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Numerical methods for time-fractional convection-diffusion problems with high-order accuracy

  • Gang Dong EMAIL logo , Zhichang Guo and Wenjuan Yao
Published/Copyright: August 16, 2021
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Open Mathematics
From the journal Open Mathematics Volume 19 Issue 1

Abstract

In this paper, we consider the numerical method for solving the two-dimensional time-fractional convection-diffusion equation with a fractional derivative of order α ( 1 < α < 2 ). By combining the compact difference approach for spatial discretization and the alternating direction implicit (ADI) method in the time stepping, a compact ADI scheme is proposed. The unconditional stability and H 1 norm convergence of the scheme are proved rigorously. The convergence order is O ( τ 3 − α + h 1 4 + h 2 4 ) , where τ is the temporal grid size and h 1 , h 2 are spatial grid sizes in the x and y directions, respectively. It is proved that the method can even attain ( 1 + α ) order accuracy in temporal for some special cases. Numerical results are presented to demonstrate the effectiveness of theoretical analysis.

Keywords: 2D time-fractional convection-diffusion equation; ADI scheme; unconditional stability; convergence
MSC 2010: 37Mxx; 65-xx

1 Introduction

Fractional differential equations have attracted considerable interest due to their ability to model many phenomena. They are widely applied in various fields of science and engineering, such as signal processing, anomalous diffusion, wave propagation, and turbulence [1,2,3]. Generally, these equations are of three types that contain derivatives of fractional order in space, time, or space-time [4]. Because of the nonlocal nature of fractional differential operators, analytical solutions of these equations are not available in most cases. Even if these solutions can be given, the part of special functions makes the computation complex. Therefore, a number of authors proposed numerical methods for solving fractional diffusion equations [5,6, 7,8,9, 10,11].

Currently, there are many algorithms designed to solve one-dimensional problems based on the memory effect in fractional derivatives. For example, many finite difference methods are developed for sub-diffusion equations [12,13,14, 15,16,17, 18,19,20]. Particularly, Yuste and Acedo [12] proposed an explicit finite difference method and a von Neumann-type stability analysis for a class of anomalous sub-diffusion equations. They pointed out the difficulty of convergence analysis when implicit methods were considered. Zhuang et al. [13] derived implicit numerical methods for an anomalous sub-diffusion equation by using the energy method. For the time-fractional diffusion equation with variable coefficients, difference schemes of second and fourth order of approximation in space and second order in time were constructed by Alikhanov in [14], where the stability and convergence were studied by the method of energy inequalities. In [15], Chen et al. proposed a numerical scheme with first-order temporal accuracy and fourth-order spatial accuracy for a variable-order anomalous sub-diffusion equation. The convergence, stability, and solvability of the numerical scheme were shown via Fourier analysis. Karatay et al. [16] extended the idea of the Crank-Nicholson method to the time-fractional heat equation. By the method of Fourier analysis, they proved that the proposed method is stable and the numerical solution converges to the exact solution with the order O ( τ 2 − α + h 2 ) . There are some other numerical methods applied for solving fractional diffusion equations. Particularly, Atanackovic et al. [17] used the expansion formula for the fractional derivative to reduce the difficulty of solving the non-linear fractional differential equations arising in mechanics. Mohebbi et al. [18] studied a modified anomalous time-fractional sub-diffusion equation with a nonlinear source term and proposed a finite difference scheme which is temporally first-order accurate and spatially fourth-order accurate. In [19], a time-stepping discontinuous Petrov-Galerkin method with a continuous conforming finite element method in space for time-fractional subdiffusion problems was proposed. In [20], two fully discrete schemes were proposed for the time-fractional sub-diffusion equation with space discretized by the finite element method and time discretized by the fractional linear multistep method.

Obviously, solving two-dimensional fractional problems numerically is difficult, which may be viewed from two aspects. In the first place, fractional derivatives are nonlocal operators and have the character of history dependence, which means that the current function value depends on all the previous values. More specifically, the storage will be very expensive if we adopt low-order methods to spatial discretization. In the second place, it spends a large amount of computational complexity and CPU time if the existing implicit schemes are applied, especially for solving multidimensional problems. Therefore, it is valuable to discuss high-order efficient methods for the high-dimensional fractional equations. As far as I know, there are some work about numerical methods of the high-dimensional fractional problem. Abbaszadeh and Mohebbi [21] established a fourth-order compact solution of a two-dimensional modified anomalous fractional sub-diffusion equation. Some other numerical methods based on compact scheme are referred to in [22,23,24, 25,26]. In addition, Zeng et al. [27] investigated the two-dimensional Riesz space fractional nonlinear reaction-diffusion equation, a new alternating direction implicit (ADI) Galerkin-Legendre spectral method was proposed. In [28], two numerical methods for solving a two-dimensional anomalous sub-diffusion equation were presented. The stability, convergence, and solvability were analyzed. Besides, combining the matrix transfer technique with finite difference method, Yang et al. [29,30] proposed a new numerical method for solving a space-time-fractional diffusion equation. Huang et al. [31,32] constructed ADI schemes for two-dimensional time-space-fractional nonlinear diffusion-wave equations by equivalently transforming the problem into their partial integro-differential forms.

This paper is devoted to designing and analyzing efficient numerical methods for multi-dimensional time-fractional convection-diffusion equations. For simplicity, we consider the two-dimensional problem. For higher dimensional cases, the convergence order of our method is better than that of the previous work (see [33]).

This paper deals with the following 2D time-fractional convection-diffusion equation:

(1) D t α 0 C u = Δ u + a u x + b u y + c u + f , ( x , y , t ) ∈ Ω × ( 0 , T ] ,

(2) u ( x , y , 0 ) = u 0 ( x , y ) , ∂ u ( x , y , 0 ) ∂ t = ψ ( x , y ) , ( x , y ) ∈ Ω ¯ = Ω ∪ ∂ Ω ,

(3) u ( x , y , t ) = ϕ ( x , y , t ) , ( x , y , t ) ∈ ∂ Ω × ( 0 , T ] ,

where D t α 0 C denotes the Caputo derivative operator defined as

(4) D t α 0 C u ( x , y , t ) = 1 Γ ( 2 − α ) ∫ 0 t ∂ 2 u ( x , y , s ) ∂ s 2 ( t − s ) 1 − α d s , 1 < α < 2 ,

where a , b , c are constants with a 2 + b 2 − 4 c ≥ 0 , Δ is the two-dimensional Laplacian, Ω = ( 0 , L 1 ) × ( 0 , L 2 ) , ∂ Ω is the boundary, and f , ϕ , u 0 , and ψ are known sufficiently smooth functions.

Equation (1) can be viewed as a generalization of the classical convection diffusion equation. Meanwhile, the case of 1 < α < 2 models a super-diffusive flow, which may occur in chaotic or turbulent processes through enhanced transport of particles [34]. The review paper by Klafter et al. [35] provided numerous references to physical phenomena in which anomalous diffusion occurs. Compared with the considerable work on sub-diffusion models, solving time-fractional convection-diffusion problems is much more difficult, and there is very little research about them. Recently, for the multidimensional time-fractional diffusion equations with the fractional order lies in ( 1 , 2 ) , some high-order compact schemes have been proposed in [36,37].

Recently, the compact difference scheme for fractional diffusion problems has been developed for promoting the spatial accuracy in [14,15,37]. The advantage of the compact difference scheme is high accuracy in the spatial direction. Therefore, we use the compact exponential difference scheme to solve the time-fractional convection-diffusion problem, and we find that this algorithm is very effective indeed.

In this paper, we aim to design effective and fast numerical methods for solving the problem (1)–(3) and establish the corresponding error estimates. As we mentioned before, we adopt a fourth-order compact difference method for spatial approximation, which needs fewer grid points to produce a highly accurate solution. Therefore, the storage requirement is reduced to some extent. On the other hand, we derive an ADI scheme by utilizing the idea of the ADI method for the parabolic problem. At each time level, only two sets of tri-diagonal linear systems need to be solved, while the size of coefficient matrices is equal to that of the one-dimensional problem. As a result, this method reduces storage requirements and computational complexities greatly.

As for the error estimate, we introduce a new norm denoted by H ˜ 2 , which is equivalent to H 1 norm. We prove that this ADI scheme is unconditionally stable to the initial value and the inhomogeneous term. The temporal convergence order of the proposed scheme is ( 3 − α ) and can attain ( 1 + α ) in some cases. In spatial direction, the scheme has fourth order accuracy. The unconditional stability and H ˜ 2 norm convergence of the scheme are proved by the discrete energy method.

The content is organized as follows. In Section 2, some notations and preliminary lemmas are introduced, and a compact ADI scheme is derived and the truncation error is analyzed. In Section 3, the unique solvability, unconditional stability, and H 1 norm convergence are proved. In Section 4, two numerical examples are given to support the theoretical result. Some comments are presented in Section 5.

2 Construction of the compact ADI scheme and the truncation error

2.1 Notations and preliminary lemmas

For the finite difference approximation, let N be a positive integer, τ = T N , and t n = n τ ( 0 ≤ n ≤ N ) . The time domain [ 0 , T ] is covered by { t n ∣ 0 ≤ n ≤ N } . In addition, for a given grid function w = { w n ∣ 0 ≤ n ≤ N } , we denote

w n − 1 2 = 1 2 ( w n + w n − 1 ) , δ t w n − 1 2 = ω n − ω n − 1 τ .

For spatial approximation, let h 1 = L 1 M 1 , h 2 = L 2 M 2 with positive integers M 1 , M 2 , h = max { h 1 , h 2 } , x i = i h 1 ( 0 ≤ i ≤ M 1 ) , and y j = j h 2 ( 0 ≤ j ≤ M 2 ) . Let Ω ¯ h = { ( x i , y j ) ∣ 0 ≤ i ≤ M 1 , 0 ≤ j ≤ M 2 } , Ω h = Ω ¯ h ∩ Ω , and ∂ Ω h = Ω ¯ h ∩ ∂ Ω . For any grid function v = { v i j ∣ 0 ≤ i ≤ M 1 , 0 ≤ j ≤ M 2 } defined on Ω ¯ h , we denote

δ x v i − 1 2 , j = v i , j − v i − 1 , j h 1 , δ x 2 v i j = δ x v i + 1 2 , j − δ x v i − 1 2 , j h 1 ,

δ y δ x v i − 1 2 , j − 1 2 = δ x v i − 1 2 , j − δ x v i − 1 2 , j − 1 h 2 , δ y δ x 2 v i , j − 1 2 = δ x 2 v i , j − δ x 2 v i , j − 1 h 2 .

Similar notations δ y v i , j − 1 2 , and δ y 2 v i j , δ x δ y 2 v i − 1 2 , j , δ x 2 δ y 2 v i j can be defined, respectively. The discrete Laplace operator is denoted as Δ h v i j = δ x 2 v i j + δ y 2 v i j .

For convenience of writing, we also define

L x v i j = 1 12 ( v i − 1 , j + 10 v i j + v i + 1 , j ) , 1 ≤ i ≤ M 1 − 1 , 0 ≤ j ≤ M 2 , v i j , i = 0 or M 1 , 0 ≤ j ≤ M 2 , L y v i j = 1 12 ( v i , j − 1 + 10 v i j + v i , j + 1 ) , 1 ≤ j ≤ M 2 − 1 , 0 ≤ i ≤ M 1 , v i j , j = 0 or M 2 , 0 ≤ i ≤ M 1 .

One verifies readily that

L x v i j = I + h 1 2 12 δ x 2 v i j , L y v i j = I + h 2 2 12 δ y 2 v i j , ( x i , y j ) ∈ Ω h ,

where I denotes the standard identical operator. Moreover, we introduce the following notations:

L v i j = L x L y v i j , Λ h v i j = ( L y δ x 2 + L x δ y 2 ) v i j , ( x i , y j ) ∈ Ω h .

The following lemmas will be used in derivation of the difference scheme.

Lemma 2.1

[38] Suppose 1 < α < 2 , y ∈ C 2 [ 0 , t n ] . It holds that

1 Γ ( 2 − α ) ∫ 0 t n y ′ ( s ) d s ( t n − s ) α − 1 − τ 1 − α Γ ( 3 − α ) a 0 y ( t n ) − ∑ k = 1 n − 1 ( a n − k − 1 − a n − k ) y ( t k ) − a n − 1 y ( 0 ) ≤ 1 Γ ( 3 − α ) 2 − α 12 + 2 3 − α 3 − α − ( 1 + 2 1 − α ) max 0 ≤ t ≤ t n ∣ y ″ ( t ) ∣ τ 3 − α ,

where

(5) a k = ( k + 1 ) 2 − α − k 2 − α , k ≥ 0 .

Lemma 2.2

[39] Let y ∈ C 3 [ t k − 1 , t k ] . It holds that

1 2 [ y ′ ( t k ) + y ′ ( t k − 1 ) ] = 1 τ [ y ( t k ) − y ( t k − 1 ) ] + τ 2 16 ∫ 0 1 y ( 3 ) t k − 1 2 + s τ 2 + y ( 3 ) t k − 1 2 − s τ 2 ( 1 − s 2 ) d s ,

where t k − 1 2 = t k − τ 2 .

Lemma 2.3

[39] Let g ( x ) ∈ C 6 [ x i − 1 , x i + 1 ] and ζ ( s ) = 5 ( 1 − s ) 3 − 3 ( 1 − s ) 5 , then

g ″ ( x i + 1 ) + 10 g ″ ( x i ) + g ″ ( x i − 1 ) 12 = g ( x i + 1 ) − 2 g ( x i ) + g ( x i − 1 ) h 1 2 + h 1 4 360 ∫ 0 1 [ g ( 6 ) ( x i − s h 1 ) + g ( 6 ) ( x i + s h 1 ) ] ζ ( s ) d s .

2.2 Derivation of the compact ADI scheme

In this subsection, we will give a compact ADI scheme for (1)–(3). In order to keep fourth-order accuracy in space and tridiagonal nature of the scheme, we first introduce a transformation, which is similar to that of Liao [40]. To eliminate the convection terms in (1), we let

(6) u ( x , y , t ) = λ 1 ( x ) λ 2 ( y ) v ( x , y , t ) ,

where λ 1 ( x ) = exp ( − a 2 x ) , λ 2 ( y ) = exp ( − b 2 y ) .

A combination of (1)–(3) and (6) leads to the following fractional diffusion-wave equation satisfied by v = v ( x , y , t ) :

(7) D t α 0 C v = Δ v + d v + g , ( x , y , t ) ∈ Ω × ( 0 , T ] ,

(8) v ( x , y , 0 ) = ( λ 1 λ 2 ) − 1 u 0 ( x , y ) = v 0 ( x , y ) , ( x , y ) ∈ Ω ¯ = Ω ∪ ∂ Ω ,

(9) ∂ v ( x , y , 0 ) ∂ t = ( λ 1 λ 2 ) − 1 ψ ( x , y ) = ψ v ( x , y ) , ( x , y ) ∈ Ω ¯ = Ω ∪ ∂ Ω ,

(10) v ( x , y , t ) = ( λ 1 λ 2 ) − 1 ϕ ( x , y , t ) = ϕ v ( x , y , t ) , ( x , y , t ) ∈ ∂ Ω × ( 0 , T ] ,

where d = c − a 2 + b 2 4 , ( λ 1 λ 2 ) − 1 = 1 λ 1 ( x ) λ 2 ( y ) = exp ( ( 1 / 2 ) ( a x + b y ) ) , and g = ( λ 1 λ 2 ) − 1 f .

After the transformation, we introduce a high-order compact difference method for (7)–(10). Let

p = v t , q = v x x , r = v y y ,

then (7) can be rewritten as

(11) 1 Γ ( 2 − α ) ∫ 0 t ∂ p ( x , y , s ) ∂ s ( t − s ) 1 − α d s = q ( x , y , t ) + r ( x , y , t ) + d v ( x , y , t ) + g ( x , y , t ) .

In addition, we define grid functions

V i j n = v ( x i , y j , t n ) , P i j n = p ( x i , y j , t n ) , Q i j n = q ( x i , y j , t n ) , R i j n = r ( x i , y j , t n ) ,

and

g i j n = g ( x i , y j , t n ) , ( x i , y j ) ∈ Ω ¯ h , 0 ≤ n ≤ N .

Considering (11) at the point ( x i , y j , t n ) , in view of Lemma 2.1, we derive that

(12) τ 1 − α Γ ( 3 − α ) a 0 P i j n − ∑ k = 1 n − 1 ( a n − k − 1 − a n − k ) P i j k − a n − 1 P i j 0 = Q i j n + R i j n + g i j n + d V i j n + ( R t α ) i j n ,

for ( x i , y j ) ∈ Ω h , 1 ≤ n ≤ N , where

∣ ( R t α ) i j n ∣ ≤ 1 Γ ( 3 − α ) 2 − α 12 + 2 3 − α 3 − α − ( 1 + 2 1 − α ) max 0 ≤ t ≤ t n ∂ 2 p ∂ t 2 ( x i , y j , t ) τ 3 − α .

For 2 ≤ n ≤ N , it holds that

τ 1 − α Γ ( 3 − α ) a 0 P i j n − 1 − ∑ k = 1 n − 2 ( a n − k − 2 − a n − k − 1 ) P i j k − a n − 2 P i j 0 = Q i j n − 1 + R i j n − 1 + g i j n − 1 + d V i j n − 1 + ( R t α ) i j n − 1 .

According to the equality

− ∑ k = 1 n − 2 ( a n − k − 2 − a n − k − 1 ) P i j k − a n − 2 P i j 0 = − ∑ k = 1 n − 1 ( a n − k − 1 − a n − k ) P i j k − 1 − a n − 1 P i j 0 ,

we get

(13) τ 1 − α Γ ( 3 − α ) a 0 P i j n − 1 − ∑ k = 1 n − 1 ( a n − k − 1 − a n − k ) P i j k − 1 − a n − 1 P i j 0 = Q i j n − 1 + R i j n − 1 + g i j n − 1 + d V i j n − 1 + ( R t α ) i j n − 1 , ( x i , y j ) ∈ Ω h , 2 ≤ n ≤ N .

Let t = 0 in (11). It holds that

0 = Q i j 0 + R i j 0 + d v i j 0 + g i j 0 ,

which means that (13) holds for n = 1 and ( R t α ) i j 0 = 0 . Due to (12)–(13), using the notation P i j k − 1 2 = 1 2 ( P i j k + P i j k − 1 ) gives

(14) τ 1 − α Γ ( 3 − α ) a 0 P i j n − 1 2 − ∑ k = 1 n − 1 ( a n − k − 1 − a n − k ) P i j k − 1 2 − a n − 1 P i j 0 = Q i j n − 1 2 + R i j n − 1 2 + g i j n − 1 2 + d V i j n − 1 2 + ( R t α ) i j n − 1 2 , ( x i , y j ) ∈ Ω h , 1 ≤ n ≤ N .

Operating L = L x L y on the above equality, we have

(15) τ 1 − α Γ ( 3 − α ) L a 0 P i j n − 1 2 − ∑ k = 1 n − 1 ( a n − k − 1 − a n − k ) P i j k − 1 2 − a n − 1 P i j 0 = L y L x Q i j n − 1 2 + L x L y R i j n − 1 2 + L g i j n − 1 2 + d L V i j n − 1 2 + L ( R t α ) i j n − 1 2 , ( x i , y j ) ∈ Ω h , 1 ≤ n ≤ N .

Lemma 2.2 implies that

(16) P i j n − 1 2 = δ t V i j n − 1 2 + ( R t ) i j n − 1 2 , ( x i , y j ) ∈ Ω h , 1 ≤ n ≤ N ,

where

( R t ) i j n − 1 2 = τ 2 16 ∫ 0 1 ∂ 3 v ∂ t 3 x i , y j , t n − 1 2 + s τ 2 + ∂ 3 v ∂ t 3 x i , y j , t n − 1 2 − s τ 2 ( 1 − s 2 ) d s .

Besides, Lemma 2.3 implies that

L x Q i j n = δ x 2 V i j n + ( R x ) i j n , ( x i , y j ) ∈ Ω h , 1 ≤ n ≤ N , L y R i j n = δ y 2 V i j n + ( R y ) i j n , ( x i , y j ) ∈ Ω h , 1 ≤ n ≤ N ,

where

( R x ) i j n = h 1 4 360 ∫ 0 1 ∂ 6 v ∂ x 6 ( x i − s h 1 , y j , t n ) + ∂ 6 v ∂ x 6 ( x i + s h 1 , y j , t n ) ζ ( s ) d s , ( R y ) i j n = h 2 4 360 ∫ 0 1 ∂ 6 v ∂ y 6 ( x i , y j − s h 2 , t n ) + ∂ 6 v ∂ y 6 ( x i , y j + s h 2 , t n ) ζ ( s ) d s .

Obviously,

(17) L y L x Q i j n − 1 2 = L y δ x 2 V i j n − 1 2 + L y ( R x ) i j n − 1 2 , ( x i , y j ) ∈ Ω h , 1 ≤ n ≤ N ,

(18) L x L y R i j n − 1 2 = L x δ y 2 V i j n − 1 2 + L x ( R y ) i j n − 1 2 , ( x i , y j ) ∈ Ω h , 1 ≤ n ≤ N .

Substituting (16)–(18) into (15), we obtain

(19) τ 1 − α Γ ( 3 − α ) L a 0 δ t V i j n − 1 2 − ∑ k = 1 n − 1 ( a n − k − 1 − a n − k ) δ t V i j k − 1 2 − a n − 1 P i j 0 = Λ h V i j n − 1 2 + L g i j n − 1 2 + d L V i j n − 1 2 + ( R 1 ) i j n − 1 2 , ( x i , y j ) ∈ Ω h , 1 ≤ n ≤ N ,

where

( R 1 ) i j n − 1 2 = L ( R t α ) i j n − 1 2 − τ 1 − α Γ ( 3 − α ) L a 0 ( R t ) i j n − 1 2 − ∑ k = 1 n − 1 ( a n − k − 1 − a n − k ) ( R t ) i j k − 1 2 + L y ( R x ) i j n − 1 2 + L x ( R y ) i j n − 1 2 .

In view of the estimate

( R t ) i j n − 1 2 ≤ τ 2 12 max 0 ≤ t ≤ T ∂ 2 v ∂ t 2 ( x i , y j , t ) ,

we derive that

− τ 1 − α Γ ( 3 − α ) L a 0 ( R t ) i j n − 1 2 − ∑ k = 1 n − 1 ( a n − k − 1 − a n − k ) ( R t ) i j k − 1 2 ≤ τ 3 − α 6 Γ ( 3 − α ) max 0 ≤ t ≤ T ∂ 2 v ∂ t 2 ( x i , y j , t ) .

As a consequence of the aforementioned facts, we find that

(20) ( R 1 ) i j n − 1 2 ≤ C 1 ( τ 3 − α + h 1 4 + h 2 4 ) , ( x i , y j ) ∈ Ω h , 1 ≤ n ≤ N ,

where C 1 is a positive constant, which is dependent on the regularity of solution v ( x , y , t ) but independent of the time step size τ , grid spacing h 1 , h 2 , and time level n .

We add small term − Γ ( 3 − α ) τ α + 1 4 − 2 Γ ( 3 − α ) d τ α δ x 2 δ y 2 δ t V i j n − 1 2 into (19) in order to rewrite (19) in a familiar ADI form, thus we have

(21) τ 1 − α Γ ( 3 − α ) L a 0 δ t V i j n − 1 2 − ∑ k = 1 n − 1 ( a n − k − 1 − a n − k ) δ t V i j k − 1 2 − a n − 1 P i j 0 = Λ h V i j n − 1 2 − Γ ( 3 − α ) τ α + 1 4 − 2 Γ ( 3 − α ) d τ α δ x 2 δ y 2 δ t V i j n − 1 2 + L g i j n − 1 2 + d L V i j n − 1 2 + ( R ) i j n − 1 2 , ( x i , y j ) ∈ Ω h , 1 ≤ n ≤ N ,

where

( R ) i j n − 1 2 = ( R 1 ) i j n − 1 2 + Γ ( 3 − α ) τ α + 1 4 − 2 Γ ( 3 − α ) d τ α δ x 2 δ y 2 δ t V i j n − 1 2 .

Furthermore, from the initial and boundary value conditions, it follows that

(22) P i j 0 = ψ v ( x i , y j ) , ( x i , y j ) ∈ Ω h ,

(23) V i j 0 = v 0 ( x i , y j ) , ( x i , y j ) ∈ Ω ¯ h ,

(24) V i j n = ϕ v ( x i , y j , t n ) , ( x i , y j ) ∈ ∂ Ω h , 1 ≤ n ≤ N .

Substituting (22) into (21), omitting the small term ( R ) i j n , and replacing the function V i j n with its numerical approximation v i j n in (21) and (23)–(24), we obtain the following difference scheme:

(25) τ 1 − α Γ ( 3 − α ) L a 0 δ t v i j n − 1 2 − ∑ k = 1 n − 1 ( a n − k − 1 − a n − k ) δ t v i j k − 1 2 − a n − 1 ( ψ v ) i j = Λ h v i j n − 1 2 − Γ ( 3 − α ) τ α + 1 4 − 2 Γ ( 3 − α ) d τ α δ x 2 δ y 2 δ t v i j n − 1 2 + L g i j n − 1 2 + d L v i j n − 1 2 , ( x i , y j ) ∈ Ω h , 1 ≤ n ≤ N ,

(26) v i j n = ϕ v ( x i , y j , t n ) , ( x i , y j ) ∈ ∂ Ω h , 1 ≤ n ≤ N ,

(27) v i j 0 = v 0 ( x i , y j ) , ( x i , y j ) ∈ Ω ¯ h ,

where a k is defined as in (5), and ( ψ v ) i j = ψ v ( x i , y j ) .

For convenience, we denote μ = Γ ( 3 − α ) τ α − 1 throughout the paper. In order to rewrite (25) in a more familiar ADI form, we multiply (25) by μ and note that a 0 = 1 . Hence,

L δ t v i j n − 1 2 − μ Λ h v i j n − 1 2 + μ 2 τ 2 4 − 2 d μ τ δ x 2 δ y 2 δ t v i j n − 1 2 = ∑ k = 1 n − 1 ( a n − k − 1 − a n − k ) L δ t v i j k − 1 2 + a n − 1 L ( ψ v ) i j + L μ g i j n − 1 2 + d μ L v i j n − 1 2 , ( x i , y j ) ∈ Ω h , 1 ≤ n ≤ N ,

i.e.,

L x − τ μ 2 − d μ τ δ x 2 L y − τ μ 2 − d μ τ δ y 2 v i j n = 2 τ 2 − d μ τ ∑ k = 1 n − 1 ( a n − k − 1 − a n − k ) L δ t v i j k − 1 2 + a n − 1 L ( ψ v ) i j + L μ g i j n − 1 2 + 2 + d μ τ 2 − d μ τ L + μ τ 2 − d μ τ Λ h + μ 2 τ 2 ( 2 − d μ τ ) 2 δ x 2 δ y 2 v i j n − 1 , ( x i , y j ) ∈ Ω h , 1 ≤ n ≤ N .

We calculate { v i j n } by solving two sets of independent one-dimensional problems. By introducing intermediate variables

v i j ∗ = L y − τ μ 2 − d μ τ δ y 2 v i j n , 0 ≤ i ≤ M 1 , 1 ≤ j ≤ M 2 − 1 ,

we firstly solve the following one-dimensional problem for fixed j ∈ { 1 , 2 , … , M 2 − 1 } to compute { v i j ∗ } :

L x − τ μ 2 − d μ τ δ x 2 v i j ∗ = 2 + d μ τ 2 − d μ τ L + μ τ 2 − d μ τ Λ h + μ 2 τ 2 ( 2 − d μ τ ) 2 δ x 2 δ y 2 v i j n − 1 + 2 τ 2 − d μ τ ∑ k = 1 n − 1 ( a n − k − 1 − a n − k ) L δ t v i j k − 1 2 + a n − 1 L ( ψ v ) i j + L μ g i j n − 1 2 , 1 ≤ i ≤ M 1 − 1 , v 0 j ∗ = L y − τ μ 2 − d μ τ δ y 2 v 0 j n , v M 1 j ∗ = L y − τ μ 2 − d μ τ δ y 2 v M 1 j n .

Once { v i j ∗ } is available, we solve the following system for fixed i ∈ { 1 , 2 , … , M 1 − 1 } :

L y − τ μ 2 − d μ τ δ y 2 v i j n = v i j ∗ , 1 ≤ j ≤ M 2 − 1 , v i 0 n = ϕ v ( x i , y 0 , t n ) , v i M 2 n = ϕ v ( x i , y M 2 , t n ) .

Hence, we obtain the wanted solution { v i j n } .

2.3 Solvability and truncation error

As we analyzed before, there are two sets of one-dimensional linear systems that need to be solved at each time level. It is clear to see that the coefficient matrices are strictly diagonally dominant; therefore, we have the following result.

Theorem 2.1

The difference scheme (25)–(27) is uniquely solvable.

It remains to give the estimate of the truncation error, which will be used in the convergence analysis.

Lemma 2.4

Suppose that v ( x , y , t ) ∈ C x , y , t 6 , 6 , 3 ( Ω × [ 0 , T ] ) is the solution of the problem (1)–(3). Then the truncation error of the scheme (25)–(27) satisfies

( R ) i j n − 1 2 ≤ C R ( τ 3 − α + h 1 4 + h 2 4 ) , ( x i , y j ) ∈ Ω h , 1 ≤ n ≤ N ,

where C R is a positive constant independent of τ , h 1 , and h 2 .

Proof

Since the estimate of ( R 1 ) i j n − 1 2 has been given in (20), it remains to estimate the small term Γ ( 3 − α ) τ α + 1 4 − 2 Γ ( 3 − α ) d τ α δ x 2 δ y 2 δ t V i j n − 1 2 .

A direct calculation by using the Taylor expansion with integral remainder leads to

δ x 2 δ y 2 V i j n = D ( x i , y j , t n ) ,

where

D ( x , y , t ) = ∫ 0 1 ∫ 0 1 ( 1 − λ ) ( 1 − ζ ) ∂ 4 v ∂ 2 x ∂ 2 y ( x − λ h 1 , y − ζ h 2 , t ) + ∂ 4 v ∂ 2 x ∂ 2 y ( x + λ h 1 , y − ζ h 2 , t ) + ∂ 4 v ∂ 2 x ∂ 2 y ( x − λ h 1 , y + ζ h 2 , t ) + ∂ 4 v ∂ 2 x ∂ 2 y ( x + λ h 1 , y + ζ h 2 , t ) d λ d ζ .

Hence, we have

δ x 2 δ y 2 δ t V i j n − 1 2 = ∫ 0 1 ∂ D ∂ s ( x i , y j , t n − 1 + s τ ) d s , ( x i , y j ) ∈ Ω h , 1 ≤ n ≤ N

and

Γ ( 3 − α ) τ α + 1 4 − 2 Γ ( 3 − α ) d τ α δ x 2 δ y 2 δ t V i j n − 1 2 = Γ ( 3 − α ) τ α + 1 4 − 2 Γ ( 3 − α ) d τ α ∫ 0 1 ∂ D ∂ s ( x i , y j , t n − 1 + s τ ) d s = O ( τ 1 + α ) .

In view of the fact that 1 < α < 2 and 1 + α > 2 > 3 − α , we complete the proof.□

Remark 2.1

From Lemmas 2.1, 2.2, and 2.4, it follows that if the solution v ( x , y , t ) ∈ C x , y , t 6 , 6 , 3 ( Ω × [ 0 , T ] ) of (7)–(10) satisfies ∂ 3 v ( x , y , t ) ∂ t 3 = 0 for ( x , y , t ) ∈ Ω × [ 0 , T ] , then the truncation error of the scheme (25)–(27) satisfies

( R ) i j n − 1 2 ≤ C ˜ R ( τ 1 + α + h 1 4 + h 2 4 ) , ( x i , y j ) ∈ Ω h , 1 ≤ n ≤ N ,

where C ˜ R is a positive constant independent of τ , h 1 , and h 2 .

3 Stability and convergence of the compact ADI scheme

3.1 Stability

In order to analyze the stability of the proposed scheme, we introduce the space of grid functions on Ω ¯ h ,

V h = { v ∣ v = { v i j ∣ ( x i , y j ) ∈ Ω ¯ h } and v i j = 0 if ( x i , y j ) ∈ ∂ Ω h } .

In addition, for any grid function w , v ∈ V h , we define the discrete inner product by

⟨ w , v ⟩ h = h 1 h 2 ∑ i = 1 M 1 − 1 ∑ j = 1 M 2 − 1 w i j ⋅ v i j

and denote ∥ v ∥ = ⟨ v , v ⟩ h . Similar notations ∥ δ x 2 v ∥ , ∥ δ y 2 v ∥ , ∥ L x v ∥ , ∥ L y v ∥ , ∥ L v ∥ , and ∥ δ x 2 δ y 2 v ∥ can be defined immediately. Furthermore, we also denote

∥ δ x v ∥ = h 1 h 2 ∑ i = 1 M 1 ∑ j = 1 M 2 − 1 δ x v i − 1 2 , j 2 , ∥ δ x δ y v ∥ = h 1 h 2 ∑ i = 1 M 1 ∑ j = 1 M 2 δ x δ y v i − 1 2 , j − 1 2 2

and

∥ δ y δ x 2 v ∥ = h 1 h 2 ∑ i = 1 M 1 − 1 ∑ j = 1 M 2 δ y δ x 2 v i , j − 1 2 2 .

Then we denote ∥ δ y v ∥ , ∥ δ x δ y 2 v ∥ in a similar way. The discrete H 1 seminorm and H 1 norm of the grid function v ∈ V h are defined as

∥ ∇ h v ∥ = ∥ δ x v ∥ 2 + ∥ δ y v ∥ 2 , ∥ v ∥ H 1 = ∥ v ∥ 2 + ∥ ∇ h v ∥ 2 .

For any grid function v ∈ V h , we denote

∥ v ∥ A = ∥ δ x v ∥ 2 − h 1 2 12 ∥ δ x 2 v ∥ 2 , ∥ v ∥ B = ∥ δ y v ∥ 2 − h 2 2 12 ∥ δ y 2 v ∥ 2

and

∥ v ∥ H ˜ 1 = ∥ L y v ∥ A 2 + ∥ L x v ∥ B 2 , ∥ v ∥ H ˜ 2 = ∥ L y v ∥ A 2 + ∥ L x v ∥ B 2 − d ∥ L v ∥ 2 .

As we will see later, the norm ∥ ⋅ ∥ H ˜ 2 is equivalent to the standard H 1 norm and is more convenient than the standard H 1 norm for the stability and convergence analysis.

Lemma 3.1

[41] For any grid function v ∈ V h , it holds that

48 ( L 1 2 + L 2 2 ) 27 [ 6 ( L 1 2 + L 2 2 ) + L 1 2 L 2 2 ] ∥ v ∥ H 1 ≤ ∥ v ∥ H ˜ 1 ≤ 4 3 ∥ v ∥ H 1 .

Lemma 3.2

For any grid function v ∈ V h , it holds that

48 ( L 1 2 + L 2 2 ) 27 [ 6 ( L 1 2 + L 2 2 ) + L 1 2 L 2 2 ] ∥ v ∥ H 1 ≤ ∥ v ∥ H ˜ 2 ≤ 16 9 − 485 d 162 ∥ v ∥ H 1 .

Proof

As a consequence of ∥ v ∥ H ˜ 2 ≥ ∥ v ∥ H ˜ 1 and Lemma 3.1, we find that

48 ( L 1 2 + L 2 2 ) 27 [ 6 ( L 1 2 + L 2 2 ) + L 1 2 L 2 2 ] ∥ v ∥ H 1 ≤ ∥ v ∥ H ˜ 2 .

The rest is to verify the inequality on the right side. Noticing that ∥ v ∥ H ˜ 2 2 = ∥ v ∥ H ˜ 1 2 − d ∥ L v ∥ 2 and applying the discrete Green formula, we get

∥ L v ∥ 2 = v + h 1 2 12 δ x 2 v + h 2 2 12 δ y 2 v + h 1 2 h 2 2 144 δ x 2 δ y 2 v , v + h 1 2 12 δ x 2 v + h 2 2 12 δ y 2 v + h 1 2 h 2 2 144 δ x 2 δ y 2 v h = ∥ v ∥ 2 + h 1 2 12 ∥ δ x v ∥ 2 + h 2 2 12 ∥ δ y v ∥ 2 + h 1 2 h 2 2 144 ∥ δ x δ y v ∥ 2 + h 1 2 12 ∥ δ x v ∥ 2 + h 1 4 144 ∥ δ x 2 v ∥ 2 + h 1 2 h 2 2 144 ∥ δ x δ y v ∥ 2 + h 1 4 h 2 2 1,728 ⟨ δ x 2 v , δ x δ y v ⟩ h + h 2 2 12 ∥ δ y v ∥ 2 + h 2 4 144 ∥ δ y 2 v ∥ 2 + h 1 2 h 2 2 144 ∥ δ x δ y v ∥ 2 + h 1 2 h 2 4 1,728 ⟨ δ y 2 v , δ x δ y v ⟩ h + h 1 2 h 2 2 144 ⟨ v , δ x 2 δ y 2 v ⟩ h + h 1 4 h 2 2 1,728 ∥ δ x 2 δ y v ∥ 2 + h 1 2 h 2 4 1,728 ∥ δ x δ y 2 v ∥ 2 + h 1 4 h 2 4 20,736 ∥ δ x 2 δ y 2 v ∥ 2 = ∥ v ∥ 2 + h 1 2 6 ∥ δ x v ∥ 2 + h 2 2 6 ∥ δ y v ∥ 2 + h 1 4 144 ∥ δ x 2 v ∥ 2 + h 2 4 144 ∥ δ y 2 v ∥ 2 + h 1 2 h 2 2 36 ∥ δ x δ y v ∥ 2 + h 1 4 h 2 2 864 ∥ δ x 2 δ y v ∥ 2 + h 1 2 h 2 4 864 ∥ δ x δ y 2 v ∥ 2 + h 1 4 h 2 4 20,736 ∥ δ x 2 δ y 2 v ∥ 2 .

In view of inverse estimates ∥ δ x 2 v ∥ ≤ 4 h 1 2 ∥ v ∥ and ∥ δ y 2 v ∥ ≤ 4 h 2 2 ∥ v ∥ , we derive that

∥ L v ∥ 2 ≤ 485 162 ∥ v ∥ 2 ≤ 485 162 ∥ v ∥ H 1 2 .

Moreover, as a direct consequence of Lemma 3.1, we have

∥ v ∥ H ˜ 2 2 ≤ 16 9 − 485 d 162 ∥ v ∥ H 1 2 .

This completes the proof.□

Lemma 3.3

For any grid function v n ∈ V h , it holds that

(28) − L x δ t v n − 1 2 , δ x 2 v n − 1 2 h = 1 2 τ ( ∥ v n ∥ A 2 − ∥ v n − 1 ∥ A 2 ) ,

(29) − L y δ t v n − 1 2 , δ y 2 v n − 1 2 h = 1 2 τ ( ∥ v n ∥ B 2 − ∥ v n − 1 ∥ B 2 ) .

Proof

A combination of the definition of operator L x and the discrete Green formula leads to

− L δ t v n − 1 2 , δ x 2 v n − 1 2 h = − δ t v n − 1 2 , δ x 2 v n − 1 2 h − h 1 2 12 δ t δ x 2 v n − 1 2 , δ x 2 v n − 1 2 h = 1 2 τ ( ∥ δ x v n ∥ 2 − ∥ δ x v n − 1 ∥ 2 ) − h 1 2 12 ( ∥ δ x 2 v n ∥ 2 − ∥ δ x 2 v n − 1 ∥ 2 ) .

Thus, the proof of equality (28) is complete. Obviously, we can prove equality (29) similarly.□

Lemma 3.4

For any grid function v ∈ V h , it holds that ⟨ δ x 2 δ y 2 v , L v ⟩ h ≥ 1 3 ∥ δ x δ y v ∥ 2 .

Proof

From the definition of operator L , it follows that

L v = v + h 1 2 12 δ x 2 v + h 2 2 12 δ y 2 v + h 1 2 h 2 2 144 δ x 2 δ y 2 v .

In view of the discrete Green formula, we derive that

⟨ δ x 2 δ y 2 v , L v ⟩ h = δ x 2 δ y 2 v , v + h 1 2 12 δ x 2 v + h 2 2 12 δ y 2 v + h 1 2 h 2 2 144 δ x 2 δ y 2 v h = ∥ δ x δ y v ∥ 2 − h 1 2 12 ∥ δ x 2 δ y v ∥ 2 − h 2 2 12 ∥ δ x δ y 2 v ∥ 2 + h 1 2 h 2 2 144 ∥ δ x 2 δ y 2 v ∥ 2 ≥ 1 3 ∥ δ x δ y v ∥ 2 ,

where inverse estimates ∥ δ x 2 δ y v ∥ ≤ 2 h 1 ∥ δ x δ y v ∥ and ∥ δ x δ y 2 v ∥ ≤ 2 h 2 ∥ δ x δ y v ∥ are used.□

Lemma 3.5

For any grid function v n ∈ V h , it holds that

(30) ⟨ Λ h v n − 1 2 , L δ t v n − 1 2 ⟩ h = − 1 2 τ ( ∥ v n ∥ H ˜ 1 2 − ∥ v n − 1 ∥ H ˜ 1 2 ) .

Proof

From the definition of difference operators Λ h and L , it follows that

(31) ⟨ Λ h v n − 1 2 , L δ t v n − 1 2 ⟩ h = ⟨ L y δ x 2 v n − 1 2 , L δ t v n − 1 2 ⟩ h + ⟨ L x δ y 2 v n − 1 2 , L δ t v n − 1 2 ⟩ h .

Since operators L x , L y , δ t , and δ x 2 are linear, we obtain

L y δ x 2 v n − 1 2 , L δ t v n − 1 2 h = L y δ x 2 v n − 1 2 , L x L y δ t v n − 1 2 h = L x δ t L y v n − 1 2 , δ x 2 L y v n − 1 2 h ,

which, together with Lemma 3.3, leads to

(32) L y δ x 2 v n − 1 2 , L δ t v n − 1 2 h = − 1 2 τ ( ∥ L y v n ∥ A 2 − ∥ L y v n − 1 ∥ A 2 ) .

Similarly, we have

(33) L x δ y 2 v n − 1 2 , L δ t v n − 1 2 h = − 1 2 τ ( ∥ L x v n ∥ B 2 − ∥ L x v n − 1 ∥ B 2 ) .

Substituting (32) and (33) into (31), we complete the proof.□

We now prove that the difference scheme (25)–(27) is stable to the initial values v 0 , ψ v and the forcing term g .

Noting that a k = ( 2 − α ) ∫ k k + 1 x 1 − α d x , and x 1 − α is a monotone decreasing function, one verifies readily that

(34) 1 = a 0 > a 1 > a 2 > ⋯ > a n > ⋯ → 0 ,

(35) ( 2 − α ) ( k + 1 ) 1 − α < a k < ( 2 − α ) k 1 − α .

Theorem 3.1

Suppose v n ∈ V h is the solution of the difference scheme (25)–(27), then it holds that

∥ v n ∥ H ˜ 2 2 ≤ ∥ v 0 ∥ H ˜ 2 2 + t n 2 − α Γ ( 3 − α ) ∥ L ψ v ∥ 2 + Γ ( 2 − α ) t n α − 1 τ ∑ k = 1 n L g k − 1 2 2 , 1 ≤ n ≤ N .

Proof

Multiplying (25) by 2 μ h 1 h 2 L δ t v i j n − 1 2 and summing over i , j for ( x i , y j ) ∈ Ω h , we have the following equation for 1 ≤ n ≤ N :

(36) 2 L δ t v n − 1 2 2 − 2 ∑ k = 1 n − 1 ( a n − k − 1 − a n − k ) L δ t v k − 1 2 , L δ t v n − 1 2 h = 2 a n − 1 L ψ v , L δ t v n − 1 2 h + 2 μ Λ h v n − 1 2 , L δ t v n − 1 2 h + 2 μ d L v n − 1 2 , L δ t v n − 1 2 h + 2 μ L g n − 1 2 , L δ t v n − 1 2 h − ( τ μ ) 2 2 − d μ τ δ x 2 δ y 2 δ t v n − 1 2 , L δ t v n − 1 2 h .

In view of Lemmas 3.4 and 3.5, we find that

(37) − ( τ μ ) 2 2 − d μ τ δ x 2 δ y 2 δ t v n − 1 2 , L δ t v n − 1 2 h ≤ − ( τ μ ) 2 6 − 3 d μ τ ∥ δ x δ y δ t v n − 1 2 ∥ 2 ≤ 0

and

(38) 2 μ Λ h v n − 1 2 , L δ t v n − 1 2 h = − μ τ ( ∥ v n ∥ H ˜ 1 2 − ∥ v n − 1 ∥ H ˜ 1 2 ) .

Substituting (37)–(38) into (36) and noticing that both a n − 1 and a n − k − 1 − a n − k are positive, we obtain

2 L δ t v n − 1 2 2 + μ τ ( ∥ v n ∥ H ˜ 1 2 − ∥ v n − 1 ∥ H ˜ 1 2 ) ≤ 2 ∑ k = 1 n − 1 ( a n − k − 1 − a n − k ) L δ t v k − 1 2 , L δ t v n − 1 2 h + 2 a n − 1 L ψ v , L δ t v n − 1 2 h + 2 μ L g n − 1 2 , L δ t v n − 1 2 h + 2 μ d L v n − 1 2 , L δ t v n − 1 2 h

≤ ∑ k = 1 n − 1 ( a n − k − 1 − a n − k ) L δ t v k − 1 2 2 + L δ t v n − 1 2 2 + a n − 1 L δ t v n − 1 2 2 + ∥ L ψ v ∥ 2 + 2 μ L g n − 1 2 , L δ t v n − 1 2 h + μ d τ ( ∥ L v n ∥ 2 − ∥ L v n − 1 ∥ 2 ) = L δ t v n − 1 2 2 + ∑ k = 1 n − 1 ( a n − k − 1 − a n − k ) L δ t v k − 1 2 2 + a n − 1 ∥ L ψ v ∥ 2 + 2 μ L g n − 1 2 , L δ t v n − 1 2 h + μ d τ ( ∥ L v n ∥ 2 − ∥ L v n − 1 ∥ 2 ) ,

i.e.,

(39) L δ t v n − 1 2 2 + μ τ ( ∥ v n ∥ H ˜ 1 2 − ∥ v n − 1 ∥ H ˜ 1 2 ) ≤ ∑ k = 1 n − 1 ( a n − k − 1 − a n − k ) L δ t v k − 1 2 2 + a n − 1 ∥ L ψ v ∥ 2 + 2 μ L g n − 1 2 , L δ t v n − 1 2 h + μ d τ ( ∥ L v n ∥ 2 − ∥ L v n − 1 ∥ 2 ) , 1 ≤ n ≤ N .

For convenience, we denote F 0 = ∥ v 0 ∥ H ˜ 1 2 − d ∥ L v 0 ∥ 2 and

F n = ∥ v n ∥ H ˜ 1 2 + τ μ ∑ k = 1 n a n − k ∥ L δ t v k − 1 2 ∥ 2 − d ∥ L v n ∥ 2 , 1 ≤ n ≤ N .

Multiplying inequality (39) by τ μ , we have

F n ≤ F n − 1 + a n − 1 τ μ ∥ L ψ v ∥ 2 + 2 τ L g n − 1 2 , L δ t v n − 1 2 h ≤ F 0 + τ μ ∑ k = 1 n a n − k ∥ L ψ v ∥ 2 + τ ∑ k = 1 n μ a n − k L g k − 1 2 2 + a n − k μ L δ t v k − 1 2 2 , 1 ≤ n ≤ N .

Hence, for 1 ≤ n ≤ N , it holds that

(40) ∥ v n ∥ H ˜ 1 2 − d ∥ L v n ∥ 2 ≤ ∥ v 0 ∥ H ˜ 1 2 − d ∥ L v 0 ∥ 2 + τ μ ∑ k = 1 n a n − k ∥ L ψ v ∥ 2 + τ ∑ k = 1 n μ a n − k L g k − 1 2 2 .

By a direct calculation, we have ∑ k = 1 n a k − 1 = n 2 − α , which means

(41) τ μ ∑ k = 1 n a n − k ∥ L ψ v ∥ 2 = t n 2 − α Γ ( 3 − α ) ∥ L ψ v ∥ 2 .

Noting that a n − k ≥ ( 2 − α ) ( n − k + 1 ) 1 − α ≥ ( 2 − α ) n 1 − α , we obtain

μ a n − k = Γ ( 3 − α ) τ α − 1 a n − k ≤ Γ ( 3 − α ) τ α − 1 ( 2 − α ) n 1 − α = Γ ( 2 − α ) t n α − 1 .

Furthermore, we have

(42) τ ∑ k = 1 n μ a n − k L g k − 1 2 2 ≤ Γ ( 2 − α ) t n α − 1 τ ∑ k = 1 n L g k − 1 2 2 .

Substituting estimates (41) and (42) into (40), we complete the proof.□

3.2 Convergence

On the basis of the analysis of stability, the convergence result of the difference scheme (25)–(27) is proposed in this subsection. Let

e i j n = V i j n − v i j n , ( x i , y j ) ∈ Ω ¯ h , 0 ≤ n ≤ N .

Subtracting (25)–(27) from (21)–(23), we get the error system

(43) τ 1 − α Γ ( 3 − α ) L a 0 δ t e i j n − 1 2 − ∑ k = 1 n − 1 ( a n − k − 1 − a n − k ) δ t e i j k − 1 2 = Λ h e i j n − 1 2 − Γ ( 3 − α ) τ α + 1 4 − 2 Γ ( 3 − α ) d τ α δ x 2 δ y 2 δ t e i j n − 1 2 + d L e i j n − 1 2 + ( R ) i j n − 1 2 , ( x i , y j ) ∈ Ω h , 1 ≤ n ≤ N ,

(44) e i j n = 0 , ( x i , y j ) ∈ ∂ Ω h , 1 ≤ n ≤ N ,

(45) e i j 0 = 0 , ( x i , y j ) ∈ Ω ¯ h .

Applying Theorem 3.1, we have that

∥ e n ∥ H ˜ 2 2 ≤ Γ ( 2 − α ) t n α − 1 τ ∑ k = 1 n ( R ) k − 1 2 2 , 1 ≤ n ≤ N .

By utilizing Lemma 2.4, we get the following convergence result.

Theorem 3.2

Assume that the problem (7)–(10) has a smooth solution v ( x , y , t ) ∈ C x , y , t 6 , 6 , 3 ( Ω ¯ × [ 0 , T ] ) , and let { v i j n ∣ ( x i , y j ) ∈ Ω h , 1 ≤ n ≤ N } be the solution of the difference scheme (25)–(27). Then it holds that

∥ V n − v n ∥ H ˜ 2 ≤ C ( τ 3 − α + h 1 4 + h 2 4 ) ,

where C = C R Γ ( 2 − α ) T α L 1 L 2 .

Based on Remark 2.1, we get the following corollary:

Corollary 3.1

Assume that the problem (7)–(10) has a smooth solution v ( x , y , t ) ∈ C x , y , t 6 , 6 , 3 ( Ω ¯ × [ 0 , T ] ) and ∂ 3 v ( x , y , t ) ∂ t 3 = 0 for ( x , y , t ) ∈ Ω × [ 0 , T ] , let { v i j n ∣ ( x i , y j ) ∈ Ω h , 1 ≤ n ≤ N } be the solution of the difference scheme (25)–(27). Then it holds that

∥ V n − v n ∥ H ˜ 2 ≤ C ˜ ( τ 1 + α + h 1 4 + h 2 4 ) ,

where C ˜ = C ˜ R Γ ( 2 − α ) T α L 1 L 2 .

4 Numerical experiments and implementation

In the following, we present a few results to numerically validate the analysis. In this section, we present implementation of the compact ADI scheme briefly and give some numerical examples to support our theory. All experiments are conducted on a Macbook Pro with a 2.9 GHz Intel i5 CPU and 8 GB RAM.

In the runs, we use the same spacing h in each direction, h 1 = h 2 = h , and compute the maximum norm errors of the numerical solution

e ( τ , h ) = max 1 ≤ n ≤ N max ( x i , y j ) ∈ Ω h ∣ v ( x i , y j , t n ) − v i j n ∣ ,

and we denote

rate 1 = log τ 1 τ 2 e ( τ 1 , h ) e ( τ 2 , h ) , rate 2 = log h 1 h 2 e ( τ , h 1 ) e ( τ , h 2 ) .

4.1 Example 1

As the first example, we consider the fractional convection-diffusion equation:

(46) D t α 0 C u = Δ u − u x − 2 u y + u 4 + f , ( x , y , t ) ∈ Ω × ( 0 , T ] ,

(47) u ( x , y , 0 ) = u 0 ( x , y ) , ∂ u ( x , y , 0 ) ∂ t = 0 , ( x , y ) ∈ Ω ¯ = Ω ∪ ∂ Ω ,

(48) u ( x , y , t ) = 0 , ( x , y , t ) ∈ ∂ Ω × ( 0 , T ] ,

where Ω = ( 0 , π ) × ( 0 , π ) and T = 1 . We choose the source term

f ( x , y , t ) = e x 2 + y sin x sin y Γ ( 3 + α ) t 2 2 + 3 t 2 + α

to obtain an exact solution u ( x , y , t ) = e x 2 + y t 2 + α sin x sin y .

We first investigate the temporal errors and convergence orders of the compact ADI scheme. In this test, we fix h = π / 16 , a value small enough such that the spatial error is negligible as compared with the temporal error. Table 1 presents the maximum norm errors and temporal convergence orders of the scheme for α = 1.25 , 1.5 , 1.75 . It is observed that the scheme generates ( 3 − α ) temporal convergence order. Then, we plot the exact solution (Figure 1) and the numerical result (Figure 2(a)) and the error between them (Figure 2(b)) at T = 1 , with α = 1.5 , M = 20 , and N = 40 .

Table 1

Maximum norm errors and temporal convergence orders of the proposed scheme when h = π / 16 for Example 1

τ e ( τ , h ) rate 1
α = 1.25 1/5 0.0212
1/10 0.0066 1.6835
1/20 0.0020 1.7225
1/40 6.1280 × 1 0 − 4 1.7065
1/80 1.8568 × 1 0 − 4 1.7226
α = 1.5 1/5 0.0693
1/10 0.0254 1.4480
1/20 0.0091 1.4809
1/40 0.0033 1.4634
1/80 0.0012 1.4594
α = 1.75 1/5 0.1721
1/10 0.0736 1.2255
1/20 0.0312 1.2382
1/40 0.0132 1.2410
1/80 0.0056 1.2370
Figure 1 The exact solution at T = 1 T=1 with α = 1.5 \alpha =1.5 , M = 20 M=20 , and N = 40 N=40 for Example 1.
Figure 1

The exact solution at T = 1 with α = 1.5 , M = 20 , and N = 40 for Example 1.

Figure 2 The plot of the numerical solution and the error between exact solution and the numerical solution computed by the proposed scheme at T = 1 T=1 with α = 1.5 \alpha =1.5 , M = 20 M=20 , and N = 40 N=40 for Example 1. (a) Numerical solution (b) Error.
Figure 2

The plot of the numerical solution and the error between exact solution and the numerical solution computed by the proposed scheme at T = 1 with α = 1.5 , M = 20 , and N = 40 for Example 1. (a) Numerical solution (b) Error.

Figure 3 shows the contour plot of the exact solution in the x – y plane and numerical solutions using the compact ADI scheme for different M at T = 1 with α = 1.5 , N = 100 . We can see that when M = 20 , the numerical solution is a good approximation of the exact solution. Finally, we observe the CPU time of the scheme, and the results are shown in Table 2. In order to illustrate the efficiency of the proposed ADI method, we compare the CPU time of the proposed ADI method with that of the direct method, which is derived from (19) by neglecting the truncation error. The comparison results are listed in Table 3. It is obvious that the proposed method is more efficient than the direct method, especially for large M and N .

Figure 3 The contour plot of the exact solution and the numerical solutions using the compact ADI scheme for different M M at T = 1 T=1 with α = 1.5 \alpha =1.5 , N = 100 N=100 for Example 1. (a) Exact solution, (b) M = 5 M=5 , (c) M = 10 M=10 , (d) M = 20 M=20 .
Figure 3

The contour plot of the exact solution and the numerical solutions using the compact ADI scheme for different M at T = 1 with α = 1.5 , N = 100 for Example 1. (a) Exact solution, (b) M = 5 , (c) M = 10 , (d) M = 20 .

Table 2

Maximum norm error and CPU time of the proposed scheme for Example 1

α N M e ( τ , h ) CPU time (s)
1.25 20 4 0.0018 0.024000
40 8 6.3446 × 1 0 − 4 0.132590
80 16 1.8568 × 1 0 − 4 1.783507
160 32 5.5383 × 1 0 − 5 23.578173
1.5 20 4 0.0094 0.032618
40 8 0.0033 0.129855
80 16 0.0012 1.586771
160 32 4.0983 × 1 0 − 4 26.438435
1.75 20 4 0.0314 0.030852
40 8 0.0132 0.149223
80 16 0.0056 1.799616
160 32 0.0023 23.534627
Table 3

CPU time (s) of the direct method and that of the proposed ADI method for Example 1

α N M Direct method ADI method
1.25 100 40 7.965835 5.015337
150 60 71.260503 25.786474
200 80 367.464825 79.461959
1.5 100 40 8.434888 5.450105
150 60 66.999567 23.968653
200 80 360.612136 79.041365

4.2 Example 2

In order to verify Corollary 3.1, the following TF convection-diffusion problem is considered:

D t α 0 C u = Δ u − 2 u x − 2 u y + u + f , ( x , y , t ) ∈ Ω × ( 0 , T ] , u ( x , y , t ) = ( t 2 + 1 ) y 3 ( 1 − y ) 2 sin ( π x ) e x + y , ( x , y , t ) ∈ Ω ¯ × ( 0 , T ] , u ( x , y , 0 ) = y 3 ( 1 − y ) 2 sin ( π x ) e x + y , ∂ u ( x , y , 0 ) ∂ t = 0 , ( x , y ) ∈ Ω ¯ = Ω ∪ ∂ Ω , u ( x , y , t ) = 0 , ( x , y , t ) ∈ ∂ Ω × ( 0 , T ] .

The domain considered here is ( 0 , 1 ) × ( 0 , 1 ) and T = 1 . The function f is specified to satisfy the given equation and exact solution. We compute the problem by using the proposed scheme (25)–(27).

In this experiment, we test the temporal errors and convergence orders of the scheme by letting τ vary and fixing the space step h sufficiently small to avoid contamination of the spatial error. Table 4 gives the maximum norm errors and temporal convergence orders of the scheme. In the runs, we let h = 1 / 20 for the compact ADI scheme. It is observed that the temporal order of the scheme is ( α + 1 ) , which verifies Corollary 3.1 successfully. Finally, the spatial error and convergence order of the scheme are presented in Table 5. It is shown that the compact ADI scheme has fourth order spatial accuracy.

Table 4

Maximum norm errors and temporal convergence orders of the proposed scheme when h = 1 / 20 for Example 2

τ e ( τ , h ) rate 1
α = 1.25 1/5 0.0021
1/10 4.6123 × 1 0 − 4 2.1868
1/20 9.8952 × 1 0 − 5 2.2207
1/40 2.0907 × 1 0 − 5 2.2558
1/80 4.3548 × 1 0 − 6 2.2633
α = 1.5 1/5 0.0014
1/10 2.5717 × 1 0 − 4 2.4446
1/20 4.5920 × 1 0 − 5 2.4855
1/40 8.0901 × 1 0 − 6 2.5049
1/80 1.3771 × 1 0 − 6 2.5545
α = 1.75 1/5 0.0010
1/10 1.5437 × 1 0 − 4 2.6955
1/20 2.3112 × 1 0 − 5 2.7397
1/40 3.3932 × 1 0 − 6 2.7679
1/80 4.5338 × 1 0 − 7 2.9039
Table 5

Maximum norm errors and spatial convergence orders of the proposed scheme when τ = 1/1,000 for Example 2

h e ( τ , h ) rate 2
α = 1.25 1/5 1.6698 × 1 0 − 5
1/10 1.0664 × 1 0 − 6 3.9689
1/15 1.9859 × 1 0 − 7 4.1451
1/20 6.3018 × 1 0 − 8 3.9899
1/25 2.4538 × 1 0 − 8 4.2269
α = 1.5 1/5 1.6447 × 1 0 − 5
1/10 1.0618 × 1 0 − 6 3.9532
1/15 2.0648 × 1 0 − 7 4.0386
1/20 6.4268 × 1 0 − 8 4.0571
1/25 2.4801 × 1 0 − 8 4.2671
α = 1.75 1/5 1.4937 × 1 0 − 5
1/10 9.6587 × 1 0 − 7 3.9509
1/15 1.8890 × 1 0 − 7 4.0245
1/20 5.9753 × 1 0 − 8 4.0009
1/25 2.4200 × 1 0 − 8 4.0506

Figure 3 shows the contour plot of the exact solution in the x – y plane and numerical solutions using the scheme for different M at T = 1 with α = 1.75 , N = 100 . We can see that the numerical solution is a good approximation of the exact solution when M = 20 .

5 Conclusion

In this paper, a compact ADI scheme for solving the convection-diffusion equation is proposed. We proved that the ADI scheme is unconditionally stable to the initial values and the inhomogeneous term. Besides, the numerical solution is convergent in the maximum norm. The coefficient matrix of the scheme is tridiagonal at each temporal level, so it can be solved by the Thomas algorithm. It is presented that the compact ADI scheme has fourth order spatial accuracy. In addition, the scheme generates ( 3 − α ) temporal convergence order. We show that when the solution of the problem satisfies certain condition, the temporal convergence order can attain ( 1 + α ) . Numerical experiments are carried out to verify the convergence order and show the efficiency of the scheme. Temporal error bounds of the solutions of the ADI scheme are achieved without losing the accuracy.

  1. Funding information: This work was partially supported by the National Natural Science Foundation of China (11971131 and 11871133), the Guangdong Basic and Applied Basic Research Foundation (2020B1515310010), and the Natural Sciences Foundation of Heilongjiang Province (LC2018001), Heilongjiang Postdoctoral Fund (LBH-Z20151), and China Postdoctoral Science Foundation (2020M670893).

  2. Conflict of interest: Authors state no conflict of interest.

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Received: 2020-06-20
Revised: 2021-02-03
Accepted: 2021-02-18
Published Online: 2021-08-16

© 2021 Gang Dong et al., published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/math-2021-0036
Keywords for this article
2D time-fractional convection-diffusion equation; ADI scheme; unconditional stability; convergence
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  2. Sharp conditions for the convergence of greedy expansions with prescribed coefficients
  3. Range-kernel weak orthogonality of some elementary operators
  4. Stability analysis for Selkov-Schnakenberg reaction-diffusion system
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  71. Fractional Hermite-Hadamard-type inequalities for interval-valued co-ordinated convex functions
  72. Montgomery identity and Ostrowski-type inequalities via quantum calculus
  73. Universal inequalities of the poly-drifting Laplacian on smooth metric measure spaces
  74. On reducible non-Weierstrass semigroups
  75. so-metrizable spaces and images of metric spaces
  76. Some new parameterized inequalities for co-ordinated convex functions involving generalized fractional integrals
  77. The concept of cone b-Banach space and fixed point theorems
  78. Complete consistency for the estimator of nonparametric regression model based on m-END errors
  79. A posteriori error estimates based on superconvergence of FEM for fractional evolution equations
  80. Solution of integral equations via coupled fixed point theorems in 𝔉-complete metric spaces
  81. Symmetric pairs and pseudosymmetry of Θ-Yetter-Drinfeld categories for Hom-Hopf algebras
  82. A new characterization of the automorphism groups of Mathieu groups
  83. The role of w-tilting modules in relative Gorenstein (co)homology
  84. Primitive and decomposable elements in homology of ΩΣℂP∞
  85. The G-sequence shadowing property and G-equicontinuity of the inverse limit spaces under group action
  86. Classification of f-biharmonic submanifolds in Lorentz space forms
  87. Some new results on the weaving of K-g-frames in Hilbert spaces
  88. Matrix representation of a cross product and related curl-based differential operators in all space dimensions
  89. Global optimization and applications to a variational inequality problem
  90. Functional equations related to higher derivations in semiprime rings
  91. A partial order on transformation semigroups with restricted range that preserve double direction equivalence
  92. On multi-step methods for singular fractional q-integro-differential equations
  93. Compact perturbations of operators with property (t)
  94. Entire solutions for several complex partial differential-difference equations of Fermat type in ℂ2
  95. Random attractors for stochastic plate equations with memory in unbounded domains
  96. On the convergence of two-step modulus-based matrix splitting iteration method
  97. On the separation method in stochastic reconstruction problem
  98. Robust estimation for partial functional linear regression models based on FPCA and weighted composite quantile regression
  99. Structure of coincidence isometry groups
  100. Sharp function estimates and boundedness for Toeplitz-type operators associated with general fractional integral operators
  101. Oscillatory hyper-Hilbert transform on Wiener amalgam spaces
  102. Euler-type sums involving multiple harmonic sums and binomial coefficients
  103. Poly-falling factorial sequences and poly-rising factorial sequences
  104. Geometric approximations to transition densities of Jump-type Markov processes
  105. Multiple solutions for a quasilinear Choquard equation with critical nonlinearity
  106. Bifurcations and exact traveling wave solutions for the regularized Schamel equation
  107. Almost factorizable weakly type B semigroups
  108. The finite spectrum of Sturm-Liouville problems with n transmission conditions and quadratic eigenparameter-dependent boundary conditions
  109. Ground state sign-changing solutions for a class of quasilinear Schrödinger equations
  110. Epi-quasi normality
  111. Derivative and higher-order Cauchy integral formula of matrix functions
  112. Commutators of multilinear strongly singular integrals on nonhomogeneous metric measure spaces
  113. Solutions to a multi-phase model of sea ice growth
  114. Existence and simulation of positive solutions for m-point fractional differential equations with derivative terms
  115. Bernstein-Walsh type inequalities for derivatives of algebraic polynomials in quasidisks
  116. Review Article
  117. Semiprimeness of semigroup algebras
  118. Special Issue on Problems, Methods and Applications of Nonlinear Analysis (Part II)
  119. Third-order differential equations with three-point boundary conditions
  120. Fractional calculus, zeta functions and Shannon entropy
  121. Uniqueness of positive solutions for boundary value problems associated with indefinite ϕ-Laplacian-type equations
  122. Synchronization of Caputo fractional neural networks with bounded time variable delays
  123. On quasilinear elliptic problems with finite or infinite potential wells
  124. Deterministic and random approximation by the combination of algebraic polynomials and trigonometric polynomials
  125. On a fractional Schrödinger-Poisson system with strong singularity
  126. Parabolic inequalities in Orlicz spaces with data in L1
  127. Special Issue on Evolution Equations, Theory and Applications (Part II)
  128. Impulsive Caputo-Fabrizio fractional differential equations in b-metric spaces
  129. Existence of a solution of Hilfer fractional hybrid problems via new Krasnoselskii-type fixed point theorems
  130. On a nonlinear system of Riemann-Liouville fractional differential equations with semi-coupled integro-multipoint boundary conditions
  131. Blow-up results of the positive solution for a class of degenerate parabolic equations
  132. Long time decay for 3D Navier-Stokes equations in Fourier-Lei-Lin spaces
  133. On the extinction problem for a p-Laplacian equation with a nonlinear gradient source
  134. General decay rate for a viscoelastic wave equation with distributed delay and Balakrishnan-Taylor damping
  135. On hyponormality on a weighted annulus
  136. Exponential stability of Timoshenko system in thermoelasticity of second sound with a memory and distributed delay term
  137. Convergence results on Picard-Krasnoselskii hybrid iterative process in CAT(0) spaces
  138. Special Issue on Boundary Value Problems and their Applications on Biosciences and Engineering (Part I)
  139. Marangoni convection in layers of water-based nanofluids under the effect of rotation
  140. A transient analysis to the M(τ)/M(τ)/k queue with time-dependent parameters
  141. Existence of random attractors and the upper semicontinuity for small random perturbations of 2D Navier-Stokes equations with linear damping
  142. Degenerate binomial and Poisson random variables associated with degenerate Lah-Bell polynomials
  143. Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part I)
  144. On the mixed fractional quantum and Hadamard derivatives for impulsive boundary value problems
  145. The Lp dual Minkowski problem about 0 < p < 1 and q > 0
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