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Journal of Computational Physics 293 (2015) 184–200

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Journal of Computational Physics

www.elsevier.com/locate/jcp

Second-order approximations for variable order fractional derivatives: Algorithms and applications

Xuan Zhao a,∗, Zhi-zhong Sun a, George Em Karniadakis ba Department of Mathematics, Southeast University, Nanjing 210096, People’s Republic of Chinab Division of Applied Mathematics, Brown University, Providence, RI 02912, USA

a r t i c l e i n f o a b s t r a c t

Article history:Received 22 May 2014Received in revised form 11 July 2014Accepted 7 August 2014Available online 13 August 2014

Keywords:Variable-order fractional operatorsHigh-orderAnomalous diffusionWave propagationBurgers equation

Fractional calculus allows variable-order of fractional operators, which can be exploited in diverse physical and biological applications where rates of change of the quantity of interest may depend on space and/or time. In this paper, we derive two second-order approximation formulas for the variable-order fractional time derivatives involved in anomalous diffusion and wave propagation. We then present numerical tests that verify the theoretical estimates of convergence rate and also simulations of anomalous sub-diffusion and super-diffusion that demonstrate new localized diffusion rates that depend on the curvature of the variable-order function. Finally, we perform simulations of wave propagation in a truncated domain to demonstrate how erroneous wave reflections at the boundaries can be eliminated by super-diffusion, and also simulations of the Burgers equation that serve as a testbed for studying the loss and recovery of monotonicity using again variable rate diffusion as a function of space and/or time. Taken together, our results demonstrate that variable-order fractional derivatives can be used to model the physics of anomalous transport with spatiotemporal variability but also as new effective numerical tools that can deal with the long-standing issues of outflow boundary conditions and monotonicity of integer-order PDEs.

© 2014 Elsevier Inc. All rights reserved.

1. Introduction

Fractional calculus allows integration and differentiation at any fractional order [1–3], and moreover the order of frac-tional integration and differentiation can be a function of space and/or time. Samko and Ross [4] and Samko [5] generalized the Riemann–Liouvile and Marchaud fractional integration and differentiation for the case of variable-order and they pre-sented interesting properties and an inversion formula. Lorenzo and Hartley [6] presented the concept of variable-order (VO) operators and investigated several potential variable-order definitions. In fact, even before the introduction of this new concept there were a few applications towards it. Smit and deVries [7] studied the stress–strain behavior of viscoelastic materials (textile fibers) with fractional order differential equations of order α (0 ≤ α ≤ 1) and showed that α depends on the strain level. Bagley [8] investigated the polymer linear viscoelastic stress relaxation described by fractional differential equations of order β for a given fixed temperature. This paper reveals a clear dependence of β on the temperature for poly-isobutlene and correlates fractional models and experiments. In addition, Metzler et al. [9] found that the order of fractional

* Corresponding author.E-mail addresses: [email protected] (X. Zhao), [email protected] (Z.-z. Sun), [email protected] (G.E. Karniadakis).

http://dx.doi.org/10.1016/j.jcp.2014.08.0150021-9991/© 2014 Elsevier Inc. All rights reserved.

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X. Zhao et al. / Journal of Computational Physics 293 (2015) 184–200 185

derivative of the fractional partial differential equations (FPDEs) describing the relaxation processes and reaction kinetics of proteins depends on temperature.

There have been several other applications with variable-order models. Ingman et al. [10] demonstrated the effectiveness of using a dynamic integro-differential operator of variable-order for cases of viscoelastic and elastoplastic spherical indenta-tion. Furthermore, the dependence of the order function on the strain and strain rate of viscoelastic material was evaluated in [14]. Later, Lorenzo and Hartley [11] explored more deeply the concept of VO integration and differentiation and created meaningful definitions for VO integration and differentiation. They also presented two forms of order distributions with applications to dynamic processes. Coimbra [12] and Diaz and Coimbra [17] investigated the dynamics and control of a nonlinear viscoelasticity oscillator via VO operators. Kobelev et al. investigated statistical and dynamical systems with fixed and variable memories, with the fractal dimension of the system being variable with time and spatial coordinate [13]. Pedro et al. studied the motion of particles suspended in a viscous fluid with drag force determined using the VO calculus [15]. Sun et al. [16] discussed four variable-order differential operators in anomalous diffusion modeling. They concluded that these four models, i.e., time/space/concentration/system parameter dependent VO models are more suitable in simulating the generalized decelerating/accelerating diffusion processes than the constant order model.

Gerasimov et al. [18] considered problems of anomalous infiltration in porous media. They proposed a fractional diffusion equation with variable-order of the time-derivative operator for describing the liquid infiltration in porous media according to the experimental data. They showed that the modified model with time fractional order depending on the concentration provides good agreement with existing experimental data for both the sub-diffusion and the super-diffusion. Sun et al. [19]analyzed mean square displacement (MSD) for characterizing anomalous diffusion and proposed an approach to establish a variable-random-order model for a given MSD function. Sun et al. [20] provided comparisons of constant-order and VO fractional models in characterizing the memory property of different systems. The advantages and potential applications of two variable-order derivative definitions were highlighted through a comparative analysis of anomalous relaxation process. Sun et al. [21] proposed a variable-index fractional-derivative model to describe the underlying transport dynamics.

The variable-order operator definitions recently proposed in the literature include the Riemann–Liouvile definition, Ca-puto definition, Marchaud definition, Coimbra definition and Grünwald definition. However, Soon et al. [22] also showed that the Coimbra definition variable-order operator satisfies a mapping requirement, and it is the only definition that correctly describes position-dependent transitions between elastic and viscous regimes because it correctly returns the appropriate derivatives as a function of space and time. Ramirez and Coimbra [23] showed that the Coimbra definition is the most appropriate definition having fundamental characteristics that are desirable for physical modeling.

Since the kernel of the variable-order operators has a variable-exponent, analytical solutions to variable-order fractional differential equations are more difficult to obtain. The solutions of the VO models are defined in fractional Besov spaces of variable-order on Rn [24]. Lin et al. [25] constructed an explicit finite difference scheme for spatial VO fractional differential equation, in which the space derivative is a generalized Riesz fractional derivative of order α(x, t) (1 < α(x, t) ≤ 2), where xand t are space and time variables, respectively, and they demonstrated stability and convergence of their scheme. The con-vergence order is one for both time and space. In a series of papers Chen et al. [26–28] studied various variable-order PDEs. Specifically, in [26], they proposed a numerical scheme with first-order temporal accuracy and fourth-order spatial accu-racy for the VO anomalous sub-diffusion equation. The convergence, stability, and solvability of the numerical scheme were shown via Fourier analysis. In [27], they investigated the finite difference method for the variable-order nonlinear Stokes’ first problem for a heated generalized second grade fluid. In [28], they considered the numerical method for a variable-order nonlinear reaction-subdiffusion equation and analyzed its stability and convergence. Recently, Chen et al. [29] proposed an alternating direct implicit method for a new two-dimensional variable-order fractional percolation equation with variable coefficients.

In [30], Valério and Sá da Costa addressed the different possible definitions of VO derivatives and their numerical ap-proximations. Sun et al. [31] developed three finite difference schemes for VO fractional sub-diffusion equation (VOFSE) with time fractional order α(x, t) (0 < α(x, t) < 1). They showed that all the schemes have first-order temporal accuracy and second-order spatial accuracy. Shen et al. [32] also studied VOFSE and proposed a finite difference scheme with ac-curacy O (τ + h2). Chen et al. [33] presented an implicit scheme for solving VOFSE in two-dimensions. The scheme was proved to be convergent with order O (�t + �2x + �2y), where �t , �x and �y are time step size, grid size in x direction and grid size in y direction, respectively. Shen et al. [34] proposed a first-order numerical scheme for spatial VO fractional advection-diffusion equation with a nonlinear source term and analyzed the stability and convergence of the scheme. Zhang et al. [35] constructed a novel implicit numerical method, which has first-order accuracy both in time and space for time fractional VO mobile–immobile advection–dispersion model.

All aforementioned numerical methods for VO FPDEs are of first-order accuracy with respect to time or space. In contrast, in the current work we derive two new approximation formulas of second-order accuracy for VO time fractional operator with order 0 < α(t) < 1 and 1 < α(t) < 2, respectively. Specifically, we adopt the following definition of VO [12]

C0D

α(t)t f (t) =

1

Γ (n − α(t))t∫

0

(t − s)n−α(t)−1 f (n)(s)ds, n − 1 < α(t) < n. (1.1)

Our objective is to apply these two new formulas to anomalous diffusion or dispersion and to exploit the advantages offered by the flexibility of VO, not only in modeling complicated and heterogeneous physics but also in accomplishing

186 X. Zhao et al. / Journal of Computational Physics 293 (2015) 184–200

certain numerical tasks nor possible with the standard calculus or with fractional constant-order models. For example, we employ the VO fractional wave-diffusion equation in order to deal with erroneous reflections in wave propagation due to the truncation of the domain. Similarly, we employ VO diffusion in the Burgers equation to impose monotonicity locally in high-order methods, which may suffer from the Gibbs phenomenon. The point we try to make is that the VO flexibility goes beyond data fitting and can be an effective new tool in controlling numerical errors or suppressing other numerical artifacts that are still an open problem in the numerical solution of standard, i.e., integer-order PDEs.

The paper is organized as follows. In the next section we present the two new formulas for nominally first and second VO time fractional derivatives and provide an error analysis. In Section 3 we present the numerical results, first on anoma-lous diffusion and subsequently on the wave equation in a truncated domain and finally on the Burgers equation whose numerical solution may exhibit large wiggles. The results are rather surprising as VO sub-diffusion or super-diffusion behave differently than the standard diffusion at early or later times but also they are different than the constant-order fractional diffusion as the specific form of the function α(x, t) plays an important role.

2. Approximation formulas

We introduce the following lemma, which will be used in the analysis of the formulas.

Lemma 2.1. (See [36].) Assuming that the derivatives of the function f (x) exist to the order of n on [a, b], and to the order of n + 1 on (a, b), a ≤ x0 < x1 < ... < xn ≤ b, pn(x) is the n-th degree interpolation polynomial of f (x) based on the points x0, x1, · · · , xn, then for x ∈ [a, b], it holds that

f (k)(x) − p(k)n (x) = f(n+1)(ξ)

(n − k + 1)!n−k∏i=0

(x − x(k)i

), 0 ≤ k ≤ n,

where ξ ∈ (a, b) depends on k and x, and xi < x(k)i < xi+k (i = 0, 1, · · · , n − k).

For a given function f (t) and an integer N > 0, let tn = nτ , f n = f (tn), n = 0, 1, ..., N , where τ = TN is the time step. Thus, the domain [0, T ] is covered by Ωτ , where Ωτ = {tn | tn = nτ , 0 ≤ n ≤ N}. For any grid function f = { f n | 0 ≤ n ≤ N}defined on Ωτ , let us introduce the following notation

δt fn+ 12 = 1

τ

(f n+1 − f n).

2.1. Mid-point formula for α(t) ∈ (0, 1)

Denote tn+ 12 = (n +12 )τ , αn+ 12 = α(tn+ 12 ), n = 0, ..., N − 1.

We evaluate the VO derivative C0Dα(t)t f (t) with order α(t) ∈ (0, 1) at the half grid point tn+ 12 , n = 0, ..., N − 1, leading to

C0D

αn+ 12

t f (tn+ 12 ) =1

Γ (1 − αn+ 12 )

tn+ 12∫0

f ′(s)

(tn+ 12 − s)α

n+ 12ds.

Evaluating the integration on each subdomain, we obtain

C0D

αn+ 12

t f (tn+ 12 ) =1

Γ (1 − αn+ 12 )

[ t 12∫t0

f ′(s)

(tn+ 12 − s)α

n+ 12ds +

n∑k=1

tk+ 12∫

tk− 12

f ′(s)

(tn+ 12 − s)α

n+ 12ds

]. (2.1)

For the interval [t0, t 12], denote the first-degree interpolation polynomial in the Lagrange form as

L10[ f ](s) = f (t0)t1 − t

τ+ f (t1) t − t0

τ.

For each interval [tk− 12 , tk+ 12 ], k = 1, · · · , n, denote the second-degree interpolation polynomial in the Lagrange form as follows

L2k [ f ](s) =k+1∑

f (t j)

(k)j (s),

j=k−1


where

(k)j (s) =

∏k−1≤m≤k+1

m �= j

s − tmt j − tm .

For the approximation formulas, we have

f (s) = L2k [ f ](s) + rk(s), s ∈ [tk− 12 , tk+ 12 ], k = 1, · · · ,n, (2.2)where

rk(s) = f′′′(ξk)

6

∏k−1≤ j≤k+1

(s − t j), ξk ∈ (tk−1, tk+1),

and

f (s) = L10[ f ](s) + r0(s), s ∈ [t0, t 12 ], (2.3)where

r0(s) = f′′(ξ0)

2

∏0≤ j≤1

(s − t j), ξ0 ∈ (t0, t1).

Substituting (2.2) and (2.3) into (2.1), yields

C0D

αn+ 12

t f (tn+ 12 ) =1

Γ (1 − αn+ 12 )

[ t 12∫t0

(L10[ f ])′(s)(tn+ 12 − s)

αn+ 12

ds +n∑

k=1

tk+ 12∫

tk− 12

(L2k [ f ])′(s)(tn+ 12 − s)

αn+ 12

]+Rn+ 12 , (2.4)

where the truncation error is

Rn+12 = 1

Γ (1 − αn+ 12 )

[ t 12∫t0

r′0(s)

(tn+ 12 − s)α

n+ 12ds +

n∑k=1

tk+ 12∫

tk− 12

r′k(s)

(tn+ 12 − s)α

n+ 12ds

](2.5)

Here we denote the discrete approximation formula for the VO derivative with order αn+ 12 as δα

n+ 12t f

n+ 12 , thus from (2.4)we have

δα

n+ 12t f

n+ 12

= 1Γ (1 − αn+ 12 )

[ t 12∫t0

(L10[ f ])′(s)(tn+ 12 − s)

αn+ 12

ds +n∑

k=1

tk+ 12∫

tk− 12

(L2k [ f ])′(s)(tn+ 12 − s)

αn+ 12

ds

]

= 1Γ (1 − αn+ 12 )

{ t 12∫t0

f 1− f 0τ

(tn+ 12 − s)α

n+ 12ds +

n∑k=1

tk+ 12∫

tk− 12

(tk+ 12 − s)f k− f k−1

τ + (s − tk− 12 )f k+1− f k

τ

(tn+ 12 − s)α

n+ 12ds

}

= τ−α

n+ 12

Γ (2 − αn+ 12 )

{c(α

n+ 12)

n(

f 1 − f 0) + n∑k=1

[a(α

n+ 12)

n−k(

f k − f k−1) + b(αn+ 12 )n−k ( f k+1 − f k)]}

= τ−α

n+ 12

Γ (2 − αn+ 12 )

[c(α

n+ 12)

n(

f 1 − f 0) + n−1∑k=0

a(α

n+ 12)

k

(f n−k − f n−k−1) + n−1∑

k=0b

(αn+ 12

)

k

(f n−k+1 − f n−k)

], (2.6)

where

a(α

n+ 12)

k = (k + 1)1−α

n+ 12 − 12 − α 1

[(k + 1)2−αn+ 12 − k2−αn+ 12 ], 0 ≤ k ≤ n − 1,

n+ 2


b(α

n+ 12)

k =1

2 − αn+ 12[(k + 1)2−αn+ 12 − k2−αn+ 12 ] − k1−αn+ 12 , 0 ≤ k ≤ n − 1,

c(α

n+ 12)

n =(

n + 12

)1−αn+ 12 − n1−αn+ 12 .

For convenience, we rewrite the formula as follows

δα

n+ 12t f

n+ 12 = τμ

n∑k=0

d(α

n+ 12)

n−k δt fk+ 12 , (2.7)

where μ = ταn+ 12 Γ (2 − αn+ 12 ),

d(α

n+ 12)

0 = b(α

n+ 12)

0 =1

2 − αn+ 12,

d(α

n+ 12)

k = a(α

n+ 12)

k−1 + b(α

n+ 12)

k =1

2 − αn+ 12[(k + 1)2−αn+ 12 − 2k2−αn+ 12 + (k − 1)2−αn+ 12 ], 1 ≤ k ≤ n − 1,

d(α

n+ 12)

n = a(α

n+ 12)

n−1 + c(α

n+ 12)

n =(

n + 12

)1−αn+ 12 − 1

2 − αn+ 12[n

2−αn+ 12 − (n − 1)2−αn+ 12 ].

We show the detailed analysis for the approximation error of the Mid-point formula in the following.

Lemma 2.2. Let αn+ 12 ∈ (0, 1), f ∈ C3([0, tn+1]), it holds that

∣∣∣∣∣ 1Γ (1 − αn+ 12 )tn+ 12∫0

f ′(s)

(tn+ 12 − s)α

n+ 12ds − τ

μ

n∑k=0

d(α

n+ 12)

n−k δt fk+ 12

∣∣∣∣∣≤ 1

Γ (1 − αn+ 12 )[Φn max

t0≤t≤t1∣∣ f ′′(t)∣∣ + 9

8(1 − αn+ 12 )max

t0≤t≤tn+1∣∣ f ′′′(t)∣∣t1−αn+ 12n

]τ 2,

where Φ0 = τ−α 1

2

(1−α 12)2

1−α 12

, Φn = t−α

n+ 12n

2 , n = 1, · · · , N − 1.

Proof. From (2.4), we have the truncation error

Rn+12 = 1

Γ (1 − αn+ 12 )

[ t 12∫t0

r′0(s)

(tn+ 12 − s)α

n+ 12ds +

n∑k=1

tk+ 12∫

tk− 12

r′k(s)

(tn+ 12 − s)α

n+ 12ds

]. (2.8)

Using Lemma 2.1, we have

r′0(s) = f ′′(η0)(s − t(0)0

), η0 ∈ (t0, t1), t(0)0 ∈ (t0, t1), (2.9)

r′k(s) =1

2f ′′′(ηk)

(s − t(k)0

)(s − t(k)1

), ηk ∈ (tk−1, tk+1), t(k)0 ∈ (tk−1, tk), t(k)1 ∈ (tk, tk+1), 1 ≤ k ≤ n. (2.10)

In what follows, we analyze the error on each interval. For the first interval, when n = 0, from (2.10), we obtain∣∣∣∣∣

t 12∫

t0

r′0(s)

(tn+ 12 − s)α

n+ 12ds

∣∣∣∣∣ ≤ maxt0≤t≤t1∣∣ f ′′(t)∣∣τ

∣∣∣∣∣t 1

2∫t0

1

(t 12

− s)α 12ds

∣∣∣∣∣ = τ−α 1

2

(1 − α 12)2

1−α 12

maxt0≤t≤t1

∣∣ f ′′(t)∣∣τ 2. (2.11)When n = 1, · · · , N − 1,

∣∣∣∣∣t 1

2∫r′0(s)

(tn+ 1 − s)α

n+ 12ds

∣∣∣∣∣ ≤ maxt0≤t≤t1∣∣ f ′′(t)∣∣τ

∣∣∣∣∣t 1

2∫1

(tn+ 1 − s)α

n+ 12ds

∣∣∣∣∣ ≤ t−α

n+ 12n

2max

t0≤t≤t1∣∣ f ′′(t)∣∣τ 2. (2.12)

t0 2 t0 2


From (2.10), leads to the estimation

∣∣∣∣∣n∑

k=1

tk+ 12∫

tk− 12

r′k(s)

(tn+ 12 − s)α

n+ 12ds

∣∣∣∣∣ =∣∣∣∣∣

n∑k=1

1

2

tk+ 12∫

tk− 12

f ′′′(ηk)(s − t(k)0 )(s − t(k)1 )(tn+ 12 − s)

αn+ 12

ds

∣∣∣∣∣

≤ 98

maxt0≤t≤tn+1

∣∣ f ′′′(t)∣∣τ 2∣∣∣∣∣

tn+ 12∫

t 12

(tn+ 12 − s)−α

n+ 12 ds

∣∣∣∣∣= 9

8(1 − αn+ 12 )max

t0≤t≤tn+1∣∣ f ′′′(t)∣∣t1−αn+ 12n τ 2. (2.13)

Substituting (2.11)–(2.13) into (2.8) leads to the lemma. �2.2. Second-order formula for α(t) ∈ (1, 2)

Denote αn = α(tn), n = 0, 1, · · · , N , f ′0 = ft(t0).We compute the VO derivative with order α(t) (1 < α(t) < 2) at tn:

C0D

α(tn)t f (tn) =

1

Γ (2 − α(tn))tn∫

0

f ′′(s)(tn − s)α(tn)−1 ds =

1

Γ (2 − α(tn))n−1∑j=0

t j+1∫t j

f ′′(s)(tn − s)α(tn)−1 ds. (2.14)

For the interval [t0, t1], we denote the cubic interpolation polynomial in the Hermite form as

L30[ f ](s) = f 0 + f ′0(s − t0) +1τ ( f

1 − f 0) − f ′0τ

(s − t0)2

+ 12τ

[ 1τ ( f

1 − f 0) − f ′0τ

−1τ ( f

2 − f 1) − 1τ ( f 1 − f 0)2τ

](s − t0)2(s − t1), (2.15)

thus we have the interpolation formula

f (s) = L30[ f ](s) + r0(s), (2.16)where the truncation error is

r0(s) = f(4)(ξ0)

24(s − t0)2(s − t1)(s − t2), ξ0 ∈ (t0, t2). (2.17)

For the interval [t j, t j+1], j = 1, · · · , n − 1, we denote the cubic interpolation polynomial at points t j−1, t j , t j+1 and t j+2in the Lagrange form as

L3j [ f ](s) =j+2∑

k= j−1f k

j+2∏m= j−1

m �=k

s − tmtk − tm ,

thus

f (s) = L3j [ f ](s) + r j(s), (2.18)where the truncation error is

r j(s) = f(4)(ξ j)

24

2∏k=−1

(s − tk+ j), ξ j ∈ (t j−1, t j+2).

Substituting (2.16) and (2.18) into (2.14), we obtain

C0D

αnt f (tn) =

1

Γ (2 − αn)n−1∑j=0

t j+1∫t

(L3j [ f ](s) + r j(s))′′(tn − s)αn−1 ds. (2.19)

j


Denote the approximation formula of the VO derivative with order αn as δαnt f n , thus

δ(αn)t f

n ≡ 1Γ (2 − αn)

n−1∑j=0

t j+1∫t j

(L3j [ f ](s))′′(tn − s)αn−1 ds =

1

Γ (2 − αn)

{ t1∫t0

(L30[ f ](s))′′(tn − s)αn−1 ds +

n−1∑j=1

t j+1∫t j

(L3j [ f ](s))′′(tn − s)αn−1 ds

}, (2.20)

where(L30[ f ](s)

)′′ = s − t0τ

· f2 − 2 f 1 + f 0

τ 2+ t1 − s

τ· 8 f

1 − f 2 − 7 f 0 − 6τ f ′02τ 2

,

(L3j [ f ](s)

)′′ = t j+1 − sτ

· fj−1 − 2 f j + f j+1

τ 2+ s − t j

τ· f

j − 2 f j+1 + f j+2τ 2

, 1 ≤ j ≤ N − 1.From mathematical calculations for (2.20), we have

δ(αn)t f

n = τ−αn

Γ (3 − αn)

[a(αn)n

(8 f 1 − f 2 − 7 f 0 − 6τ f ′0

) + n−1∑j=1

a(αn)n− j(

f j−1 − 2 f j + f j+1)

+n−1∑j=0

b(αn)n− j(

f j − 2 f j+1 + f j+2)], (2.21)

where

a(αn)j =1

3 − αn(

j3−αn − ( j − 1)3−αn) − ( j − 1)2−αn , 1 ≤ j ≤ n, (2.22)b(αn)j = j2−αn −

1

3 − αn(

j3−αn − ( j − 1)3−αn), 1 ≤ j ≤ n. (2.23)We present the detailed analysis for the truncation error of formula (2.21) in the following lemma.

Lemma 2.3. For α(tn) ∈ (1, 2), f ∈ C4([0, tn+1]), it holds that∣∣∣∣∣ 1Γ (2 − α(tn))tn∫

0

f ′′(s)(tn − s)α(tn)−1 ds − δ

(αn)t f

n

∣∣∣∣∣ ≤ 2t2−αnn

Γ (3 − αn) maxt0≤t≤tn+1∣∣ f (4)(t)∣∣τ 2.

Proof. From (2.19), we have the truncation error

Rn = 1Γ (2 − αn)

[ t1∫t0

[r30(s)]′′(tn − s)αn−1 ds +

n−1∑j=1

t j+1∫t j

[r3j (s)]′′(tn − s)αn−1 ds

]. (2.24)

Apply Lemma 2.1, we have[r30(s)

]′′ = f (4)(η0)2

(s − t(0)0

)(s − t(0)1

), η0 ∈ (t0, t2), t(0)0 ∈ (t0, t1), t(0)1 ∈ (t0, t2), (2.25)

[r3j (s)

]′′ = f (4)(η j)2

(s − t( j)0

)(s − t( j)1

), η j ∈ (t j−1, t j+2), t( j)0 ∈ (t j−1, t j+1), t( j)1 ∈ (t j, t j+2). (2.26)

Substituting (2.25) and (2.26) into (2.24), yields the estimation

∣∣Rn∣∣ = 1Γ (2 − αn)

∣∣∣∣∣t1∫

t0

f (4)(η0)2 (s − t(0)0 )(s − t(0)1 )

(tn − s)αn−1 ds +n−1∑j=1

t j+1∫t j

f (4)(η j)2 (s − t( j)0 )(s − t( j)1 )

(tn − s)αn−1 ds∣∣∣∣∣

≤ 1Γ (2 − αn)

[max

t0≤t≤t2∣∣ f (4)(t)∣∣τ 2

t1∫t0

(tn − s)1−αn ds + 2 maxt0≤t≤tn+1

∣∣ f (4)(t)∣∣τ 2tn∫

t1

(tn − s)1−αn ds]

≤ 2Γ (2 − αn) maxt0≤t≤tn+1

∣∣ f (4)(t)∣∣τ 2tn∫

t0

(tn − s)1−αn ds

≤ 2t2−αnn max

t ≤t≤t∣∣ f (4)(t)∣∣τ 2. �

Γ (3 − αn) 0 n+1


Table 1Maximum errors and convergence rates for Example 2.1 at T = 2.

α(t) h Error Rate

e−t 1/40 4.834e−02 1.97241/80 1.232e−02 1.98431/120 5.509e−03 1.98891/160 3.109e−03 1.99141/200 1.993e−03 *


α(t) h Error Rate

t2+12 1/40 1.726e−03 1.9354

1/80 4.512e−04 1.94211/120 2.053e−04 1.94381/160 1.173e−04 1.94681/200 7.600e−05 *


α(t) h Error Ratesin(t)+2

4 + 1 1/40 4.187e−01 1.98401/80 1.058e−01 1.98891/120 4.725e−02 1.99101/160 2.665e−02 1.99221/200 1.708e−02 *


α(t) h Error Rate

sin(t) + 1 1/40 9.163e−03 1.95321/80 2.366e−03 1.96171/120 1.068e−03 1.96551/160 6.068e−04 1.96791/200 3.911e−04 *

2.3. Numerical verification

In this subsection, we report on two numerical examples for checking the accuracy of the Mid-point formula (2.7) and the second-order formula (2.21).

Example 2.1 (Accuracy of formula (2.7)). Take f (t) = t5 − t3, 0 < t ≤ T .

Table 1 and Table 2 give the approximation errors and rates for two different orders: they both verify the second-order accuracy of the Mid-point formula (2.7) obtained in the analysis.

Example 2.2 (Accuracy of formula (2.21)). Take f (t) = t5, f ′(0) = 0, 0 < t ≤ T .

Table 3 and Table 4 list the maximum errors and convergence rates for the variable-order α(t) = sin(t)+24 + 1 and α(t) =sin(t) + 1 respectively. The numerical results demonstrate that the formula (2.21) leads to second-order accuracy.

3. Applications

In this section, we present numerical examples of anomalous sub-diffusion and super-diffusion and compare results of variable order and constant order. We then use these results to first control erroneous reflections in wave propagation and also to impose monotonicity of the solution in Burgers equation.


3.1. Anomalous diffusion

Example 3.1 (Sub-diffusion). We consider the following problem

0 Dα(x)t u(x, t) = uxx(x, t) + g(x, t), x ∈ [−1,1], t ∈ (0, T ], (3.1)

u(x,0) = (1 − x2)4, (3.2)with zero boundary conditions and 0 < α(x) ≤ 1.

We solve the equation with g(x, t) = 0 and several different order functions α(x).α1(x) = 1, (3.3)α2(x) = 1 − x2, (3.4)α3(x) = 2/3, (3.5)α4(x) = −0.9|x| + 0.95, (3.6)

α5(x) = 1|x| +√

5−12

−√

5 − 12

. (3.7)

We use Chebyshev collocation method for space discretization with 30 collocation points, and the Mid-point for-mula (2.7) for time discretization with time step τ = 1/100. The equation is solved in the domain x ∈ [−1, 1], t ∈ (0, T ]. Fig. 1 presents the numerical solutions of the problem at different times. Surprisingly, we observe that at early times there is a super-diffusion effect although we expect sub-diffusion, which is established after time T = 0.8. This is true both for variable and constant order but we also observe that the curvature of the variable order influences significantly the diffusion rate. These results are interesting as they can be used to model variable diffusion rate in time or space with the short term and the long term effects exactly opposite.

Example 3.2 (Super-diffusion). Next we consider the following diffusion-wave problem

0 Dα(x)t u(x, t) = uxx(x, t) + g(x, t), x ∈ [−1,1], t ∈ (0, T ], (3.8)

u(x,0) = (1 − x2)4, (3.9)ut(x,0) = 0, (3.10)u(−1, t) = 0, u(1, t) = 0, (3.11)

where 1 < α(x) < 2.

We solve the equation with g(x, t) = 0 and several different order functions α(x).α1(x) = 1.05, (3.12)α2(x) = 1.0001 + 0.3x2, (3.13)α3(x) = −0.9|x| + 1.95, (3.14)α4(x) = 1.0001 + 0.6x2, (3.15)α5(x) = 1.1. (3.16)

We use Chebyshev collocation method for space discretization with 30 collocation points, and the second-order method (2.21) we developed for time discretization with time step size τ = 1/100. Fig. 2 presents numerical solutions of the problem at different times. Similarly to the previous examples, at early times we observe a subdiffusive effect but after time T = 0.6 super-diffusion sets in for constant order but for variable order the effect is more complex as the rate of diffusion depends strongly on the form of the variable order function.

Example 3.3 (Convergence rate for sub-diffusion). We solve the above problem (3.1)–(3.2) with the exact solution u(x, t) =t2 sin(2πx) in the domain [0, 1] × [0, 1] in order to verify numerically the second-order accuracy of the proposed approxi-mation formula (2.7). We employ the variable-order α(x) = 1−x21.01 . The convergence order in time is listed in Table 5, which obviously shows asymptotically second-order accuracy in temporal direction.


Fig. 1. Sub-diffusion: (a) Fractional order functions (3.3)–(3.7); (b)–(f) Numerical solution of problem (3.1)–(3.2) at time: (b) t = 1/20, (c) t = 1/5, (d) t = 1/2, (e) t = 4/5, (f) t = 6/5.


Fig. 2. Super-diffusion: (g) Fractional order functions (3.12)–(3.16); (h)–(l) Numerical solution of problem (3.8)–(3.11) at time: (h) t = 1/20, (i) t = 1/10, (j) t = 3/5, (k) t = 4/5, (l) t = 1.


Table 5Maximum errors and convergence orders for Example 3.3; colloca-tion points N = 10.

α(x) τ Error Rate

1−x21.01 1/20 1.357e−03 1.9276

1/40 3.567e−04 1.95201/60 1.617e−04 2.00901/80 9.070e−05 2.01361/100 5.787e−05 *

Table 6Maximum errors and convergence orders for Example 3.4; collocation points N = 10.

α(x) τ Error Rate

α(x) = 12 tanh(2x − 2) + 12 tanh(2π + 3 − 2x) + 1 1/40 4.940e−04 2.05791/60 2.145e−04 2.05661/80 1.187e−04 2.05571/100 7.503e−05 2.04021/120 5.172e−05 *

Example 3.4 (Convergence rate for super-diffusion). Next we solve the above problem (3.8)–(3.11) with the exact solution u(x, t) = t3 sin(x) in the domain [0, 2π ] × [0, 1] in order to verify numerically the second-order accuracy of the proposed approximation formula (2.21). We employ the variable-order

α(x) = 12

tanh(2x − 2) + 12

tanh(2π + 3 − 2x) + 1.The convergence order in time is listed in Table 6, which obviously shows asymptotically second-order accuracy.

3.2. Wave propagation in truncated computational domains

The purpose of this numerical test is to provide an alternative and seamless way in dealing with the erroneous wave reflections in truncated computational domains of integer-order wave equation. This can be accomplished by “gracefully” switching from a wave to a diffusion-dominated governing equation at the boundaries using the concept of VO. In particular, we consider the second-order wave equation in one-dimension,{ utt = uxx, x ∈R, t ∈ (0, T ],

u(x,0) = φ(x), x ∈R,ut(x,0) = ψ(x), x ∈R.

(3.17)

This equation is defined in an unbounded domain and hence no physical boundary conditions are required but in sim-ulations we have to truncate the domain and hence we have to provide mathematically correct boundary conditions; see references [37,38] for typical boundary conditions. Such boundary conditions depend strongly on the type of wave propaga-tion and the computational domain and not universal. Here we present some preliminary results based on the VO concept. Specifically, we truncate the computational domain to a bounded domain Ω = [a, b] and replace the second-order derivative of time by a variable order fractional derivative, to obtain:⎧⎪⎪⎨

⎪⎪⎩0 D

α(x,t)t u = uxx, x ∈ Ω, t ∈ (0, T ],

u(x,0) = φ(x), x ∈ Ω,ut(x,0) = ψ(x), x ∈ Ω,u(x, t) = 0, x ∈ ∂Ω.

(3.18)

For the space discretization of the equation, we employ the Chebyshev collocation method with 80 points, and for the time discretization, we use second-order formula (2.21) with time step τ = 1/10.

Next, we choose the different order functions α(x, t) and observe the behavior of the numerical solutions. We consider the equation in the domain Ω = [0, 20]. In order to solve the equation numerically and compare the solution with the exact solution, we set ψ = 0 and assume that φ(x) is a single peak smooth function, defined as

u(x,0) = φ(x) = 2 exp(−(x − 10)2).Then the exact solution of Eq. (3.17) is

u(x, t) = exp(−(x − t − 10)2) + exp(−(x + t − 10)2).Eq. (3.18), when α(x, t) = 2, becomes the original wave equation with Dirichlet boundary condition. Under these conditions, the wave traveling outside will be reflected back when it reaches the boundaries. We will convert the wave equation into a


Fig. 3. Wave propagation: Numerical solutions at time t = 6. Upper: α1(x, t), Middle: α2(x, t), Lower: α3(x, t).

Fig. 4. Wave propagation: Numerical solutions at time t = 13. Upper: α1(x, t), Middle: α2(x, t), Lower: α3(x, t).

diffusion-wave equation invoking larger diffusivity by taking α to be smaller. So, in order to keep the solution accurate in the domain of interest, we set α = 2 in the inner part of the domain and α < 2 near the boundaries.

We consider the equation with two smooth order functions α2(x, t) and α3(x, t), defined as α1(x, t) = 2,

α2(x, t) = 12

{tanh

[2π

(x − t

4tanh(πt − 31) − t

4

)]+ tanh

[2π

(20 − t

4tanh(πt − 31) − t

4− x

)]}+ 1,

α3(x, t) = 12

{tanh

[π

2

(x − t

4tanh(πt − 31) − t

4

)]+ tanh

[π

2

(20 − t

4tanh(πt − 31) − t

4− x

)]}+ 1.

Figs. 3–6 list the results for waves. We observe that the maximum errors are less than 2% and they decay in time unlike the standard wave equation for which the errors are order one and they are increasing in time.

3.3. Burgers equation and loss of monotonicity

The purpose of this numerical test is to demonstrate how to impose monotonicity in cases that the solution is under-resolved or in the presence of discontinuities. To this end, we consider the viscous Burgers equation as a model problem:⎧⎨

⎩∂u

∂t+ 1

2

∂u2

∂x= μ∂

2u

∂x2, x ∈ R, t > 0,

u(x,0) = sin(πx)(3.19)

with periodic boundary conditions.


Fig. 5. Wave propagation: Numerical errors at time t = 6.

Fig. 6. Wave propagation: Numerical errors at time t = 13.

The exact solution is given by

u(x, t) = −∫ ∞−∞ sin(π(x − η)) f (x − η)exp(−η2/(4μt))dη∫ ∞

−∞ f (x − η)exp(−η2/(4μt))dη, x ∈ R, (3.20)

where f (y) = exp[− cos(π y)/(2πμ)] and μ is a very small positive number.It is well-known that a very steep front will appear at x = 0 if μ approaches 0 causing wiggles and hence loss of

monotonicity [39]. Here, we will replace the first-order time derivative by fractional order time derivative of order α (0 <α < 1) and observe the impact of fractional order on the solution. At first it is perhaps more intuitive to attempt to increase the order α > 1 in order to impose super-diffusion but since we are interested to make this change only locally in space/time and given our findings of the section on anomalous diffusion (Example 3.1), here we choose to impose α(x, t) < 1 as follows:⎧⎨

⎩ 0 Dα(x,t)t u +

1

2

∂u2

∂x= μ∂

2u

∂x2, x ∈ R, t > 0,

u(x,0) = sin(πx).(3.21)

To solve the fractional Burgers equation numerically, we employ formula (2.7) for the time discretization. Considering the periodic boundary conditions, we employ Fourier collocation method for the space discretization with 150 points and the Mid-point formula (2.7) for time discretization with step size τ = 1/300. In our test, we let μ = 1150π and the numerical results of the classical Burgers equation and the variable-order Burgers equation at different times are shown in Figs. 7–11. Specifically, we obtain solutions for α1(x, t) = 1 (classical) and also for α2(x, t) = 1 − 0.9 exp(−8|x| − 7000(t − 0.8)12), which deviates from one only at the time instant of the steepest solution at x = 0. Figs. 7–11 show the results at different times.


Fig. 7. Numerical solutions of Burgers equation at T = 0.3. Upper: classical Burgers equation; Lower: fractional Burgers equation.



Although the wiggles are not totally eliminated they are suppressed substantially hence demonstrating the flexibility of the VO formulation. Further tuning may be required to achieve a smoother solution.

4. Summary

The main algorithmic contribution of this paper is the extension of variable-order approximation formulas from first-order to second-order accuracy, which is particularly important for long-time integration of time-dependent fractional PDEs. An equally important contribution is our demonstration tests that point to the effectiveness of the variable-order fractional


Fig. 10. Numerical solutions of Burgers equation at T = 1. Upper: classical Burgers equation; Lower: fractional Burgers equation.


derivatives as a totally new approach to control errors in the classical integer order PDEs arising from truncated domains and erroneous boundary conditions or from the loss of monotonicity of the numerical solution either due to under-resolution or due to the presence of discontinuities. The use of super-diffusion/sub-diffusion in controlling such numerical artifacts is particularly useful in large-scale practical simulations, where there is never enough resolution to capture all fine scales everywhere of a heterogeneous field in a complex-geometry domain or the outflow boundary conditions are not readily available. While our examples here are for one-dimensional problems, the extension of these ideas to two- and three-dimensions is rather straightforward.

We also want to note that we have observed that the schemes presented here seem to be unconditionally stable through long-time runs, however, we could not analyze rigorously the stability at the present time but we hope to report on this important issue in future work.

Acknowledgements

We would like to acknowledge support by National Natural Science Foundation of China (No. 11271068), the US NSF/DMS-1216437, China Scholarship Council (No. 201206090058), the Research and Innovation Project for College Gradu-ates of Jiangsu Province (No. CXLX11_0093).

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Second-order approximations for variable order fractional derivatives: Algorithms and applications1 Introduction2 Approximation formulas2.1 Mid-point formula for α(t)∈(0,1)2.2 Second-order formula for α(t)∈(1,2)2.3 Numerical veriﬁcation

3 Applications3.1 Anomalous diffusion3.2 Wave propagation in truncated computational domains3.3 Burgers equation and loss of monotonicity

4 SummaryAcknowledgementsReferences

journal of computational physics - brown university · alternating direct implicit method for a new...

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