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Zhuang, Pinghui, Gu, YuanTong, Liu, Fawang, Turner, Ian, & Yarlagadda,Prasad(2011)Time-dependent fractional advection-diffusion equations by an implicitMLS meshless method.International Journal for Numerical Methods in Engineering, 88(13), pp.1346-1362.

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https://doi.org/10.1002/nme.3223

1

Time-dependent fractional advection-diffusion equations by an implicit MLS meshless method

P. Zhuang1, Y. T. Gu2* , F. Liu3, I. Turner3, and P. K.D.V. Yarlagadda2

1. School of mathematical sciences, Xiamen University, 361005, Xiamen, China 2. School of Engineering Systems, Queensland University of Technology, G. P. O. Box 2434,

Brisbane, QLD 4001, Australia 3. School of Mathematical Sciences, Queensland University of Technology, G. P. O. Box 2434,

Brisbane, QLD 4001, Australia

Abstract

Recently, many new applications in engineering and science are governed by a series of

fractional partial differential equations (FPDEs). Unlike the normal partial differential

equations (PDEs), the differential order in a FPDE is with a fractional order, which will lead

to new challenges for numerical simulation, because most existing numerical simulation

techniques are developed for the PDE with an integer differential order. The current dominant

numerical method for FPDEs is Finite Difference Method (FDM), which is usually difficult to

handle a complex problem domain, and also hard to use irregular nodal distribution. This

paper aims to develop an implicit meshless approach based on the moving least squares (MLS)

approximation for numerical simulation of fractional advection-diffusion equations (FADE),

which is a typical FPDE. The discrete system of equations is obtained by using the MLS

meshless shape functions and the meshless strong-forms. The stability and convergence

related to the time discretization of this approach are then discussed and theoretically proven.

Several numerical examples with different problem domains and different nodal distributions

are used to validate and investigate accuracy and efficiency of the newly developed meshless

formulation. It is concluded that the present meshless formulation is very effective for the

modeling and simulation of the FADE.

Key words: Fractional partial differential equation, Fractional advection-diffusion equation,

Moving least squares, Meshless method, Meshfree method, Implicit numerical

schemes

* Corresponding author: yuantong.gu@qut.edu.au

2

1. INTRODUCTION

The application of fractional calculus to deal with engineering and science problems has

become increasingly popular in recent years [1][2][3][4][5][6]. For example, fractional kinetic

equations, such as fractional diffusion equation, fractional advection-diffusion equation,

fractional Fokker-Planck equation, fractional cable equation etc., were recognized as a useful

approach for the description of transport dynamics in complex systems. Examples include

systems exhibiting Halmiltonian chaos, disordered medium, plasma and fluid turbulence,

underground water pollution, dynamics of protein molecules, motions under the influence of

optical tweezers, reactions in complex systems, and more[7][8][9][10][11] . When the fractional

differential equations describe the asymptotic behavior of continuous time random walks,

their solutions correspond to the Lévy walks. The advantage of the fractional model basically

lies in straightforward way of including external force terms and of calculating boundary

value problems. In practical physical applications, Brownian motion, the diffusion with an

additional velocity field and the diffusion under the influence of a constant external force field

are modeled by the advection-dispersion equation. However, in the case of anomalous

diffusion this is no longer true, i.e., the fractional generalization may be different for the

advection case and the transport in an external force field[8]. A straightforward extension of

the continuous time random walk (CTRW) model leads to a fractional advection-dispersion

equation. The fraction advection-dispersion equation known as its non-local dispersion, has

been proved as a useful tool for modeling the transport of passive tracers carried by fluid flow

in a porous medium for the groundwater hydrology research[2] and is also used to describe the

transport dynamics in complex systems which are governed by anomalous diffusion and non-

exponential relaxation patterns[8].

If a fractional calculus is included in the governing equation, this equation is usually called

the fractional ordinary differential equation (FODE) or the fractional partial differential

equation (FPDE). At present, most of fractional differential equations are solved numerically

using Finite Difference Method (FDM) [12][13][39][40][41], Finite Element Method (FEM)[14][15][16],

and spectral approximation[17]. For example, Liu and Turner and other colleagues studied

several different FPDEs by FDM[39][40][41], which is currently a dominant numerical method in

simulation of FPDE. FDM, however, depends on pre-defined grids/meshes, which lead to

inherited issues or shortcomings including: a) difficulty in handling a complicated problem

3

domain; b) difficulty in handling the Neumann boundary conditions, c) difficulty in handling

irregular nodal distribution; d) difficulty in conducting adaptive analysis, and e) low accuracy.

Therefore, these shortcomings become the main barrier for the development of a powerful

simulation tool for real applications governed by FPDE. In addition, most current research in

this field is still limited in some one-dimensional (1-D) or two-dimensional (2-D) benchmark

problems with very simple problem domains (i.e., squares and rectangles) and Dirichlet

boundary conditions. It already becomes crucial to develop an alternative numerical technique

for modeling and simulation of FPDE.

In recent years, meshless or meshfree methods have attracted increasing attention for the

research community, and regarded as promising numerical methods for computational

mechanics. These meshless methods do not require a mesh to discretize the problem domain,

because the aim of meshless methods is to eliminate at least the structure of the mesh and

approximate the solution entirely using the nodes/points as a quasi-random set of points rather

than nodes of an element/grid-based discretization[18][19]. In recent years, a group of meshless

(or meshfree) methods have been developed and successfully used in many fields of

engineering and science. Detailed reviews of meshfree methods can be found in, e.g.,

monograph[18]. The meshless methods based on collocation techniques, which can be

developed by using Dirac- delta-test function, have a relatively long history, and they include

smooth particle hydrodynamics (SPH) [20], the meshless collocation methods [21], Finite point

method (FPM) [19], etc. The so-called meshless methods using various global weak-forms and

local weak-forms were proposed about twenty years ago. This category of meshless methods

includes the diffuse element method (DEM) [22], the element-free Galerkin (EFG) method[18],

the reproducing kernel particle method [23], and the point interpolation method (PIM)[24][25],

the meshless local Petrov-Galerkin (MLPG) method[26], the local radial point interpolation

method (LRPIM)[27][28], the boundary node method (BNM)[29], and the boundary point

interpolation method (BPIM)[30][31].

The above discussed meshless methods have demonstrated possessing some distinguished

advantages [32] including: they do not use a mesh (at least in field approximation), so that the

burden of mesh generation in FDM and FEM is overcome. Hence, an adaptive analysis is

easily achievable; they are usually more accurate than FDM and FEM due to the use of higher

order meshless trial functions; and they are capable of solving complex problems that are

difficult for the conventional FDM and FEM. Because of these unique advantages, meshless

methods seem to have a good potential for the simulation of FPDE. Although the meshless

4

methods have been successfully applied to a wide range of problems, for which, however, the

governing equations are conventional PDE with an integer differential order, very limited

work was reported to handle fractional partial differential equations (FPDE) by the meshless

techniques[33][34]. Gu et al. [34] developed a local radial basis function (RBF) meshless

technique for simulation of anomalous subdiffusion equations, which are a type of FPDEs.

Though some encouraged results were reported in this paper[34], there were still some

technical issues in using RBF for FPDEs including the determination of suitable RBF shape

parameters and the poor computational efficiency. The moving least squares (MLS)

approximation is one of the most widely used methods for construction of meshless shape

functions. It has been found that the MLS is accurate and stable for arbitrarily distributed

nodes[32] for many problems in computational mechanics. However, to the best of authors’

knowledge, this is no research to develop MLS meshless techniques for FPDEs. This topic

calls for a significant development.

The objective of this paper is to develop an implicit meshless formulation based on the

moving least squares (MLS) for numerical simulation of time-dependent fractional advection-

diffusion equation (FADE). The discrete equations for two-dimensional time-dependent

FADE are obtained, by using the meshless MLS shape functions and the strong-forms. The

essential boundary conditions are enforced by the direct meshless collocation method[35][36].

The stability and convergence related to time discretization of this method are then discussed

and theoretically proven. Several numerical examples with different problem domains and

different nodal distributions are used to validate and investigate accuracy and efficiency of the

newly developed meshless formulation. Some key parameters, which affect the performance

of this meshless technique, are thoroughly investigated and the optimized parameters are

recommended.

The paper is organized as follows: a finite difference scheme for temporal discretization of

FADE is proposed in Section 2, where the stability and convergence analysis is given. In

Section 3, the MLS interpolation approximation is briefly discussed, and the meshless

scheme for spatial discretization is also developed. Numerical examples are studied and

discussed in Section 4. Finally, conclusions are presented in Section 5.

2. Time Fractional advection-diffusion equation

In this section, we present the collocation schemes to temporally discretize the time-

dependent fractional convection-diffusion equation. The time fractional advection-diffusion

equation can be written in the following form

20 ( , ) ( , ) ( , ) ( , ), , 0C

tD u t u t u t f t tα κ= ∆ − ⋅∇ + ∈Ω ⊂ >x x v x x x R ， (1)

together with the general boundary and initial conditions

5

2( , ) ( , ), , 0,u t g t t= ∈∂Ω ⊂ >x x x R (2)

0( ,0) ( ), ,u u= ∈Ωx x x (3)

where ∆ is the Laplace differential operator, i.e., 2 2

2 2

u uu

x y

∂ ∂∆ = +∂ ∂

, and ∇ the gradient

differential operator. Ω is a bounded domain in 2R , ∂Ω the boundary of Ω , κ the diffusion

coefficient, T[ , ]x yv v=v the advection coefficient (or velocity) vector, and ( , ), ( , )f t g tx x and

0( )u x are known functions.

In Equation (1), 0 ( , )CtD u tα x is the Caputo fractional derivative of order α ( 0 1α< < ), which

is defined as

0 0

1 ( , )( , ) ( ) d

(1 )

tCt

uD u t tα α ηη η

α η− ∂= −

Γ − ∂∫x

x . (4)

2.1. Discretization of time

Let

, 0,1,2, ,kt k t k n= ∆ = ⋯⋯⋯⋯ , (5)

where /t T n∆ = is time step. The time fractional derivative at 1kt t += can be approximated by

the following approximation

1

0 1 10

1 ( , )( , ) ( ) d

(1 )j

j

k tCt k kt

j

uD u t tα α ηη η

α η+ −

+ +=

∂= −Γ − ∂∑∫

xx

111 1

0

( , ) ( , )1( ) d

(1 )

j

j

k tj jk kt

j

u t u tt R

tαη η

α++ −

+ +=

−= − +

Γ − ∆∑ ∫x x

ɶɶɶɶ .

(6)

We can obtain[17]

21 ( )kR C t α−

+ ≤ ∆ɶɶɶɶ (7)

Let

1 1( 1)jb j jα α− −= + − , 0,1,2, ,j n= ⋯⋯⋯⋯ (8)

then Equation (6) can be rewritten as

0 1 1 10

( )( , ) [ ( , ) ( , )]

(2 )

kC

t k k j j j kj

tD u t b u t u t R

αα

α

−

+ − + +=

∆= − +Γ − ∑x x x ɶ

(9)

or

6

0 1 1 10

( )( , ) [ ( , ) ( , )]

(2 )

kC

t k j k j k j kj

tD u t b u t u t R

αα

α

−

+ − + − +=

∆= − +Γ − ∑x x x ɶ

(10)

Substituting Equation (10) into Equation (1), we can obtain

1 1 1( , ) [ ( , ) ( , )]k k ku t u t u tµ κ+ + +− ∆ − ⋅∇x x v x

1 1 11

( , ) [ ( , ) ( , )] ( , )k

k j k j k j k kj

u t b u t u t f t Rµ− + − + +=

= − − + +∑x x x x

(11)

where

( ) (2 )t αµ α= ∆ Γ − and 21 ( )kR C t+ ≤ ∆ɶ (12)

In which Cɶ is a positive constant.

Let ( )k ku u= x be the numerical approximation to ( , )ku tx , and 1 1( , )k kf f t+ += x , then the

Equation (1) can be discretized as the following scheme

1 1 1 1 1

1

[ ] ( )k

k k k k k j k j kj

j

u u u u b u u fµ κ µ+ + + − + − +

=

− ∆ − ⋅∇ = − − +∑v (13)

2.2. Stability and Convergence

In order to discuss the stability and convergence of Equation (13), let us introduce to the

following inner product

( , ) ( ) ( )d dv w v w x yΩ

= ∫∫ x x (14)

and norm in L2

1/2

1/ 2 2

2[( , )] ( )d dv v v v x y

Ω

= =

∫∫ x

(15)

The equation (13) can be rewritten as

1 11 1[ ]

k kk k

x y

u uu u v v

x yµ κ

+ ++ + ∂ ∂− ∆ − −

∂ ∂1

0 11 1

1

(1 ) ( )k

k k j kj j k

j

b u b b u b u fµ−

− ++

=

= − + − + +∑

(16)

For this difference scheme, we have the following result.

Lemma 1 If ( ), 0,1,2, ,ku k n=x ⋯ is the solution of Equation (16) and ( ) 0ku∂Ω

=x , then

7

0 112 2 20

maxk lk

l nu u b fµ−

− ≤ ≤≤ + (17)

Proof: For 0k = , we have

1 1 0 1x y

u uu u v v u f

x yµ κ µ ∂ ∂− ∆ − − = + ∂ ∂

(18)

By multiplying Equation (18) by 1u and integrating on Ω , we obtain

1 121 1 1 1 1 0 1 1 1

2( , ) , , ( , ) ( , )x x

u uu u u v u v u u u f u

x yµ κ µ ∂ ∂− ∆ − − = + ∂ ∂

(19)

Note that 1( ) 0u∂Ω

=x , then

2 21 11 1

2 2

( , )u u

u ux y

∂ ∂∆ = − + ∂ ∂

, 1 1

1 1, , 0u u

u ux y

∂ ∂= = ∂ ∂

(20)

hence,

2 21 121 0 1 1 1

22 2

( , ) ( , )u u

u u u f ux y

µκ µ ∂ ∂+ + = +

∂ ∂

(21)

Using Schwarz inequality, we have

1 0 1 0 102 2 2 2 20

max l

l nu u f u b fµ µ−

≤ ≤≤ + ≤ + (22)

Suppose now we have proven

0 112 2 20

max , 1,2, , .j lj

l nu u b f j kµ−

− ≤ ≤≤ + = ⋯ (23)

Multiplying Equation (16) by 1ku + and integrating on Ω , we can obtain

1 121 1 1 1 1

2( , ) , ,

k kk k k k k

x x

u uu u u v u v u

x yµ κ

+ ++ + + + + ∂ ∂− ∆ − − ∂ ∂

11 1 0 1 1 1

1 11

(1 )( , ) ( )( , ) ( , ) ( , )k

k k k j k k k kj j k

j

b u u b b u u b u u f uµ−

+ − + + + ++

=

= − + − + +∑

(24)

Using Schwarz inequality, and the inequality[13]

1, 0,1, , 1j jb b j n+≥ = −⋯ (25)

we have

8

11 0 1

1 12 2 2 2 21

(1 ) ( )k

k k k j kj j k

j

u b u b b u b u fµ−

+ − ++

=

≤ − + − + +∑ (26)

Hence, by using Equation (23), we obtain

1 0 11 12 2 20

(1 )[ max ]k lk

l nu b u b fµ+ −

− ≤ ≤≤ − +

10 1 0 1

1 12 2 2 201

( )[ max ]k

l kj j k j k

l nj

b b u b f b u fµ µ−

− ++ − − ≤ ≤=

+ − + + +∑

Note that 1 11 , 0,1, , 1k j kb b j k− −

− − ≤ = −⋯ , we have

1 11 0 1

1 12 2 200 0

[ ( )] [ ( )]max ]k k

k lk j j k k j j

l nj j

u b b b u b b b b fµ− −

+ −+ + ≤ ≤= =

≤ + − + + −∑ ∑

0 1

2 20max l

kl n

u b fµ −

≤ ≤= +

(27)

Further, we have the following stability result of this new approach.

Theorem 1 The fractional implicit numerical method defined by Equation (16) is

unconditionally stable.

Proof: Suppose that ɶ ( )k

u x , 1,2, ,k n= ⋯ is the solution of the Eq. (16) with the initial

condition ɶ0

( ,0)u u=x , then the error ɶ( ) ( ) ( )kk ku uε = −x x x satisfies

1 1 11 1 0

10

( ) .k k k

k k k jx y j j k

j

v v b b bx x

ε εε µ κ ε ε ε+ + −

+ + −+

=

∂ ∂− ∆ − − = − + ∂ ∂ ∑

(28)

From Lemma 1, we can obtain

0

2 2, 1,2, , .k k nε ε≤ = ⋯ (29)

Now we carry out an error analysis for the solution of the time-discretized problem Equation

(16). We denote from now by C a generic constant which may not be the same at different

occurrences.

Theorem 2 Let 0 ( , ) nk ku t =x be the exact solution of (1)-(3), 0 ( )k n

ku =x be the time-discrete

solution of Equation (16) with initial condition 0( ) ( ,0)u u=x x , then we have the following

error estimates

2

2( , ) ( ) ( )k

ku t u C t α−− ≤ ∆x x (30)

where C is a positive constant.

9

Proof：：：： Let ( ) ( , ) ( )k k kku t uξ ξ= = −x x x , from Equations (11) and (16), we obtain

1 11 1[ ]

k kk k

x yv vx y

ξ ξξ µ κ ξ+ +

+ + ∂ ∂− ∆ − −∂ ∂

10

1 1 11

(1 ) ( ) ,k

k k jj j k k

j

b b b b Rξ ξ ξ−

−+ +

=

= − + − + +∑

(31)

and 0( ) 0ξ =x .

Hence, from Lemma 1, we have

1 1 21 12 20max ( ) .k l

k kl n

b R Cb tξ − −− −≤ ≤

≤ ≤ ∆ (32)

Because 11( )kb t α−

− ∆ is bounded[13], thus

2

2( ) .k C t αξ −≤ ∆ɶ (33)

3. The meshless approach

3.1 Moving least squares shape functions

Several meshless approximation formulations have been proposed[32]. The moving least

squares (MLS) approximation is one of the widely used meshless methods because of its’

attractive properties of accuracy, robustness, and higher-order of continuity.

Consider an unknown scalar function of a field variable u(x) in the domain, Ω. The MLS

approximation of u(x) is defined at x as

T

1

( ) ( ) ( ) ( ) ( )m

hj j

j

u p a=

= =∑x x x p x a x (34)

where p(x) is the basis function of the spatial coordinates, xT=[x, y] for two-dimensional

problem, and m is the number of the basis functions.

In Equation (34), ( )a x is a vector of coefficients, which is a function of x. The coefficients a

can be obtained by minimizing the following weighted discrete L2 norm.

T 2

1

( )[ ( ) ( ) ]n

i i ii

J W u=

= − −∑ x x p x a x

(35)

where n is the number of nodes in the support domain of x for which the weight function

( ) 0iW − ≠x x

, when the weight function ( )iW −x x

can be also denoted as ( )iW x

; and ui is the

nodal parameter of u at x=xi, which can be considered as the smoothed values of u, and the

detailed discussions for it can be found in Liu and Gu’s book[32]. Because the number of

10

nodes, n, used in the MLS approximation is usually much larger than the number of unknown

coefficients, m, the approximated function, uh, does not pass through the nodal values[32].

The stationarity of J with respect to a(x) gives / 0∂ ∂ =J a which leads to the following set of

linear relations.

A(x)a(x)=B(x)Us (36)

where sU is the vector that collects the nodal parameters of field function for all the nodes in

the support domain. A(x) is called the weighted moment matrix defined by

T

1

( ) ( ) ( ) ( )n

i i ii

W=

=∑A x x p x p x

(37)

The matrix B in Equation (36) is defined as

B(x)=[ 1W

(x)p(x1) 2W

(x)p(x2) … nW

(x)p(xn)] (38)

Solving Equation (36) for a(x), we have

1( ) ( ) ( ) s−=a x A x B x U (39)

Substituting the above equation back into Equation (34), we obtain

T

1

( ) ( ) ( )n

hi i s

i

u uφ=

= =∑x x Φ x U (40)

where Φ (x) is the vector of MLS shape functions corresponding n nodes in the support

domain of the point x, and can be written as,

T T 11 2( ) ( ) ( ) ( ) ( ) ( ) ( )nφ φ φ −= =Φ x x x x p x A x B x⋯ (41)

For the convenience in obtaining the partial derivatives of the shape functions, Equation (41)

is re-written as [38]

T T( ) ( ) ( )=Φ x γ x B x (42)

where

T T 1−=γ p A , then =Aγ p (43)

The partial derivatives of γγγγ can then be obtained by solving the following equations.

, , ,i i i= −Aγ p A γ , , , , , , , ,( )ij ij i j j i ij= − + +Aγ p A γ A γ A γ

, , , , , , , , , ,

, , , , ,

(

)ijk ijk i jk j ik k ij ij k

ik j jk i ijk

= − + + +

+ + +

Aγ p A γ A γ A γ A γ

A γ A γ A γ

(44)

where i, j and k denote coordinates x and y, and a comma designates a partial derivative with

11

respect to the indicated spatial coordinate that follows. The partial derivatives of the shape

function ΦΦΦΦ can be obtained using the following expressions.

T T T, , ,i i i= +Φ γ B γ B , T T T T T

, , , , , , ,ij ij i j j i ij= + + +Φ γ B γ B γ B γ B

T T T T, , , , , , , ,

T T T T, , , , , , ,

ijk ijk ij k ik j jk i

i jk j ik k ij ijk

= + + +

+ + + +

TΦ γ B γ B γ B γ B

γ B γ B γ B γ B

(45)

Equation (45) shows that the continuity of the MLS shape function ΦΦΦΦ is governed by the

continuity of the basis function p as well as the smoothness of the matrices A and B. The

latter is governed by the smoothness of the weight function. Therefore, the weight function

plays an important role in the performance of the MLS approximation.

There are a number of weight functions can be used. The detailed discussions for selection of

weight functions in MLS can be found in the book by Liu and Gu[32]. The exponential

function and spline functions are often used in practice. Liu and Gu[32] reported and

recommended that the spline weight function is easier to use than the exponential weight

function, because the spline function has better accuracy and can be easily constructed with

required continuity. Hence, we employ the spline weight functions in this paper. Among them,

the most commonly used spline weight functions are listed below. The cubic spline function

(W1) has the following form of

2 3

2 3

2/3 4 4

( ) 4 /3 4 4 4/3

0

i i

i i i i

r r

W r r r

− += − + −

x

0.5

0.5 1

1

i

i

i

r

r

r

≤

< ≤

>

(46)

The quartic spline function (W2) is given by

2 3 41 6 8 3( )

0i i i

i

r r rW

− + −=

x 1

1

i

i

r

r

≤

>

(47)

The above two spline functions have been found [32] to lead to very similar results for the

problems studied. Hence, we will used the quartic spline function, given in Eq. (47), as the

weight function in the following numerical studies.

3.2 Meshless approach for FADE

Let καµκ α )2()( −Γ∆=−= tr and ( ) (2 )t αµ α= = ∆ Γ −q v v . Equation (13) can be rewritten as

the following form

11 1 1 0 1

10

( )k

k k k k j kj j k

j

u r u u b b u b u fµ−

+ + + − ++

=

+ ∆ + ⋅∇ = − + +∑q (48)

We define the new operator by 1 .H r= + ∆ + ⋅∇q

12

The problem domain is discretized by properly distributed field nodes. The MLS shape

functions obtained in Equation (40) are used to approximate ( , )u tx . Assume that there are

dN internal (domain) points and bN boundary points. Suppose that 1 2 , , , , il nl i i i=x ⋯ are

the nodes selected in the support domain of ix to construct MLS shape functions, and

1 2 , , , ii ni i i=D ⋯ . Then the following bd NN + equations are satisfied,

11 1 1 1 1

0

( ) , 1,2, ,k

k k k k k j k j ki i i i j i i i d

j

u r u u u b u u f i Nµ−

+ + + − + − +

=

+ ∆ + ⋅∇ = − − + =∑q ⋯

(49)

.,,2,1),,( 11bddd

ki

ki NNNNitXgu +++== ++

⋯ (50)

where

( )

( )

( )

( ),

( ),

( ).

i

i

i

j j ii l l i

l

j j ii l l i

l

j j ii l l i

l

u X

u X

u X

λ

λ

λ

∈

∈

∈

= Φ

∆ = ∆Φ

∇ = ∇Φ

∑

∑

∑

D

D

D

(51)

The partial derivatives of the MLS shape function ΦΦΦΦ can be obtained using Equation (45).

Thus, we obtain the following discrete equation for field point i.

1 ( ) 1 ( )1

1

( ) ( , ) , 1 ,i i

kk i k k l k l ij j j l j j j i k d

j j l

H b f X t i Nλ ψ λ λ λ ψ µ+ − + −+

∈ ∈ =

= − − + ≤ ≤

∑ ∑ ∑D D

1 ( )1( , ) , 1 .

i

k ij j i k d d b

j

g X t N i N Nλ ψ++

∈= + ≤ ≤ +∑

D

(52)

It should be mentioned here that although only Dirichlet boundary conditions are considered

in the above development, a similar algorithm can be developed to handle the Neumann

boundary conditions. For example, a simple method to deal with Neumann boundary

conditions is to add new discretized derivatives equations by replacing ψ in Eq. (52) by its

required derivatives. The detailed algorithm of implementation of Neumann boundary

conditions in the strong-form based meshless methods have been developed and discussed in

details in the book by Liu and Gu[32].

4 Numerical examples

13

In this section, two-dimensional numerical examples are solved using regular and irregular

nodes to illustrate the efficiency and accuracy of the proposed meshless approach. Following

error norm are defined as error indicators in this paper:

∑

∑

∑

∑

∑

∑

=

=

=

=

=

=

−=

−=

−=

d

d

d

d

d

d

N

i

exactyi

N

i

numyi

exactyi

yN

i

exactxi

N

i

numxi

exactxi

xN

i

exacti

N

i

numi

exacti

u

uu

e

u

uu

e

u

uu

e

1,

1,,

1,

1,,

1

10 ,,

(53)

In which 0, xe e and ye are the error norms for the solution and its derivatives with respect

to x andy , respectively, exactiu and num

iu are the exact and the numerical solutions at node i,

respectively, ,i xu and ,i yu are derivatives of the solution with respect to x andy , respectively,

and dN is the internal (domain) points.

Consider the following 2-D time-dependent fractional advection-diffusion equation

0 ( , , ), [0,1] [0,1], 0Ct

u uD u u f x y t x t

x yα ∂ ∂= ∆ − − + ∈Ω = × >

∂ ∂,

(54)

where 2( , , ) 0.5 (3 ) x yf x y t t eα += Γ + , with the following Dirichlet boundary condition

2( , , ) e , ( , ) , 0x yu x y t t x y tα+ += ∈∂Ω > (55)

and initial condition

( , ,0) 0, ( , )u x y x y= ∈Ω (56)

The exact solution to this problem can be easily obtained as

exact 2 e .x yu t α+ += (57)

In construction of MLS shape functions, the second order polynomial basis is used, i.e.,

T 2 2[1 ].x y x xy y=P

The α in Eq. (54) can be any value of 0<α <1, and the simulation algorithm developed in this

paper is suitable for any α. In the following studies, we select α close to 1.0 because it makes

this FPDE close to a parabolic equation.

4.1 A case with a rectangular problem domain

4.1.1 Results using regular nodal distribution

14

To examine the convergence rate of this method, we choose 41 41n = × regularly distributed

nodes with nodal space of 1/ 40x y h∆ = ∆ = = . The nodal distribution is shown in Fig. 1. The

radius of the influence domain for MLS approximation is 2h . Table 1 and Table 2 present

results obtained using different time steps, and the numerical convergent rate is also listed in

the same table for different time steps. The convergent rates, R, of these four errors regarding

to time steps are also listed in the same table. The convergent rate is calculated as:

1

2

( )

( )

tR

tεεε

∆=∆

(54)

From Table 1 and Table 2, we can conclude that the convergence order of the newly

developed meshless technique is ))(( 2 α−∆tO in time.

For comparison, Equation (54) has been numerically solved by the newly developed meshless

approach and the traditional finite difference method (FDM)[13], respectively. The results

obtained by the meshless method and FDM, respectively, are compared in Table 3 and Figure

2. From Table 3 and Figure 2, it has been found the proposed meshless approach leads to

more accurate results and better convergent rate than FDM. In addition, comparing with

results obtained by the RBF method[34], it has been found that the presented MLS meshless

method has better accuracy and computational efficiency. It should be mentioned here that to

determine the convergent rate for a meshless technique is much more complicated than for

FDM due to the complicities of meshless shape functions and algorithm. For a MLS meshless

technique, its convergent rate will depend on many factors including the order of polynomial

basis functions and the continuity of weight functions. Liu and Gu[32] reported the

approximate convergent rate for a MLS meshless method is in the range of 1.4-2.0, which

agrees with the results presented in Table 3.

4.1.2 Results using irregular nodes

Total 1025 irregularly distributed nodes are also used, as shown in Figure 3. The radius of the

influence domain for MLS approximation is 4d , where /( 1)d A N= − [37], A is the area

of domain and N is the number of nodes. The convergence processes regarding to time steps

are plotted in Figure 4. It can be found that irregular nodal distributions also lead to good

convergence for time-dependent fractional advection-diffusion equation for different α .

15

The computational errors for the analytical solution ( , , )u x y t and its derivatives

( , , ) ( , , ),

u x y t u x y t

x y

∂ ∂∂ ∂

are shown in Figure 5-Figure 7. The maximum error for ( , , )u x y t is

order of 310− , the maximum errors for ,u u

x y

∂ ∂∂ ∂

are order of 210− . These figures have proven

that the newly proposed meshless approach has very good accuracy even using irregular nodal

distributions. It should mention here that the irregular grid will lead to a big difficulty for the

conventional FDM.

4.2 A case with a circular problem domain

Let us consider the following circular problem domain

2 2( , ) 1x y x yΩ = + ≤ (58)

Total 1791 irregularly distributed nodes for this circular domain are used, as shown in Fig. 8.

The radius of the influence domain for MLS approximation is 4d , where /( 1)d A N= − ,

A is the area of domain and N is the number of nodes. The convergence processes

regarding to time steps are plotted in Figure 9. It can be found that irregular nodal

distributions lead to good convergence for time-dependent fractional advection-diffusion

equation for different α on circular problem domain.

The computational errors are plotted in Figure 10. The maximum error for ( , , )u x y t is order of

310− . It has proven that the newly proposed meshless approach has very good accuracy even

using irregular nodal distributions on non-rectangular problem domain.

5 Conclusions

This paper aims to develop an implicit meshless approach, based on the moving least squares

(MLS) and the meshless strong-form, for numerical simulation of fractional advection-

diffusion equation (FADE), which is a typical fractional partial differential equation(FPDE).

The stability and convergence of this meshless approach are proven theoretically and

numerically. Several numerical examples with different problem domains and nodal

16

distributions are used to study accuracy and efficiency of the newly developed meshless

approach. From above studies, we have concluded that: a) the presented meshless approach is

un-conditionally stable; b) the convergence order of this present method regarding to time

is 2( )O t α−∆ ; c) the presented MLS meshless method has better accuracy and convergence than

FDM, and it also has better computational accuracy and efficiency than the RBF meshless

method; and d) the newly developed meshless approach is robust for arbitrarily distributed

nodes and complex problem domains, for which the conventional Finite Difference Method

(FDM) is difficult to handle.

In summary, the present meshless formulation is very effective for modeling and simulation

of fractional partial differential equations, and it has good potential in development of a

robust simulation tool for problems in engineering and science which are governed by the

various types of fractional partial differential equations.

Acknowledgement

This work is supported by the ARC Future Fellowship grant (FT100100172) and the NSF of Fujian

(Grant No. 2010J01011).

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Table1: The errors of different time step using the meshless approach with 0.7, 1/ 40hα = = when 1.0t =

t∆ 0e Convergent rate yx ee = Convergent rate

20

1/10 2.54e-3 - 8.06e-3 -

1/20 1.06e-3 1.32.40 (20 /10)≈ 3.37e-3 3.1)10/20(39.2 ≈

1/30 6.38e-4 1.31.66 (30 / 20)≈ 2.02e-3 3.1)20/30(67.1 ≈

1/40 4.44e-4 1.31.44 (40 /30)≈ 1.40e-3 3.1)30/40(44.1 ≈

Table 2: The errors of different time step using the meshless approach with 0.85, 1/ 40hα = = when 1.0t =

t∆ 0e Convergent rate yx ee = Convergent rate

1/10 5.24e-3 - 1.66e-2 -

1/20 2.41e-3 15.1)10/20(17.2 ≈ 7.63e-3 15.1)10/20(18.2 ≈

1/30 1.53e-3 15.1)20/30(58.1 ≈ 4.83e-3 15.1)20/30(58.1 ≈

1/40 1.10e-3 15.1)30/40(39.1 ≈ 3.49e-3 15.1)30/40(39.1 ≈

Table3: Numerical error e0 for the meshless technique and FDM for different nodal spacing h, at time 0.1=t ( 0.85α = )

h MLS meshless technique FDM

e0 Convergent rate e0 Convergent rate

1/8 1.089e-4 - 4.389e-3 -

1/12 4.663e-5 2.335 2.846e-3 1.880

1/16 2.539e-5 1.837 2.105e-3 1.352

1/20 1.562e-5 1.625 1.670e-3 1.260

21

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

y

Figure 1: Regular nodal distribution for the rectangular domain

0.00E+00

5.00E-04

1.00E-03

1.50E-03

2.00E-03

2.50E-03

3.00E-03

3.50E-03

4.00E-03

4.50E-03

5.00E-03

0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12 0.13Time Step

Err

or

Meshless

FDM

Figure 2: Comparison of meshless approach and FDM For 0.85α =

22

Figure 3: Irregular distribution of points on rectangular domain

Figure 4: Errors as a function of the time step t∆ for various α

23

Figure 5: Numerical error for ( , )u tx at 1.0t = (Irregular nodal distribution on Rectangular

domain)

Figure 6: Numerical error for /u x∂ ∂ at 1.0t = (Irregular nodal distribution on Rectangular

domain)

24

Figure 7: Numerical error for /u y∂ ∂ at 1.0t = (Irregular nodal distribution on Rectangular

domain)

Figure 8: Scattered distribution of points on circular domain

25

Figure 9: Errors as a function of the time step t∆ for various α

Figure 10: Numerical error at 1.0t = (Irregular nodal distribution on circular domain)