σ u = js=0 (us+1 − s τ α Γ(2−α) us )aj+1 s . PN −1 2 f (xi , tj+σ ). Then the implicit difference scheme Theorem 2.2 Denote kf j+σ k2 = h i=1 (2.12) is unconditionally and the following a priori estimate holds: kuj+1k2 ≤ ku0 k2 +

T α Γ(1 − α) j+σ 2 kf k , 2c ln 2

0 ≤ j ≤ M − 1,

(2.28)

j+1 j+1 T where uj+1 = (uj+1 1 , u2 , . . . , uN −1 ) .

Proof. To make an inner product of (2.12) with u(σ) , we have (∆α0,tj+σ u, u(σ) ) = (δhβ u(σ) , u(σ) ) + (f j+σ , u(σ) ).

(2.29)

It follows from Theorem 2.1 and Lemma 2.6 that (δhβ u(σ) , u(σ) ) ≤ −c ln 2ku(σ) k2 , 1 (∆α0,tj+σ u, u(σ) ) ≥ ∆α0,tj+σ (kuk2 ). 2

(2.30) (2.31)

Inserting (2.30)-(2.31) into (3.6) and using the Cauchy-Schwarz inequality, we obtain 1 α ∆0,tj+σ (kuk2 ) ≤ −c ln 2ku(σ) k2 + (f j+σ , u(σ) ) 2 ≤ −c ln 2ku(σ) k2 + c ln 2ku(σ) k2 + ≤

1 kf j+σ k2 . 8c ln 2

1 kf j+σ k2 8c ln 2

(2.32)

Next, it holds that j+1 2 aj+1 k j ku

≤

j X s=1

j+1 s 2 0 2 (aj+1 − aj+1 s s−1 )ku k + a0 ku k +

Then, to notice that aj+1 > 0 j+1 2 aj+1 k j ku

≤

j X s=1

(aj+1 s

1 2T α Γ(1−α)

−

1 kf j+σ k2 . 4c ln 2

(2.33)

(cf. [10]), we obtain cf.

s 2 aj+1 s−1 )ku k

+

aj+1 0

The targeted result then follows by induction.

T α Γ(1 − α) j+σ 2 ku k + kf k . 2c ln 2 0 2

(2.34) ✷

Theorem 2.3 Suppose that u(x, t) is the solution of (1.1) and {uji | xi ∈ ̟h , 0 ≤ j ≤ M}, is the solution of the implicit difference scheme (2.12). Denote eji = u(xi , tj ) − uji ,

xi ∈ ̟h ,

0 ≤ j ≤ M.

(2.35)

Then there exists a positive constant c˜ such that kej k ≤ c˜(τ 2 + h2 ),

0 ≤ j ≤ M.

(2.36)

Proof. It can easily obtain that ej satisfies the following error equation  β (σ) j+σ α  ∆0,tj+σ ei = δh ei = Ri , 1 ≤ i ≤ N − 1, 0 ≤ j ≤ M − 1, e0i = 0, 1 ≤ i ≤ N − 1,   j j e0 = eN = 0, 0 ≤ j ≤ M. where Rij+σ = O(τ 2 + h2 ). By exploiting Theorem 2.2, we get kej+1k2 ≤

T α Γ(1 − α) j+σ 2 kR k ≤ c˜(τ 2 + h2 ), 2c ln 2

0 ≤ j ≤ M − 1,

which proves the theorem. ✷ From above obtained conclusions it follows that if the solution and input data of problem (1.1) are sufficiently smooth, the solution of implicit difference scheme (2.12) converges to the solution of the time-space fractional differential equation with the rate equal to the order of the approximation error O(h2 + τ 2 ).

3

Numerical results

In this section we carry out three numerical examples with both constant and time variable coefficients in Eq. (1.1) to illustrate the performance and powerfulness of the proposed implicit difference scheme for solving the TSFCDE (2.12). All experiments were performed on a Windows 7 (64 bit) PC-Intel(R) Core(TM) i5-3740 CPU 3.20 GHz, 8 GB of RAM using MATLAB 2014a with machine epsilon 10−16 in double precision floating point arithmetic. For the numerical schemes of the fractional diffusion equation, we present some numerical results in one and two dimension cases to verify the theoretical results including the convergence orders and unconditional stability. For the tempered fractional diffusion equation, the numerical simulations are also performed which show the effectiveness of the proposed scheme; and the desired fourth order convergence is also obtained Example 1. In this example, we consider the equation (1.1) on space interval [a, b] = [0, 1] and time interval [0, T ] = [0, 1] with diffusion coefficients d+ (t) = d+ = 0.8, d− (t) = d− = 0.5, convection coefficient γ(t) = γ = −0.1, initial condition u(x, 0) = x2 (1 − x)2 , and

source term f (x, t) =

n Γ(3 + α) 2 Γ(3) x (1 − x)2 t2 − (t2+α + 1) 2γx(1 − x)(1 − 2x) + [d+ x2−β + 2 Γ(3 − β) Γ(5) 2Γ(4) [d+ x3−β + d− (1 − x)3−β ] + [d+ x4−β + d− (1 − x)2−β ] − Γ(4 − β) Γ(5 − β) o 4−β d− (1 − x) ] .

The exact solution of this example is u(x, t) = (t2+α + 1)x2 (1 − x)2 . For the finite difference discretization, the space step and time step are taken to be h = 1/N and τ = h, respectively. The errors (z = U − u) and convergence order (CO) in the norms k · k0 and k · kC(¯ωhτ ) , where kUkC(¯ωhτ ) = max(xi ,tj )∈¯ωhτ |U|, are given in Table 1-2. Here these notations are used throughout this section. Table 1: L2 -norm and maximum norm error behavior versus grid size reduction when τ = h and β = 1.8 in Example 1.

h max0≤n≤M kz n k0 1/32 2.7954e-4 1/64 6.6775e-5 1/128 1.6010e-5 1/256 3.8514e-6 0.50 1/32 2.6670e-4 1/64 6.3583e-5 1/128 1.5219e-5 1/256 3.6558e-6 0.90 1/32 2.4972e-4 1/64 5.9441e-5 1/128 1.4206e-5 1/256 3.4078e-6 0.99 1/32 2.5899e-4 1/64 6.2121e-5 1/128 1.4944e-5 1/256 3.6057e-6 α 0.10

CO in k · k0 – 2.0657 2.0603 2.0556 – 2.0685 2.0628 2.0576 – 2.0708 2.0650 2.0596 – 2.0598 2.0555 2.0512

kzkC(¯ωhτ ) CO in k · kC(¯ωhτ ) 4.0880e-4 – 9.8580e-5 2.0520 2.3815e-5 2.0494 5.7630e-6 2.0470 3.8874e-4 – 9.3590e-5 2.0544 2.2573e-5 2.0518 5.4539e-6 2.0492 3.6255e-4 – 8.7173e-5 2.0562 2.0993e-5 2.0540 5.0762e-6 2.0481 3.7959e-4 – 9.1923e-5 2.0460 2.2275e-5 2.0450 5.4042e-6 2.0433

As seen from Table 1, it finds that as the number of the spatial subintervals and time steps is increased keeping h = τ , a reduction in the maximum error takes place, as expected and the convergence order of the approximate scheme is O(h2 ) = O(τ 2 ), kz1 k (zi is the error where the convergence order is given by the formula: CO = logh1 /h2 kz 2k corresponding to hi ). On the other hand, Table 2 illustrates that if h = 1/1000, then as the number of time steps of our approximate scheme is increased, a reduction in the maximum error takes place, as expected and the convergence order of time is O(τ 2 ), where 1k . the convergence order is given by the following formula: CO = logτ1 /τ2 kz kz2 k

Table 2: L2 -norm and maximum norm error behavior versus τ -grid size reduction when h = 1/1000 and β = 1.8 in Example 1.

α τ max0≤n≤M kz n k0 0.10 1/10 1.9209e-5 1/20 4.6741e-6 1/40 1.0134e-6 0.50 1/10 1.2639e-4 1/20 3.1564e-5 1/40 7.7315e-6 0.90 1/10 2.4927e-4 1/20 6.2151e-5 1/40 1.5356e-5 0.99 1/10 2.7402e-4 1/20 6.8333e-5 1/40 1.6913e-5

CO in k · k0 – 2.0390 2.2054 – 2.0015 2.0295 – 2.0039 2.0170 – 2.0036 2.0145

kzkC(¯ωhτ ) CO in k · kC(¯ωhτ ) 3.0437e-5 – 7.4069e-6 2.0389 1.6095e-6 2.2023 1.9985e-4 – 4.9914e-5 2.0014 1.2232e-5 2.0288 3.9380e-4 – 9.8203e-5 2.0036 2.4272e-5 2.0165 4.3269e-4 – 1.0791e-4 2.0035 2.6714e-5 2.0142

Example 2. Let us consider the equation (1.1) on space interval [a, b] = [0, 1] and time interval [0, T ] = [0, 1] with diffusion coefficients d+ (t) = d+ = 0.5, d− (t) = d− = 0.7, convection coefficient γ(t) = γ = −0.1, initial condition u(x, 0) = 64x3 (1 − x)3 , and source term n Γ(4) 64Γ(4 + α) 3 3 t x (1 − x)3 − (64t3+α + 64) 3γx2 (1 − x)2 (1 − 2x) + [ f (x, t) = Γ(4) Γ(4 − β) 3Γ(6) 3Γ(5) [d+ x4−β + d− (1 − x)4−β ] + [d+ x5−β d+ x3−β + d− (1 − x)3−β ] − Γ(5 − β) Γ(6 − β) o Γ(7) + d− (1 − x)5−β ] − [d+ x6−β + d− (1 − x)6−β ] . Γ(7 − β) The exact solution of this example is u(x, t) = 64(t3+α + 1)x3 (1 − x)3 . For the finite difference discretization, the space step and time step are taken to be h = 1/N and τ = h, respectively. According to results shown in Table 3, we can conclude that as the number of the spatial subintervals and time steps is increased keeping h = τ , a reduction in the maximum error takes place, as expected and the convergence order of the approximate scheme is 1k O(h2 ) = O(τ 2 ), where the convergence order is given by the formula: CO = logh1 /h2 kz kz2 k (zi is the error corresponding to hi ). On the other hand, Table 4 illustrates that if h = 1/1000, then as the number of time steps of our approximate scheme is increased, a reduction in the maximum error takes place, as expected and the convergence order of time is O(τ 2 ), where the convergence order is given by the following formula: CO = 1k . logτ1 /τ2 kz kz2 k Example 3. In the last test, we investigate the equation (1.1) on the space interval [a, b] = [0, 1] and the time interval [0, T ] = [0, 1] with diffusion coefficients d+ (t) =

Table 3: L2 -norm and maximum norm error behavior versus grid size reduction when τ = h and β = 1.6 in Example 2.


CO in k · k0 – 2.0107 2.0032 2.0010 – 2.0112 2.0036 2.0012 – 2.0105 2.0032 2.0010 – 2.0104 2.0033 2.0011




CO in k · k0 – 1.9928 2.0406 – 1.9922 2.0081 – 2.0008 2.0097 – 1.9990 2.0053


9 sin(t), d− (t) = 4 sin(t), convection coefficient γ(t) = −t, initial condition u(x, 0) = x2 (1 − x)2 , and source term n Γ(3 + α) 2 2 Γ(3) sin(t) 2−β 2 2+α f (x, t) = t x (1 − x) − (t + 1) − 2tx(1 − x)(1 − 2x) + [9x + 2 Γ(3 − β) o Γ(5) sin(t) 4−β 2Γ(4) sin(t) 3−β [9x + 4(1 − x)3−β ] + [9x + 4(1 − x)4−β ] . 4(1 − x)2−β ] − Γ(4 − β) Γ(5 − β) The exact solution of this example is defined as u(x, t) = (t2+α )x2 (1−x)2 . For the implicit finite difference discretization, the space step and time step are taken to be h = 1/N and τ = h, respectively. Table 5: L2 -norm and maximum norm error behavior versus grid size reduction when τ = h and β = 1.3 in Example 3.


CO in k · k0 – 2.0657 2.0521 2.0414 – 2.0668 2.0527 2.0414 – 2.0636 2.0524 2.0430 – 2.0638 2.0522 2.0426


According to the numerical results illustrated in Table 5, it follows that as the number of the spatial subintervals and time steps is increased keeping h = τ , a reduction in the maximum error takes place, as expected and the convergence order of the approximate scheme is O(h2 ) = O(τ 2 ), where the convergence order is given by the formula: CO = kz1 k logh1 /h2 kz (zi is the error corresponding to hi ). On the other hand, Table 6 illustrates 2k that if h = 1/1200, then as the number of time steps of our approximate scheme is increased, a reduction in the maximum error takes place, as expected and the convergence order of time is O(τ 2 ), where the convergence order is given by the following formula: CO kz1 k . = logτ1 /τ2 kz 2k



4

CO in k · k0 – 2.0269 2.1312 – 1.9994 2.0180 – 2.0014 2.0103 – 2.0023 2.0095


Conclusions

In this paper, the stability and convergence of an implicit difference schemes approximating the TSFCDE of a general form is studied. Sufficient conditions for the unconditional stability of such difference schemes are obtained. For proving the stability of a wide class of difference schemes approximating the time fractional diffusion equation, it is simple enough to check the stability conditions obtained in this paper. Meanwhile, the new difference schemes of the second approximation order in space and the second approximation order in time for TSFCDEs with variable coefficients (in terms of t) are constructed as well. The stability and convergence of these schemes in the mesh L2 -norm with the rate equal to the order of the approximation error are proved. The method can be easily adopted to other TSFCDEs with other boundary conditions. Numerical tests completely confirming the obtained theoretical results are carried out. In future work, we will focus on the extension of the proposed implicit difference scheme for handling two/three-dimension TSFCDEs subject to various boundary value conditions. Additionally, the implementation of the implicit difference scheme (2.12) really requires the Toeplitz matrix computation, so we also contribute us to build the suitable fast numerical methods for (2.12) based on Toeplitz matrix properties, see e.g. [15–17, 19, 42] for a discussion of these issues.

Acknowledgement We are grateful to Dr. Zhao-Peng Hao and Prof. Zhi-Zhong Sun for his fruitful discussion about the convergence analysis of the implicit difference scheme. This research is supported by 973 Program (2013CB329404), NSFC (61370147, 61170309, 11301057, and 61402082), the Fundamental Research Funds for the Central Universities

(ZYGX2013J106, ZYGX2013Z005, and ZYGX2014J084).

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