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$: A characteristic difference method for fractional ...ashm-journal.com/test/vol3-1/72.pdf · Applied mathematics in Engineering, Management and Technology 3(1) 2015:618-630 618 Abstract$
Applied mathematics in Engineering, Management and Technology 3(1) 2015:618-630

www.amiemt-journal.com

618

Abstract

In this paper, Fractional advection-dispersion equations, which is important in

modeling the transport of passive tracers carried by fluid flow in a porous medium, is

revisited. It is applied a technique so-called characteristic finite difference method

(CFDM) to analyze 2-D fractional advection-dispersion equations with variable

coefficients on a finite domain. This method is in fact the combination of

characteristic methods and fractional finite difference techniques. Also, stability,

consistency and convergence of CFDM are fully discussed and more, an error

estimate is given. Furthermore, some numerical illustrative examples are carried out

and compared with other known methods.

Key words:Finite difference approximation; Radial dispersion; Shifted Grünwald-Letnikov

formula; Method of characteristics; Fractional advection-dispersion.

1. Introduction and basic formulation

The history of fractional calculus and integer-order calculus is almost the same [1, 2]. However, fractional

calculus was not developed very fast in comparison to integer-order calculus until the late 1970s due to lack of

application background. Since then, it has progressively found applications in many different fields of

engineering and sciences. For instance, fractional derivatives have recently been applied to some problems in

physics [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], finance [16, 17, 18] and hydrology [19, 20, 21, 22, 23, 24, 25,

26]. Fractional space derivatives usually are used to model anomalous diffusion or dispersion. When second

derivative in a diffusion or dispersion model is replaced by a fractional derivative, it leads to superdiffusion or

enhanced diffusion. When one dimensional advection-dispersion model has constant coefficients, analytical

solutions can be obtained by using Fourier transform methods [7, 21]. However, many practical problems lead to

a model with variable coefficients [4, 27].

In current paper, we develop the basic theory of characteristic finite difference method (CFDM) for the

space-fractional advection-dispersion equation

∂𝑢(𝑥 ,𝑡)

∂𝑡 + 𝑉(𝑥, 𝑡)

∂𝑢(𝑥 ,𝑡)

∂𝑥 − 𝐷 (𝑥, 𝑡)

∂𝛼𝑢(𝑥 ,𝑡)

∂𝑥𝛼 = 𝑠(𝑥, 𝑡) (1)

on a finite domain 𝑥𝐿 < 𝑥 < 𝑥𝑅. Physical considerations need 1 < 𝛼 ≤ 2 , see [25]. We suppose 𝑉(𝑥, 𝑡) ≥ 0

and 𝐷(𝑥, 𝑡) ≥ 0 so that the flow is from left to right. We also suppose an initial condition 𝑢(𝑥, 0) = 𝑢0(𝑥) for

𝑥𝐿 < 𝑥 < 𝑥𝑅 and a natural set of boundary conditions for mentioned problem: 𝑢(𝑥𝐿 , 𝑡) = 0 for all 𝑡 ≥ 0 and ∂𝑢(𝑥𝑅 ,𝑡)

∂𝑡= 0 for all 𝑡 ≥ 0. From the physical point of view, the boundary conditions mean that no tracer leaks

past the left boundary, and that the tracer moves freely through the right boundary. With these assumptions, it has

been shown in [28] that the implicit Euler method is unconditionally stable when a modified form of the

Grünwald formula is used for the fractional derivative. Otherwise, the implicit Euler method is unstable and so its

solution does not converge to the exact solution. The explicit Euler method is unstable as well when the standard

Grünwald formula is used. The term ∂𝛼𝑢(𝑥 ,𝑡)

∂𝑥𝛼 in Eq.(1) is the Riemann-Liouville fractional derivative of order 𝛼,

which is defined by

A characteristic difference method for fractional

advection-dispersion flow equations

Elyas Shivanian*, Hamid Reza Khodabandehlo

Department of Mathematics, Imam Khomeini International University, Qazvin, 34149-16818, Iran



619

∂𝛼𝑓(𝑥)

∂𝑥𝛼 =

1

Γ(𝑛−𝛼)

𝑑𝑛

𝑑𝑥𝑛 ‍𝑥

𝑥𝐿

𝑓(𝜉)

(𝑥−𝜉)𝛼+1−𝑛 𝑑𝜉 , (2)

where 𝑛 is an integer such that 𝑛 − 1 < 𝛼 ≤ 𝑛 and Γ(. ) is the well known Gamma function. In most of the

related literature, the case 𝑥𝐿 = 0 is called the Riemann-Liouville form, and the case 𝑥𝐿 = −∞ is called the

Liouville definition. Since 1 < 𝛼 ≤ 2 then

∂𝛼𝑓(𝑥)

∂𝑥𝛼 =

1

Γ(2−𝛼)

𝑑2

𝑑𝑥2 ‍𝑥

𝑥𝐿

𝑓(𝜉)

(𝑥−𝜉)𝛼−1 𝑑𝜉. (3)

The Grünwald-Letnikov definition of fractional derivatives of order 𝛼 of a function 𝑓(𝑥) for 𝑥âˆˆ ∈ [𝑥𝐿 ,𝑥𝑅]

is probably more computationally feasible, which are defined by [29]

∂𝛼𝑓(𝑥)

∂𝑥𝛼 = lim

𝑕→ 0

1

𝑕𝛼 ‍

[𝑥−𝑥𝐿𝑕

]

𝑘=0 𝑔𝑘(𝛼)

𝑓(𝑥 − 𝑘𝑕), (4)

where 𝑕 is the space step and 𝑔𝑘(𝛼)

= (−1)𝑘 𝛼𝑘 =

Γ(𝑘 − 𝛼)

Γ(− 𝛼) Γ(𝑘 + 1) with

𝛼𝑘 being the binomial

coefficients.

The Riemann-Liouville definition and the Grünwald-Letnikov definition coincide under relatively weak

conditions [30]: If 𝑓(𝑥) has second-order Sobolev derivative on the interval [𝑥𝐿 ,𝑥𝑅], then for every 1 < 𝛼 ≤ 2

both the Riemann-Liouville and the Grünwald-Letnikov derivatives exist and coincide on the interval [𝑥𝐿 ,𝑥𝑅].

This equivalence enables us to use the computationally more feasible Grünwald-Letnikov definition (4) in the

numerical discretization. The following shifted Grünwald approximations in discrete forms at each node 𝑥 were

introduced in [31]

∂𝛼𝑓(𝑥)

∂𝑥𝛼 =

1

𝑕𝛼 ‍

[𝑙1] + 1𝑘=0 𝑔𝑘

(𝛼) 𝑓(𝑥 − (𝑘 − 1)𝑕) + 𝑎1

∂𝛼+1𝑓(𝑥)

∂𝑥𝛼+1 𝑕 + 𝑂(𝑕2), (5)

where 𝑙1 = (𝑥−𝑥𝐿)

𝑕 , with [𝑙1] being the floor of 𝑙1 , respectively. The constants 𝑎1 is independent of 𝑕,𝑓 and

𝑥 . Moreover, the coefficients 𝑔𝑘(𝛼)

can be evaluated recursively:

𝑔0(𝛼)

= 1 , 𝑔𝑘(𝛼)

= (1 − 𝛼+1

𝑘) 𝑔𝑘−1

(𝛼). (6)

2. A fractional characteristic finite difference method

In this section, we develop a fractional characteristic finite difference method [32, 33] for problem (1) base upon

the Eulerian fractional finite difference method for problem (1) with the characteristic method which has been

well developed for second-order transient advection-diffusion equations [34]. Let 𝜓(𝑥, 𝑡) = 1 + 𝑉2(𝑥, 𝑡)

and denote the characteristic direction associated with the operator 𝑢𝑡 + 𝑉(𝑥, 𝑡) 𝑢𝑥 , by 𝜏 = 𝜏(𝑥, 𝑡) , then the

derivative along the characteristic direction can be written as [34]

𝑑

𝑑𝜏 =

1

𝜓(𝑥 ,𝑡) ∂

∂𝑡 +

𝑉(𝑥 ,𝑡)

𝜓(𝑥 ,𝑡) ∂

∂𝑥,

and the hyperbolic part of Eq.(1) becomes

∂𝑢 (𝑥 ,𝑡)

∂𝑡 + 𝑉(𝑥, 𝑡)

∂𝑢 (𝑥 ,𝑡)

∂𝑥 = 𝜓(𝑥, 𝑡)

𝑑𝑢 (𝑥 ,𝑡)

𝑑𝜏.

Thus, the fractional advection-diffusion equation in (1) is rewritten as a fractional parabolic equation along the

characteristics

𝜓(𝑥, 𝑡) 𝑑𝑢 (𝑥 ,𝑡)

𝑑𝜏 − 𝐷 (𝑥, 𝑡)


∂𝑥𝛼 = 𝑠(𝑥, 𝑡), (7)

In other words, the nonsymmetry due to advection in the governing fractional advection-diffusion equation (1) is

symmetrized in (7) in the Lagrangian coordinates. Let 𝑀 be a positive integer and Δ𝑡 = 𝑇

𝑀. We define a

uniform partition of the time interval [ 0 ,𝑇 ] by 𝑡𝑚 . For any 𝑥 ∈ [𝑥𝐿 ,𝑥𝑅] at time step 𝑡𝑚+1, we define a

backward characteristic tracking by

𝑟(𝑡; 𝑥, 𝑡𝑚+1) = 𝑥 − 𝑉(𝑥, 𝑡𝑚 + 1) (𝑡𝑚+1 − 𝑡),

𝑥 = 𝑟(𝑡𝑚 ; 𝑥, 𝑡𝑚+1) = 𝑥 − 𝑉(𝑥, 𝑡𝑚+1) Δ𝑡,

Δ𝑡 = 𝑡𝑚+1 − 𝑡𝑚 .

(8)



620

Then the characteristic derivative is approximated by a backward difference quotient along the approximate

characteristics (8) in the time stepping procedure [34]

𝜓(𝑥, 𝑡𝑚+1) 𝑑𝑢 (𝑥 ,𝑡𝑚+1)

𝑑𝜏 = 𝜓(𝑥, 𝑡𝑚+1)

𝑢(𝑥 ,𝑡𝑚+1) − 𝑢(𝑥 ,𝑡𝑚 )

(𝑥 − 𝑥)2 +(Δ𝑡)2 + 𝑅(1)(𝑥, 𝑡𝑚+1)

= 𝜓(𝑥, 𝑡𝑚+1) 𝑢(𝑥 ,𝑡𝑚+1) − 𝑢(𝑥 ,𝑡𝑚 )

(𝑥 − 𝑥 +𝑉(𝑥 ,𝑡𝑚+1) Δ𝑡)2 +(Δ𝑡)2 + 𝑅(1)(𝑥, 𝑡𝑚+1)

= 𝜓(𝑥, 𝑡𝑚+1) 𝑢(𝑥 ,𝑡𝑚+1) − 𝑢(𝑥 ,𝑡𝑚 )

(𝑉2(𝑥 ,𝑡𝑚+1) +1)(Δ𝑡)2 + 𝑅(1)(𝑥, 𝑡𝑚+1)

= 𝜓(𝑥, 𝑡𝑚+1) 𝑢(𝑥 ,𝑡𝑚+1) − 𝑢(𝑥 ,𝑡𝑚 )

𝜓(𝑥 ,𝑡𝑚+1)Δ𝑡 + 𝑅(1)(𝑥, 𝑡𝑚+1)

= 𝑢(𝑥 ,𝑡𝑚+1) − 𝑢(𝑥 ,𝑡𝑚 )

Δ𝑡 + 𝑅(1)(𝑥, 𝑡𝑚+1)

(9)

where 𝑅(1)(𝑥, 𝑡𝑚+1) is the local truncation error defined by

𝑅(1)(𝑥, 𝑡𝑚+1) = 1

Δ𝑡 ‍𝑡𝑚+1

𝑡𝑚 (𝑟(𝑡; 𝑥, 𝑡𝑚+1) − 𝑥)2 + (𝑡 − 𝑡𝑚 )2 𝑑2 𝑢

𝑑𝜏2 (𝑟(𝑡; 𝑥, 𝑡𝑚+1), 𝑡) 𝑑𝜏

= 1


𝑡𝑚 ((𝑥 − 𝑉(𝑥, 𝑡𝑚+1))(𝑡𝑚 − 𝑡) − 𝑥 + 𝑉(𝑥, 𝑡𝑚+1)Δ𝑡)2 + (𝑡 − 𝑡𝑚 )2 𝑑2 𝑢

𝑑𝜏2 (𝑟(𝑡; 𝑥, 𝑡𝑚+1), 𝑡)𝑑𝜏

= 1


𝑡𝑚 𝑉2(𝑥, 𝑡𝑚+1)(Δ𝑡 − (𝑡𝑚+1 − 𝑡))2 + (𝑡 − 𝑡𝑚 )2 𝑑2 𝑢


= 1


𝑡𝑚 (𝑉2(𝑥, 𝑡𝑚+1) + 1) (𝑡 − 𝑡𝑚 )2 𝑑2 𝑢


= 1


𝑡𝑚 (𝑉2(𝑥, 𝑡𝑚+1) + 1) (𝑡 − 𝑡𝑚 )

𝑑2 𝑢

𝑑𝜏2 (𝑟(𝑡; 𝑥, 𝑡𝑚+1), 𝑡) 𝑑𝜏.

(10)

Then a semi-discrete form of Eq.(7) can be written as

𝑢(𝑥 ,𝑡𝑚+1) − 𝑢(𝑥 ,𝑡𝑚 )

Δ𝑡 − 𝐷 (𝑥, 𝑡𝑚+1)

∂𝛼𝑢(𝑥 ,𝑡𝑚+1)

∂𝑥𝛼 = 𝑠(𝑥, 𝑡𝑚+1) − 𝑅(1)(𝑥, 𝑡𝑚+1). (11)

To develop a characteristic finite difference method, we are now in a position to discretize the fractional

diffusion terms. At each node 𝑥𝑖 = 𝑥𝐿 + 𝑖𝑕 for 𝑖 = 1,2, . . . ,𝑁 − 1 with 𝑕 = (𝑥𝑅 − 𝑥𝐿)

𝑁, we use the first-order

shifted Grünwald approximations (5) to discretize the fractional diffusion term as follows [31]

∂𝛼𝑢(𝑥𝑖 ,𝑡

𝑚+1)

∂+𝑥𝛼 =

1

𝑕𝛼 ‍𝑖 + 1𝑘=0 𝑔𝑘

(𝛼) 𝑢(𝑥𝑖 −𝑘 +1, 𝑡𝑚+1) + 𝑅+

(2)(𝑥𝑖 , 𝑡

𝑚+1), (12)

where 𝑅+(2)

(𝑥𝑖 , 𝑡𝑚+1) is local truncation term of order 𝑂(𝑕) defined by

𝑅+(2)

(𝑥𝑖 , 𝑡𝑚+1) = 𝑎1

∂𝛼+1𝑢(𝑥𝑖 ,𝑡𝑚+1)

∂+𝑥𝛼+1 𝑕 + 𝑂(𝑕2). (13)

Then the semi-discrete form (11) of Eq.(1) can be written as the following fully discrete form

𝑢(𝑥𝑖 ,𝑡

𝑚+1) − 𝑢(𝑥𝑖 ,𝑡𝑚 )

Δ𝑡 −

𝐷 (𝑥 ,𝑡𝑚+1)

𝑕𝛼 ‍𝑖 + 1𝑘=0 𝑔𝑘

(𝛼) 𝑢(𝑥𝑖 −𝑘 +1 , 𝑡𝑚+1) = 𝑠(𝑥𝑖 , 𝑡

𝑚+1) − 𝑅(𝑥𝑖 , 𝑡𝑚+1), (14)

where 𝑅(𝑥𝑖 , 𝑡𝑚+1) is the local truncation error defined by

𝑅(𝑥𝑖 , 𝑡𝑚+1) = − 𝑅(1)(𝑥𝑖 , 𝑡

𝑚+1) + 𝐷 (𝑥, 𝑡𝑚+1) 𝑅+(2)

(𝑥𝑖 , 𝑡𝑚+1). (15)

We let the node functions 𝑈𝑖𝑚 be the numerical approximation to the true solution 𝑢(𝑥𝑖 , 𝑡

𝑚 ) , 𝑉𝑖𝑚 =

𝑉(𝑥𝑖 , 𝑡𝑚 ), 𝑠𝑖

𝑚 = 𝑠(𝑥𝑖 , 𝑡𝑚 ) , 𝐷𝑖

𝑚 = 𝐷(𝑥(𝑖), 𝑡𝑚 ). The characteristic finite difference method for problem (1)) is

formulated as follows: At each time step 𝑡𝑚+1, find 𝑈𝑖𝑚+1 for 𝑖 = 1,2, . . . ,𝑁 − 1 such that

𝑈𝑖𝑚+1 − 𝑈𝑖

𝑚+1

Δ𝑡 −

𝐷 𝑚+1

𝑕𝛼 ‍𝑖 + 1𝑘=0 𝑔𝑘

(𝛼) 𝑈𝑖 −𝑘 +1

𝑚+1 = 𝑠𝑖𝑚+1 . (16)



621

Note that in general 𝑥𝑖 does not necessarily coincide with a node in the space, so 𝑈𝑖𝑚

is not defined. In this

case, 𝑈𝑖𝑚

can be evaluated via a linear interpolation. More specifically, let 𝐶𝑟𝑖 = 𝑉𝑖𝑚

Δ𝑡

𝑡 be the Courant

number and [ 𝐶𝑟𝑖] be its floor. When the Courant number 𝐶𝑟𝑖 is an integer, 𝑥𝑖 will intersect a node at time

step 𝑡𝑚 , so the evaluation of 𝑈𝑖𝑚

is straightforward. Otherwise, 𝑥𝑖 falls into the interval [𝑥𝑖0𝑚 , 𝑥𝑖0𝑚 +1] where

𝑖0𝑚 = 𝑖𝑚 − [𝐶𝑟𝑖] − 1.

Thus, 𝑈𝑖𝑚

can be evaluated by

𝑈𝑖𝑚

= (1 − 𝐶𝑟𝑖𝑚 ,∗ )𝑈𝑖0𝑚 +1 + 𝐶𝑟𝑖

𝑚 ,∗ 𝑈𝑖0𝑚𝑚 + 𝑂(𝑕2), (17)

with 𝐶𝑟𝑖𝑚 ,∗ being the fractional part of the Courant number. In practice, the diffusion is often symmetric.

In this case, the characteristic finite difference scheme (16) symmetrizes the governing fractional

advection-diffusion equations and generates a symmetric and positive-definite coefficient matrix with a greatly

improved, almost well conditioned number. In contrast, the fractional finite difference methods with standard

temporal discretization tend to generate nonsymmetric and ill-conditioned coefficient matrices.

3. Stability and error analysis of the fractional characteristic finite difference method

In this section we analyze the stability and convergence behavior of the characteristic finite difference method.

3.1. Three useful lemmas

For the purpose of the analysis, we rewrite the numerical scheme (16) into the following form

𝑈𝑖𝑚+1 − 𝜉𝑚+1 ‍𝑖 +1

𝑘=0 𝑔𝑘(𝛼)

𝑈𝑖 −𝑘 +1𝑚+1 = (1 − 𝐶𝑟𝑖

𝑚 ,∗ )𝑈𝑖0𝑚 +1 + 𝐶𝑟𝑖𝑚 ,∗ 𝑈𝑖0𝑚

𝑚 + Δ𝑡𝑠𝑖𝑚+1 ,

𝑖 = 1,2, . . . ,𝑁 − 1 ,𝑚 = 0,1, . . . ,𝑀 − 1 ,

𝑈𝑖0 = 𝑈0(𝑥𝑖) , 𝑖 = 1,2, . . . ,𝑁 − 1,

(18)

where 𝜉𝑖𝑚+1 = 𝐷𝑖

𝑚+1 Δ𝑡

𝑕𝛼 are associated with the diffusion coefficients. We carry out the analysis via the

approach of matrix analysis. To do so, we first recall some basic concepts in the theory of matrix analysis. The

following two lemmas play an important role in the analysis [35, 36]:

Lemma 3.1 Assume A = [ai,j] is diagonally dominant by rows. Then, the following estimate holds

∥ A ∥∞ ≤ max1 ≤ i ≤ n

1

|ai ,j | − ‍1≤j≤n ,j≠i |ai ,j |. (19)

Lemma 3.2 The coefficients gk(α)

given in (6) with 1 < α „ 2 satisfy the following properties:

g0

(α) = 1 , g1

(α) = − α < 0,

1 ≥ g2(α)

≥ g3(α)

≥ . . .≥ 0

‍∞k = 0 gk

(α) = 0 ,

‍mk = 0 gk

(α) ≤ 0 , (m ≥ 1).

(20)

To proving lemma(3.2) it is enough present following lemma that was proved in [28].

Lemma 3.3 It is well known that

( 1 + z)α = ‍∞k = 0

kα zk ,

for any complex |z| ≤ 1 and any α ≥ 0 , where

kα =

(−1)α Γ(k − α)

Γ( − α) Γ(k + 1),



622

kα (−1)k =

(−α) (−α +1) ...(−α + k − 1)

k! =

Γ(k − α)

Γ( − α) Γ(k + 1) .

3.2. Stability analysis

Theorem 3.4 The characteristic finite difference scheme (16) is unconditionally stable in the L∞ norm for

1 < α „ 2 . In particular, the matrices Am+1 and Bm defined in (23)-(25) below satisfy

∥ (Am+1)−1 Bm ∥∞≤ 1, ∥ (Am+1)−1 ∥∞≤ 1. (21)

Proof. We notice that the scheme (18) can be written as a matrix form

𝐴𝑚+1 𝑈𝑚+1 = 𝐵𝑚 𝑈𝑚 + Δ𝑡 𝑆𝑚+1 , 𝑚 = 0,1, . . . ,𝑀 − 1. (22)

Here

𝑈𝑚 = [𝑈1𝑚 ,𝑈1

𝑚 , . . . ,𝑈1𝑚 ]𝑇 , 𝑆𝑚 = [𝑆1

𝑚 ,𝑆2𝑚 , . . . , 𝑆𝑁−1

𝑚 ]𝑇 ,

𝐴𝑚+1 = [𝑎𝑖,𝑗𝑚+1]𝑖 ,𝑗 =1

𝑁−1 ,𝐵𝑚 = [𝑏𝑖,𝑗𝑚 ]𝑖 ,𝑗 =1

𝑁−1 , (23)

with

𝑎𝑖 ,𝑗𝑚+1 =

1 − 𝜉𝑖

𝑚+1 𝑔1(𝛼)

, 𝑗 = 𝑖,

− 𝜉𝑖𝑚+1 𝑔2

(𝛼), 𝑗 = 𝑖 − 1,

− 𝜉𝑖𝑚+1 𝑔0, 𝑗 = 𝑖 + 1,

− 𝜉𝑖𝑚+1 𝑔𝑖 −𝑗 +1

(𝛼), 𝑗 < 𝑖 − 1,

0, 𝑗 > 𝑖 + 1,

(24)

𝑏𝑖,𝑗𝑚 =

1 − 𝐶𝑟𝑖𝑚 ,∗, 𝑗 = 𝑖 − [𝐶𝑟𝑖] ,

𝐶𝑟𝑖𝑚 ,∗, 𝑗 = 𝑖 − [𝐶𝑟𝑖] − 1,

0, 𝑜𝑡𝑕𝑒𝑟𝑤𝑖𝑠𝑒.

(25)

It is clear that 𝐵𝑚 ≥ 0 elementwise. Furthermore, obviously we have

∥ 𝐵𝑚 ∥∞ = max1 ≤ 𝑖 ≤ 𝑁−1

‍𝑁−1𝑗 = 1 |𝑏𝑖,𝑗

𝑚 | ≤ 1. (26)

As for matrix 𝐴𝑚+1 , we use Lemma (3.2) to conclude that 𝑎𝑖,𝑗𝑚+1 ≤ 0 for all 𝑖 ≠ 𝑗 , and

𝑎𝑖 ,𝑖𝑚+1 = 1 − 𝜉𝑖

𝑚+1 𝑔1(𝛼)

= 1 + 𝜉𝑖𝑚+1 𝛼 > 0. (27)

Thus, we have

‍𝑁−1𝑗 = 1,𝑖≠𝑗 |𝑎𝑖 ,𝑗

𝑚+1| = 𝜉𝑖𝑚+1 ‍𝑖

𝑘=0,𝑘≠ 1 𝑔𝑘(𝛼)

≤ 𝜉𝑖𝑚+1 𝛼. (28)

We combine (27) and (28) to obtain

|𝑎𝑖 ,𝑖𝑚+1| − ‍𝑁−1

𝑗 = 1,𝑖≠𝑗 |𝑎𝑖 ,𝑗𝑚+1| ≥ 1. (29)

So, 𝐴𝑚+1 is diagonally dominant by rows, and from Lemma (3.1) we have

∥ (𝐴𝑚+1)−1 ∥∞ ≤ max1 ≤ 𝑖 ≤ 𝑁−1

1

|𝑎𝑖,𝑗𝑚+1| − ‍1≤𝑗≤𝑁−1 ,𝑗≠𝑖 |𝑎𝑖,𝑗

𝑚+1| = 1. (30)

Thus, the second estimate of (21) is proved.

Finally, we combine (26) with (30) to conclude

∥ (𝐴𝑚+1)−1 𝐵𝑚 ∥∞ ≤ ∥ (𝐴𝑚+1)−1 ∥∞ ∥ 𝐵𝑚 ∥∞ ≤ 1. (31)

Therefore, we have proved the first estimate in (4.4). This concludes the proof.

4. Convergence analysis and error estimates

Let 𝑊∞𝑘 (𝑥𝐿 ,𝑥𝑅) be the Sobolev spaces defined by [37]

𝑊∞𝑘 (𝑥𝐿 ,𝑥𝑅) = 𝑓(𝑥) ∈ 𝐿∞(𝑥𝐿 ,𝑥𝑅) ;

𝑑𝑚 𝑓(𝑥)

𝑑𝑥𝑚 ∈ 𝐿∞(𝑥𝐿 ,𝑥𝑅) , 0 ≤ 𝑚 ≤ 𝑘 , (32)

with the norm given by



623

∥ 𝑓 ∥𝑊∞𝑘(𝑥𝐿 ,𝑥𝑅) = max

0 ≤ 𝑚 ≤ 𝑘 ∥

𝑑𝑚 𝑓(𝑥)

𝑑𝑥𝑚 ∥𝐿∞(𝑥𝐿 ,𝑥𝑅) . (33)

In addition, we also need fractional-order Sobolev spaces 𝑊∞𝑠 (𝑥𝐿 ,𝑥𝑅) for any positive real number 𝑠 > 0,

which can be defined, e.g., via the interpolation of spaces.

For any Banach space 𝑋, we introduce Sobolev spaces involving time

𝑊𝑞𝑙 (𝑡1 , 𝑡2 ,𝑋) = 𝑓(𝑥 , 𝑡) ∶ ∥

∂𝑠 𝑓

∂𝑡𝑠 (. , 𝑡) ∥𝑋 ∈ 𝐿𝑞(𝑡1 , 𝑡2) , 0 ≤ 𝑠 ≤ 1 , 1 ≤ 𝑞 ≤ ∞ , (34)

with the norm defined by

∥ 𝑓 ∥𝑊𝑞𝑙 (𝑡1 ,𝑡2 ,𝑋) =

( ‍𝑙𝑠=0 ‍

𝑡2

𝑡1 ∥

∂𝑠 𝑓

∂𝑡𝑠 (. , 𝑡) ∥𝑋

𝑞 𝑑𝑡 )

1

𝑞 , 1 ≤ 𝑞 < ∞,

𝑚𝑎𝑥0 ≤ 𝑠 ≤ 1 𝑒𝑠𝑠𝑠𝑢𝑝𝑡 ∈ (𝑡1 ,𝑡2) ∥∂𝑠 𝑓

∂𝑡 𝑠 (. , 𝑡) ∥𝑋 , 𝑞 = ∞.

(35)

In addition, we define the following discrete norms

∥ 𝑓 ∥𝐿 𝑝 (𝑥𝐿 ,𝑥𝑅) = ( ‍𝑁−1𝑖=1 |𝑓( 𝑥𝑖 )|𝑝)

1

𝑝 , 1 ≤ 𝑝 < +∞,

𝑚𝑎𝑥0 ≤ 𝑖 ≤ 𝑁−1 |𝑓( 𝑥𝑖 )|, 𝑝 = +∞, (36)

∥ 𝑓 ∥𝐿 ∞ (0 ,𝑇 ,𝑋) = max0 ≤ 𝑚 ≤𝑀

∥ 𝑓( 𝑥 , 𝑡𝑚𝑝

) ∥𝑋 .

With these preparations we now prove the following theorem:

Theorem 4.1 Assume that the true solution u to problem (1) satisfies

𝑢 ∈ 𝐿1(0,𝑇;𝑊∞𝛼+1 (𝑥𝐿 , 𝑥𝑅)) ∩ 𝑊1

2(0,𝑇; 𝐿∞ (𝑥𝐿 ,𝑥𝑅)).

Let U be the numerical solution of the characteristic finite difference scheme (16). Then the following error

estimate holds

∥ 𝑈 − 𝑢 ∥𝐿 ∞(0,𝑇;𝐿∞ (𝑥𝐿 ,𝑥𝑅)) ≤ Δ𝑡 1 + 𝑉𝑚𝑎𝑥

2 ∥𝑑2𝑢

𝑑𝜏2 ∥𝐿1(0,𝑇;𝐿∞ (𝑥𝐿 ,𝑥𝑅))

+ 𝐶 𝑕 ∥ 𝑢 ∥𝐿1(0,𝑇;𝑊∞𝛼+1 (𝑥𝐿 ,𝑥𝑅)).

(37)

Here the constant 𝐶 is independent of 𝑕, 𝑡, the true solution 𝑢 and the end time 𝑇 but depends on 𝑎1 in (5)

and the maximum of 𝐷𝑖𝑚 .

Proof. Let 𝑒𝑖𝑚 = 𝑢(𝑥𝑖 , 𝑡

𝑚 ) − 𝑈𝑖𝑚 and 𝑒𝑖

𝑚 = 𝑢(𝑥𝑖 , 𝑡

𝑚 ) − 𝑈𝑖𝑚

be the global truncation errors. We

subtract the numerical scheme (16) from Eq. (14) to derive the following error equation for the global truncation

error 𝑒𝑚+1

(𝑈𝑖

𝑚+1 − 𝑢(𝑥𝑖 ,𝑡𝑚+1)) − (𝑈𝑖

𝑚 − 𝑢(𝑥𝑖 ,𝑡

𝑚 ))

Δ𝑡 −

𝐷𝑖𝑚+1

𝑕𝛼 ‍𝑖 + 1𝑘=0 𝑔𝑘

(𝛼) (𝑈𝑖 − 𝑘 + 1

𝑚+1 − 𝑢(𝑥𝑖 − 𝑘 + 1 , 𝑡𝑚+1))

= 𝑠𝑖𝑚+1 − 𝑠(𝑥𝑖 , 𝑡

𝑚+1) + 𝑅(𝑥𝑖 , 𝑡𝑚+1).

With assumption that we performed above

𝑒𝑖𝑚+1 − 𝑒𝑖

𝑚

Δ𝑡 −

𝐷𝑖𝑚+1

𝑕𝛼 ‍𝑖 + 1𝑘=0 𝑔𝑘

(𝛼) 𝑒𝑖 − 𝑘 + 1𝑚+1 = 𝑅(𝑥𝑖 , 𝑡

𝑚+1), (38)

Like Eq.(22) this equation can be written as a matrix form

𝐴𝑚+1 𝐸𝑚+1 = 𝐵𝑚 𝐸𝑚 + Δ𝑡𝑅𝑚+1 , 𝑚 = 0,1, . . . ,𝑀− 1, (39)

where

𝐸𝑚 = [ 𝑒1𝑚 , 𝑒2

𝑚 𝑒3𝑚 , . . . , 𝑒𝑁−1

𝑚 ]𝑇 , 𝑅𝑚 = [ 𝑅1𝑚 ,𝑅2

𝑚 ,𝑅3𝑚 , . . . ,𝑅𝑁−1

𝑚 ]𝑇 . (40)

We apply Theorem (4.1) to bound the global truncation error 𝐸𝑚+1 as follows:



624

∥ 𝐸𝑚+1 ∥𝐿 ∞ (𝑥𝐿 ,𝑥𝑅) = ∥ (𝐴𝑚+1)−1 ( 𝐵𝑚 𝐸𝑚 + Δ𝑡𝑅𝑚+1 ) ∥𝐿 ∞ (𝑥𝐿 ,𝑥𝑅)

≤∥ (𝐴𝑚+1)−1 𝐵𝑚 ∥𝐿 ∞ (𝑥𝐿 ,𝑥𝑅) ∥ 𝐸𝑚 ∥𝐿 ∞ (𝑥𝐿 ,𝑥𝑅)

+ Δ𝑡 ∥ (𝐴𝑚+1)−1 ∥𝐿 ∞ (𝑥𝐿 ,𝑥𝑅)∥ 𝑅𝑚+1 ∥𝐿 ∞ (𝑥𝐿 ,𝑥𝑅)

≤ ∥ 𝐸𝑚 ∥𝐿 ∞ (𝑥𝐿 ,𝑥𝑅) + Δ𝑡 ∥ 𝑅𝑚+1 ∥𝐿 ∞ (𝑥𝐿 ,𝑥𝑅)

≤ ∥ 𝐸𝑚−1 ∥𝐿 ∞ (𝑥𝐿 ,𝑥𝑅) + Δ𝑡 (∥ 𝑅𝑚 ∥𝐿 ∞ (𝑥𝐿 ,𝑥𝑅) + ∥ 𝑅𝑚+1 ∥𝐿 ∞ (𝑥𝐿 ,𝑥𝑅))

≤ . . .

≤ ∥ 𝐸0 ∥𝐿 ∞ (𝑥𝐿 ,𝑥𝑅) + ∥ 𝑅 ∥𝐿 1 ( 0,𝑡𝑚+1;𝐿∞(𝑥𝐿 ,𝑥𝑅)),

(41)

according to the initial condition, we see ∥ 𝐸0 ∥𝐿 ∞ (𝑥𝐿 ,𝑥𝑅) = 0 .

Then, (41) is reduced to

∥ 𝐸 ∥𝐿 ( 0,𝑇;𝐿 ∞(𝑥𝐿 ,𝑥𝑅)) ≤ ∥ 𝐸0 ∥𝐿 ∞(𝑥𝐿 ,𝑥𝑅) + ∥ 𝑅 ∥𝐿 1 ( 0,𝑇;𝐿 ∞(𝑥𝐿 ,𝑥𝑅))

≤ ∥ 𝑅 ∥𝐿 1 ( 0,𝑇;𝐿 ∞(𝑥𝐿 ,𝑥𝑅)).

(42)

In short, to prove the error estimate (36) we need only to bound the local truncation error

∥ 𝑅 ∥𝐿 1 ( 0,𝑇;𝐿 ∞(𝑥𝐿 ,𝑥𝑅)).

From (10) we get

∥ 𝑅(1) ∥𝐿 1 ( 0,𝑇;𝐿 ∞(𝑥𝐿 ,𝑥𝑅)) ≤ Δ𝑡 1 + 𝑉𝑚𝑎𝑥2 ∥

𝑑2𝑢

𝑑𝜏2 ∥𝐿1(0,𝑇;𝐿∞ (𝑥𝐿 ,𝑥𝑅)), (43)

By (13) we obtain

∥ 𝑅−(2) ∥𝐿 1 ( 0,𝑇;𝐿 ∞(𝑥𝐿 ,𝑥𝑅)) + ∥ 𝑅+

(2)∥𝐿 1 ( 0,𝑇;𝐿 ∞(𝑥𝐿 ,𝑥𝑅)) ≤ 𝐶𝑕 ∥ 𝑢 ∥𝐿1(0,𝑇;𝑊∞

𝛼+1 (𝑥𝐿 ,𝑥𝑅)). (44)

Now, we need to combine the estimates (42)-(44) with (15) to conclude the proof.

5. Radial flow application

A tracer solute is introduced into an aquifer at an injection well, and then pumped out at a second extraction well.

We adopt a radial coordinate system centered at the extraction well, and assume that the medium is radially

homogeneous. Tracer concentration 𝑢(𝑥, 𝑡) in the aquifer is related to tracer flux 𝑞(𝑥, 𝑡) and injection rate

𝑓(𝑥, 𝑡) according to the conservation equation

∂𝑢(𝑥 ,𝑡)

∂𝑡 = −

1

𝑥 ∂

∂𝑥 (𝑥 𝑞(𝑥, 𝑡)) + 𝑠(𝑥, 𝑡) , (45)

where the flux is described by

𝑞(𝑥, 𝑡) = 𝜈0

𝑥 𝑢(𝑥, 𝑡) −

𝑑0

𝑥 ∂𝑢(𝑥 ,𝑡)

∂𝑥 . (46)

The first term is the advective flux, inversely proportional to 𝑥 because of the radial geometry. The second term

is the dispersive flux, and this empirical formula derives from the fact that dispersion is the effect of differential

advection in a porous media, so that advection and dispersion coefficients are roughly proportional. In a recent

field study [38, 39, 40] at a Nevada test aquifer, 20.81𝑘𝑔 of bromide used as a tracer solute, at an average

concentration of 3600𝑚𝑔/𝑙, was introduced at the injection well for a period of 85𝑕 at the rate of 67.8𝑙/𝑕. The



625

distance from injection well to the extraction well was 30𝑚, and the radius of extraction well was 0.127𝑚. The

velocity coefficient and the dispersion constant were estimated to be of the same order of magnitude. Measured

concentrations over time at the extraction well show early breakthrough that cannot be explained by the classical

radial flow model. To address this situation, we employ a model with a fractional dispersive flux:

𝑞(𝑥, 𝑡) = 𝜈0

𝑥 𝑢(𝑥, 𝑡) −

𝑑0

𝑥 ∂𝛽𝑢(𝑥 ,𝑡)

∂𝑥𝛽 , (47)

for some 0 < 𝛽 ≤ 1. The fractional term models anomalous dispersion due to velocity contrasts resulting from

the interaction with a porous medium. Note that 𝜈0 and 𝑑0 in Eq.(47) are not the same as in the classical flux

Eq.(46) and the 𝑑0s do not even have the same dimensions. We have emphasized the numerical approximation

method here. See [40] for the details of the tested aquifer and associated parameters. Also, see [25] for a physical

derivation of the fractional flux based on statistical mechanics. The classical model corresponds to a value of

𝛽 = 1. Values of 𝛽 < 1 lead to superdispersion, in which solute spreads faster than the classical model predicts.

See [22, 41] for some practical methods of estimating the order 𝛽 of the fractional derivative from data. In the

limiting case of 𝛽 = 0, the dispersion process is replaced by an advection process in the differential equation.

Substituting (47) with 𝛼 = 𝛽 + 1 into the radial conservation law (45) we get

∂𝑢(𝑥 ,𝑡)

∂𝑡 = −

𝜈0


∂𝑥 +

𝑑0

𝑥 ∂𝛼𝑢(𝑥 ,𝑡)

∂𝑥𝛼 𝑠(𝑥, 𝑡) . (48)

Note that Eq.(48) is the same as Eq. (1) with 𝑣( 𝑥) =𝑣0

𝑥 and 𝑑( 𝑟) =

𝑑0

𝑥. We consider this equation and its

numerical solution in the case 1 ≤ 𝛼 ≤ 2, for 𝑥𝐿 < 𝑥 < 𝑥𝑅 and 0 < 𝑡 ≤ 𝑇.

6. Numerical experiments

In this section we carry out numerical experiments to investigate the performance and convergence behavior of

the fractional characteristic finite difference method (CFDM).

6.1. Performance of the fractional characteristic finite difference method for fractional advection -

dispersion flow equations

In this section, We consider following fractional advection - dispersion flow equation:

∂𝑢(𝑥 ,𝑡)

∂𝑡 + 𝑉(𝑥, 𝑡)

∂𝑢(𝑥 ,𝑡)

∂𝑥 − 𝐷 (𝑥, 𝑡)


∂𝑥𝛼 = 𝑠(𝑥, 𝑡) ,

with the coefficients

𝑉(𝑥, 𝑡) = 1 ,𝐷(𝑥, 𝑡) = Γ(3 − 𝑎) 𝑥𝛼 , 𝛼 = 1.8 , 0 ≤ 𝑥 ≤ 2 ,0 < 𝑡 ≤ 1 ,

forcing function

𝑠(𝑥, 𝑡) = −32 𝑒−𝑡 𝑥2 − 2.5 𝑥3 + 25

22 𝑥4 −

1

8(4 𝑥3 − 12 𝑥2 + 8 𝑥) +

1

8𝑥2 (2 − 𝑥)2 ,

the initial condition

𝑢(𝑥, 0) = 4 𝑥2 (2 − 𝑥)2 ,

and the boundary condition

𝑢(0, 𝑡) = ∂𝑢(2,𝑡)

∂𝑡 = 0, 𝑡 ≥ 0.

The exact solution of this fractional advection-dispersion flow equation is given by

𝑢(𝑥, 𝑡) = 4 𝑒−𝑡 𝑥2 (2 − 𝑥)2 .

We have shown the exact and numerical solutions in figures 1 and 2. Comparison of Tables 1 and 2 reveals the

reliability of CFDM.



626

Table 1: Maximum error behavior for Implisit Finite Difference Method (IFDM) and the effect of the grid size

reduction for the problem 6.1 at tim 𝑡 = 1.0.

𝑀 𝑁 𝑀𝑎𝑥𝑖𝑚𝑢𝑚 𝐸𝑟𝑟𝑜𝑟 𝐸𝑟𝑟𝑜𝑟 𝑟𝑎𝑡𝑒

10 10 0.157438 −

20 20 0.098924 1.59

40 40 0.053277 1.85

80 80 0.027706 1.92

160 160 0.014104 1.96

320 320 0.007114 1.98

640 640 0.003572 1.99

Fig. 1: Numerical solutions and exact solution at time 𝑡 = 1.0 for the problem 6.1 with 𝑀 = 𝑁 = 80. The solid

line corresponds to the exact solution, the stared line corresponds to numerical solution of the IFDM.

Table 2: Maximum error behavior for Charateristic Finite Difference Method (CFDM) and the effect of the grid

size reduction for the problem 6.1 at tim 𝑡 = 1.0.


10 10 0.253162 −

20 20 0.081188 3.11

40 40 0.040091 2.02

80 80 0.021085 1.90

160 160 0.010802 1.95

320 320 0.005467 1.97

640 640 0.002750 1.99



627


line corresponds to the exact solution, the stared line corresponds to numerical solution of the CFDM.

6.2. Performance of the fractional characteristic finite difference method for Radial flow equations

In this section, we consider following fractional advection-dispersion Radial flow equation

∂𝑢(𝑥 ,𝑡)

∂𝑡 = −

𝜈0


∂𝑥 +

𝑑0

𝑥 ∂𝛼𝑢(𝑥 ,𝑡)

∂𝑥𝛼 + 𝑠(𝑥, 𝑡),

with the coefficients

𝜈0 = 1 , 𝑑0 = 3 , 𝛼 = 1.8 , 0 ≤ 𝑥 ≤ 2 ,0 < 𝑡 ≤ 1,

forcing function

𝑠(𝑥, 𝑡) = −4 𝑒−𝑡 3 8

Γ(1.2) 𝑥−0.8 −

24

Γ(2.2) 𝑥0.2 +

24

Γ(3.2) 𝑥1.2 − 4 𝑥2 − 12 𝑥 + 8 + 𝑥2 (2 − 𝑥)2 ,

the initial condition

𝑢(𝑥, 0) = 4 𝑥2 (2 − 𝑥)2 ,

and the boundary condition

𝑢(0, 𝑡) = ∂𝑢(2,𝑡)

∂𝑡 = 0, 𝑡 ≥ 0 .

The exact solution of this fractional advection-dispersion flow equation is given by

𝑢(𝑥, 𝑡) = 4 𝑒−𝑡 𝑥2 (2 − 𝑥)2 .

We have plotted the exact and numerical solutions in figures 3 and 4. Comparison of Tables 3 and 4 confirms

again the reliability of CFDM.

Table 3: Maximum error behavior for Implisit Finite Difference Method (IFDM) and the effect of the grid size

reduction for the problem 6.2 at tim 𝑡 = 1.0.


10 10 0.053563 −

20 20 0.034956 1.53

40 40 0.020437 1.71

80 80 0.010960 1.86

160 160 0.005660 1.94

320 320 0.002873 1.97

640 640 0.001447 1.99



628


line corresponds to the exact solution, the stared line corresponds to numerical solution of the IFDM.

Table 4: Maximum error behavior for Characteristic Finite Difference Method (CFDM) and the effect of the

grid size reduction for the problem 6.2 at tim 𝑡 = 1.0.


10 10 0.143419 −

20 20 0.050043 2.86

40 40 0.018767 2.67

80 80 0.009014 2.08

160 160 0.004723 1.91

320 320 0.002401 1.97

640 640 0.001204 1.99



629


line corresponds to the exact solution, the stared line corresponds to numerical solution of the CFDM

7. Conclusion

Fractional derivatives in space can be used to model anomalous dispersion/diffusion, where particles spread

faster than the classical models predict. Fractional advection-dispersion equations with variable coefficients

allows velocity to vary over the domain, which is important in applications. An implicit Euler method, based on a

modified Grünwald approximation to the fractional derivative, is consistent and unconditionally stable. If the

usual Grünwald approximation is used, the implicit Euler method is always unstable. This simple numerical

method is useful for solving the fractional radial flow equation, where the fluid velocity increases as flow

converges on an extraction well. The fractional derivative allows the model to more accurately describe early

arrival, which is important in modeling groundwater contamination.

we propose a new finite difference method, called the fractional CFDM, and prove that this new method is

unconditionally stable, consistent and convergent, whose accuracy is of order 𝑂(𝑕 + Δ𝑡). At the meantime, an

error estimate is given. Numerical experiments are carried out and shown that our new method is much better than

the IFDM. The main advantage of the CFDM is that large time steps can be used without the loss of accuracy,

leading to significantly improved efficiency.

Moreover, the CFDM is especially efficient and superior for the high-dimensional convection-dominated

diffusion equations.

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