An optimal error estimate for the two-dimensional nonlinear time fractional advection–diffusion equation with smooth and non-smooth solutions

Computers and Mathematics with Applications xxx (xxxx) xxx

Contents lists available at ScienceDirect

Computers and Mathematics with Applications journal homepage: www.elsevier.com/locate/camwa

An optimal error estimate for the two-dimensional nonlinear time fractional advection–diffusion equation with smooth and non-smooth solutions ∗

Hui Zhang, Junqing Jia , Xiaoyun Jiang

∗

School of Mathematics, Shandong University, Jinan 250100, PR China

article

info

a b s t r a c t In this paper, a linearized L1 Legendre–Galerkin spectral method for the two-dimensional nonlinear time fractional advection–diffusion equation is derived. The numerical method is stable without the CFL conditions based on the error splitting argument technique and the discrete fractional Gronwall type inequality. We also consider the case of nonsmooth solutions by adding some correction terms. The numerical experiments are given to verify the theoretical analysis and the effectiveness of the correction method. © 2019 Elsevier Ltd. All rights reserved.

Article history: Received 1 May 2019 Received in revised form 9 December 2019 Accepted 10 December 2019 Available online xxxx Keywords: Two-dimensional nonlinear time fractional advection–diffusion equation Linearized L1 Legendre–Galerkin spectral method Optimal error estimate Error splitting argument technique Correction method

1. Introduction In this paper, the following two-dimensional nonlinear time fractional advection–diffusion equation [1–5] is considered,

⎧C α ⎨0 Dt u = −k1 ∇ u + k2 ∆u + G(u) + f (x, y, t), (x, y, t) ∈ Ω × I , u(x, y, 0) = u0 (x, y), (x, y) ∈ Ω , ⎩ u(· , t) = 0, (x, y) ∈ ∂ Ω , t ∈ I ,

(1)

which was first proposed by Zaslavsky [6] for Hamiltonian chaos, where u is an unknown probability density function, k1 , k2 denote the fractional velocity and fractional dispersion coefficient, respectively. G(u) is a nonlinear term satisfying the local Lipschitz condition. Ω = (a, b) × (c , d), I = (0, T ], and C0 Dαt (0 < α < 1) represents the Caputo fractional derivative defined as [7], C α 0 Dt u(t)

=

1

Γ (1 − α )

t

∫ 0

u′ (s)ds (t − s)α

.

(2)

Numerical methods for the time fractional equation have been extensively studied [8–15]. In addition, many researchers have devoted themselves to the numerical solution of the fractional advection–diffusion equation, see [16–19] and references therein. Due to the high computational efficiency and high accuracy, spectral method has been widely ∗ Corresponding authors. E-mail addresses: [email protected] (H. Zhang), [email protected] (J. Jia), [email protected] (X. Jiang). https://doi.org/10.1016/j.camwa.2019.12.013 0898-1221/© 2019 Elsevier Ltd. All rights reserved.

Please cite this article as: H. Zhang, J. Jia and X. Jiang, An optimal error estimate for the two-dimensional nonlinear time fractional advection–diffusion equation with smooth and non-smooth solutions, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.12.013.

2

H. Zhang, J. Jia and X. Jiang / Computers and Mathematics with Applications xxx (xxxx) xxx

used in fractional integral and differential equations, such as [20–24]. Lin and Xu [25] investigated a finite difference spectral method to research the time fractional diffusion equation. In [26], Zhang and Jiang proposed a Legendre spectral scheme to obtain the numerical solution of the space fractional coupled nonlinear Schrödinger equations. In this paper, we develop an L1 Legendre–Galerkin spectral method to solve the nonlinear time fractional advection–diffusion equation. The usual convergence analysis for high-dimensional nonlinear problems often have the time step restrictions dependent on the spatial mesh size. Ewing and Wheeler [27] used Galerkin method to approximate the miscible displacement problems in porous media, under the time step restriction τ = O(h), an optimal L2 -error estimate was derived, where τ is the time step size, h is the space mesh size. In [28], Luskin presented an optimal H 1 -error estimate with a requirement τ 4 = O(hµ ), where µ is the dimension of the defined domain. Zeng et al. [29] obtained the stability and the convergence of a two-dimensional Riesz space fractional nonlinear reaction–diffusion equation when τ 2 N ≤ C , where N is the polynomial degree. For more details we can see [30,31] and references therein. In [32], Li first proposed the temporal–spatial error splitting argument technique to weak the time step restriction conditions such as τ = O(h), τ 4 = O(hµ ) and τ 2 N ≤ C in integral order partial differential equations. Here, we extend the temporal–spatial error splitting argument technique to analyze the two-dimensional nonlinear time fractional advection–diffusion equation. In real applications, the analytical solutions of time fractional differential equations are not smooth and may have strong singularity at t = 0, see e.g. [33,34]. In order to solve the singularity at t = 0, Li [33] developed the graded mesh method. Zeng [35,36] introduced some correction terms to deal with the non-smooth solutions. The novelty of this paper is the numerical method is stable without the CFL conditions for the two-dimensional nonlinear time fractional advection–diffusion equation, while previous works required certain time step restrictions. Our proof is based on a time–space error splitting argument technique and the discrete fractional Gronwall type inequality [37]. And we deal with the non-smooth solutions by introducing the correction method. This paper is organized as follows. We develop a linearized L1 Legendre–Galerkin spectral method to solve the two-dimensional nonlinear time fractional advection–diffusion equation in Section 2. In Section 3, by introducing a timediscrete system we obtain an optimal error estimate for the fully discrete scheme. Correction method is presented to deal with singularity in Section 4. Numerical examples are provided in Section 5 to verify the theoretical analysis and the effectiveness of the correction method. Finally, Section 6 gives some conclusions and discussions. 2. Numerical method In this paper, the exact solution is supposed to satisfy

∥u0 ∥H r + ∥u∥C (I ;H r ) + ∥ut ∥C (I ;H r ) + ∥utt ∥C (I ;H 2 ) ≤ η,

(3)

where η is a constant which is independent of N, n, and τ . Remark 2.1. We intend to prove the optimal error estimate unconditionally for our numerical method with no time step restriction dependent on the polynomial degree N. For convenience, we assume (3) holds, and it does not involve the regularity of the solution at the initial value at t = 0. Let tn = nτ , 0 ≤ n ≤ K , the time step is τ = T /K . The Caputo derivative can be approximated as follows

⎛ C α n 0 Dt u

=

1

µ

⎝ a0 un −

n−1 ∑

⎞

(an−j−1 − an−j )uj − an−1 u0 ⎠ + E n ,

(4)

j=1

here µ = τ α Γ (2 − α ) and aj = (j + 1)1−α − j1−α , j ≥ 0, the truncation error E n satisfies |E n | ≤ O(τ 2−α ) [25]. Denote

⎛ D τ un =

1

µ

⎝a0 un −

n−1 ∑

⎞

(an−j−1 − an−j )uj − an−1 u0 ⎠ , n = 1, 2, . . . , K .

(5)

j=1

It is obvious that Dτ un = −k1 ∇ un + k2 ∆un + 2G(un−1 ) − G(un−2 ) + f n + Rn ,

(6)

where Rn = Dτ un −

C α n 0 Dt u

− (2G(un−1 ) − G(un−2 ) − G(un )).

(7)

1

For u , we have u1 − u0

µ

= −k1 ∇ u1 + k2 ∆u1 + G(u0 ) + f 1 + R1 ,

(8)

and R1 =

u1 − u0

µ

− C0 Dαt u1 − (G(u0 ) − G(u1 )).

(9)

Please cite this article as: H. Zhang, J. Jia and X. Jiang, An optimal error estimate for the two-dimensional nonlinear time fractional advection–diffusion equation with smooth and non-smooth solutions, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.12.013.

H. Zhang, J. Jia and X. Jiang / Computers and Mathematics with Applications xxx (xxxx) xxx

3

Clearly, by Taylor’s theorem

∥Rn ∥2 ≤ Cη τ 2−α , n = 2, 3, . . . , K ,

(10)

∥R1 ∥2 ≤ Cη τ ,

where Cη represents a positive constant depending on η. For the spatial approximation, we use the Legendre–Galerkin spectral method. Let VN0 be the two-dimensional subspace of H01 (Ω ), where VN0 = (PN (Ix ) ⊗ PN (Iy )) ∩ H01 (Ω ), PN (Θ ) be the space of polynomials of degree at most N on the interval Θ , Ix = (a, b), Iy = (c , d). The basis functions in the x and y directions are given by,

φi (x) = Li (xˆ ) − Li+2 (xˆ ), xˆ ∈ (−1, 1), x = ϕj (y) = Lj (yˆ ) − Lj+2 (yˆ ), yˆ ∈ (−1, 1), y =

(b − a)xˆ + a + b 2 (d − c)yˆ + c + d 2

∈ (a, b), (11)

∈ (c , d),

where Ll (zˆ ) is the Legendre polynomial [38]. So the function space can be expressed by VN0 = span{φi (x)ϕj (y), i, j = 0, 1, 2, . . . , N − 2}. We write the fully discrete approximation of (1) as: find UNn ∈ VN0 Dτ UNn , v = − k1 (∇ UNn , v ) + k2 ∆UNn , v + 2G(UNn−1 ) − G(UNn−2 ), v

(

(

)

)

(

)

+ (f n , v ),

(12)

for n ≥ 2, ∀v ∈ VN0 . When n = 1,

(

UN1 − UN0

µ

,v

)

( ) ( ) = − k1 (∇ UN1 , v ) + k2 ∆UN1 , v + G(UN0 ), v

(13)

+ (f 1 , v ), ∀v ∈ VN0 , with 1,0

UN0 = ΠN u0 ,

(14)

ΠN1,0

where is the projection operator see in Section 3. Next, we derive the matrix representation. The numerical solution can be denoted by UNn =

N −2 N −2 ∑ ∑

cijn φi (x)ϕj (y).

(15)

i=0 j=0

Let v = φh ϕl (h, l = 0, 1, 2, . . . , N − 2) in (12), we have N −2 N −2 { ∑ ∑

(φi ϕj , φh ϕl ) + k1 µ(∂x φi ϕj , φh ϕl ) + k1 µ(φi ∂y ϕj , φh ϕl )

i=0 j=0

(16)

+ k2 µ(∂x φi ϕj , ∂x φh ϕl ) + k2 µ(φi ∂y ϕj , φh ∂y ϕl )

}

cijn = H n−1 (φh ϕl ),

where H n−1 (φh ϕl ) =

(∑ n−1

j

(an−j−1 − an−j )UN + an−1 UN0 , φh ϕl

)

j=1

(17)

(

) n−1 n−2 n + µ 2G(UN ) − G(UN ) + f , φh ϕl . Then the matrix representation is obtained,

(

Mx ⊗ My + k1 µ(Ax ⊗ My + Mx ⊗ Ay )

+ k2 µ(Sx ⊗ My + Mx ⊗ Sy )

)

(18) C n = H n−1 ,

where (Mx )h,i = (φi , φh ), (My )l,j = (ϕj , ϕl ), (Ax )h,i = (∂x φi , φh ), (Ay )l,j = (∂y ϕj , ϕl ), Please cite this article as: H. Zhang, J. Jia and X. Jiang, An optimal error estimate for the two-dimensional nonlinear time fractional advection–diffusion equation with smooth and non-smooth solutions, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.12.013.

4

H. Zhang, J. Jia and X. Jiang / Computers and Mathematics with Applications xxx (xxxx) xxx

(Sx )h,i = (∂x φi , ∂x φh ), (Sy )l,j = (∂y ϕj , ∂y ϕl ), (C n )i,j = cijn , i, j = 0, 1, 2, . . . , N − 2. 3. Error estimate In the following, Cq , C q , ˆ Cq , ˜ Cq (q = 1, 2, . . .) are positive constants which are independent of τ , n, and N. 1,0 2 To analyze the L error estimate, we introduce the projection operator ΠN : H01 (Ω ) → VN0 defined by 1,0

(∇ (u − ΠN u), ∇v ) = 0, ∀v ∈ VN0 .

(19)

The following estimate [39] holds,

∥u − ΠN1,0 u∥l ≤ C1 N l−r ∥u∥r , 0 ≤ l ≤ r .

(20)

Next, we present the following lemmas used in the numerical analysis. Lemma 3.1 ([29]). For any u ∈ PN (Θ ), the inverse inequality is obtained,

∥u∥L∞ ≤ C2 N ∥u∥. Lemma 3.2 ([29]). For u ∈ H01 (Ω ), there exists a constant C4 such that

∥u∥L2 (Ω ) ≤ C4 |u|H 1 (Ω ) . 0

Lemma 3.3 ([37]). Let {un , g n |n ≥ 0} be a nonnegative sequence of functions defined in Ω satisfies Dτ u1 ≤ λ1 u1 + λ2 u0 + g 1 and Dτ un ≤ λ1 un + λ2 un−1 + λ3 un−2 + g n , 2 ≤ n ≤ K , where λ1 ≥ 0, λ2 ≥ 0 and λ3 ≥ 0 are presented constants independent of τ . There exists a positive constant τ ∗ such that, when τ ≤ τ ∗ , un ≤

(

Here Eα (z) =

u0 +

tnα

max g j

)

Γ (1 + α ) 1≤j≤n

zk k=0 Γ (1+kα )

∑∞

Eα (2λtnα ), 1 ≤ n ≤ N .

is the Mittag-Leffler function, and λ = λ1 +

λ2 a0 −a1

+

λ3 a1 −a2

.

Lemma 3.4 ([40]). Suppose Ω is a domain in Rd satisfying the cone condition. When mp > d, take p ≤ q ≤ ∞; when mp = d, let p ≤ q < ∞; when mp < d, let p ≤ q ≤ q∗ = dp/(d − mp). There exists a constant C3 depending on m, d, p, q, and the dimensions of Ω ,

∥u∥q ≤ C3 ∥u∥εm,p ∥u∥1p−ε , where ε = (d/mp) − (d/mq). 3.1. Temporal error analysis A time-discrete system is given by, Dτ U n = −k1 ∇ U n + k2 ∆U n + 2G(U n−1 ) − G(U n−2 ) + f n , n ≥ 2.

(21)

When n = 1, U1 − U0

µ

= −k1 ∇ U 1 + k2 ∆U 1 + G(U 0 ) + f 1 ,

(22)

with the initial–boundary conditions U 0 (x, y) = u0 (x, y), (x, y) ∈ Ω ,

(23)

U n (x, y) = 0, (x, y) ∈ ∂ Ω , n = 0, 1, . . . , K .

(24)

Thus, the error can be divided into two parts,

∥un − UNn ∥ ≤ ∥un − U n ∥ + ∥U n − UNn ∥.

(25)

The boundedness of (25) can be obtained by estimating ∥un − U n ∥ and ∥U n − UNn ∥ separately. Please cite this article as: H. Zhang, J. Jia and X. Jiang, An optimal error estimate for the two-dimensional nonlinear time fractional advection–diffusion equation with smooth and non-smooth solutions, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.12.013.

H. Zhang, J. Jia and X. Jiang / Computers and Mathematics with Applications xxx (xxxx) xxx

5

Let en = un − U n , n = 0, 1, . . . , K , Dτ en = −k1 ∇ en + k2 ∆en + F1n−1 + Rn , n ≥ 2,

(26)

where F1n−1 = 2G(un−1 ) − G(un−2 ) − (2G(U n−1 ) − G(U n−2 )).

(27)

We subtract (8) from (22), e1

µ

= −k1 ∇ e1 + k2 ∆e1 + F10 + R1 ,

(28)

in which F10 = G(u0 ) − G(U 0 ).

(29)

We define

η1 = max ∥un ∥L∞ + max ∥Dτ un ∥2 + max ∥un ∥2 + 1. 1≤n≤K

1≤n≤K

(30)

1≤n≤K

Now, we develop an optimal error estimate of en for the time-discrete system (21)–(24). Theorem 3.1. Assume that u ∈ H01 (Ω ) ∩ H r (Ω )(r ≥ 2) satisfying (3) and U n are the solutions of problems (1) and (21)–(24), respectively. There exists τ1∗ > 0 such that when τ ≤ τ1∗ ,

∥en ∥2 ≤ τ ,

(31)

∥U ∥L∞ + ∥Dτ U ∥2 ≤ 2η1 . n

n

(32)

Proof. When n = 1, taking the inner product (28) with e1 in L2 ,

∥e1 ∥2 + k2 ∥∇ e1 ∥2 = k1 (e1 , ∇ e1 ) + (F10 , e1 ) + (R1 , e1 ). µ

(33)

Since e0 = u0 − U 0 = 0, from (33), we have

∥e1 ∥2 + µk2 ∥∇ e1 ∥2 = µk1 (e1 , ∇ e1 ) + µ(R1 , e1 ) k1 ∥e1 ∥2 k2 ≤ µk1 ( + ∥∇ e1 ∥2 ) + µ(∥R1 ∥2 + ∥e1 ∥2 ), k2

4k1

thus

µk21

(1 −

k2 µk21

letting 1 −

k2

− µ)∥e1 ∥2 ≤ µ∥R1 ∥2 ,

− µ > 0, by the definition of µ, then τ <

(µk2 − 2µ2 k21 )∥∇ e1 ∥2 ≤ thus τ <

(

)1/α

k2 2Γ (2−α )k21

µ 2

(

k2

Γ (2−α )(k21 +k2 )

)1/α

, similarly, we have

∥R1 ∥2 ,

, then

∥e1 ∥ ≤ C 1 τ 1+α/2 ,

(34)

∥∇ e1 ∥ ≤ C 2 τ 1+α/2 , (35) {( )1/α ( )1/α } k2 2 where τ ≤ τ1 = min , 2Γ (2k−α , 1 . Taking the inner product (28) with ∆e1 in L2 , it is easy to Γ (2−α )(k2 +k ) )k2 1

2

1

have the following estimate

∥∆e1 ∥ ≤ C 2 τ 1+α/2 ,

(36)

when τ ≤ τ1 . Thus,

∥e1 ∥2 ≤ τ , where τ ≤ τ2 =

(37)

2 (C 1

+

2 2C 2 )−1/α ,

and using Lemma 3.4,

+ C3 ∥e ∥2 ≤ ∥u1 ∥L∞ + C3 τ ≤ η1 , τ 1−α ∥Dτ U 1 ∥2 ≤ ∥Dτ u1 ∥2 + ∥Dτ e1 ∥2 ≤ ∥Dτ u1 ∥2 + ≤ η1 , Γ (2 − α ) 1

∥U ∥

L∞

1

≤ ∥u ∥

L∞

1

(38)

Please cite this article as: H. Zhang, J. Jia and X. Jiang, An optimal error estimate for the two-dimensional nonlinear time fractional advection–diffusion equation with smooth and non-smooth solutions, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.12.013.

6

H. Zhang, J. Jia and X. Jiang / Computers and Mathematics with Applications xxx (xxxx) xxx 1

when τ ≤ τ3 = min {(C3 )−1 , (Γ (2 − α )) 1−α }. So (31)–(32) hold for n = 1. We suppose (31)–(32) hold for n ≤ k − 1. Therefore,

∥U n ∥L∞ ≤ ∥un ∥L∞ + ∥en ∥L∞ ≤ ∥un ∥L∞ + C3 ∥en ∥2 ≤ ∥un ∥L∞ + C3 τ ≤ ∥un ∥L∞ + 1 ≤ η1 ,

(39)

where τ ≤ τ3 . Next, we prove that (31) holds for n = k. Taking the inner product of (26) with ek in L2 , (Dτ ek , ek ) + k2 ∥∇ ek ∥2 = k1 (ek , ∇ ek ) + (F1k−1 , ek ) + (Rk , ek ).

(40)

It is shown that

∥F1k−1 ∥ =∥2G′ (ξ k−1 )ek−1 − G′ (ξ k−2 )ek−2 ∥ ( ) ≤ C 3 ∥G′ (ξ k−1 )∥∞ ∥ek−1 ∥ + ∥G′ (ξ k−2 )∥∞ ∥ek−2 ∥

(41)

∥) , ] where ξ l ∈ min{ul , U l }, max{ul , U l } , |ξ l | ≤ max {η, η1 }, l = n − 2, n − 1. It infers that, ≤ C 4 (∥ e

k−1

k−2

∥ + ∥e

[

(Dτ ek , ek ) ≤ C 5 (∥ek−1 ∥2 + ∥ek−2 ∥2 ) + Cη2 τ 4−2α + C 6 ∥ek ∥2 .

(42)

By the definition of Dτ and the fact 1 = a0 > a1 > · · · > an−1 > 0, we have (Dτ ek , ek )

=

≥

= =

(

1

µ µ 1 2µ 2

k−1 ∑

(ak−j−1 − ak−j )ej − ak−1 e0 , ek

)

j=1

(

1

1

a0 ek −

k 2

a0 ∥ e ∥ −

k−1 ∑

(ak−j−1 − ak−j )

∥ej ∥2 + ∥ek ∥2 2

j=1

(

a0 ∥ek ∥2 −

− ak−1

k−1 ∑

(ak−j−1 − ak−j )∥ej ∥2 − ak−1 ∥e0 ∥2

∥e0 ∥2 + ∥ek ∥2

)

2

)

j=1

Dτ ∥ek ∥2 ,

then from Lemma 3.3 (Discrete fractional Gronwall inequality), let λ1 = C 6 , λ2 = λ3 = C 5 , g k = Cη2 τ 4−2α , then there exists a positive constant τ ∗ = τ4 , when τ ≤ τ4 , we obtain

∥ek ∥ ≤ C 7 τ 2−α .

(43) k

2

k

Taking the inner product of (26) with Dτ e in L to get the estimate ∥e ∥1 ,

∥Dτ ek ∥2 + k2 (Dτ ∇ ek , ∇ ek )

(44)

= − k1 (∇ ek , Dτ ek ) + (F1k−1 , Dτ ek ) + (Rk , Dτ ek ). Then Dτ ∥∇ ek ∥2 ≤ C5 ∥∇ ek ∥2 + C6 τ 4−2α .

Similar to (43), by Lemma 3.3, let λ1 = C5 , λ2 = λ3 = 0, g k = C6 τ 4−2α , then there exists a positive constant τ ∗ = τ5 , when τ ≤ τ5 , we obtain

∥∇ ek ∥2 ≤ C 8 τ 4−2α .

(45)

Moreover, taking the inner product of (26) with Dτ ∆e in L , k

∥Dτ ∇ ek ∥2 + k2 (Dτ ∆ek , ∆ek ) = − k1 (∆ek , Dτ ∇ ek ) + (∇ F1k−1 , Dτ ∇ ek ) + (∇ Rk , Dτ ∇ ek ),

2

(46)

Adding (44) and (46), we arrive at

∥∆ek ∥2 ≤ C 9 τ 4−2α ,

(47)

here τ ≤ τ5 > 0. Combining (43), (45) and (47) yields, 2

1

∥ek ∥2 ≤ (C 7 + C 8 + C 9 ) 2 τ 2−α ≤ τ ,

(48)

Please cite this article as: H. Zhang, J. Jia and X. Jiang, An optimal error estimate for the two-dimensional nonlinear time fractional advection–diffusion equation with smooth and non-smooth solutions, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.12.013.

H. Zhang, J. Jia and X. Jiang / Computers and Mathematics with Applications xxx (xxxx) xxx

7

−1

2

when τ ≤ τ6 = (C 7 + C 8 + C 9 ) 2(1−α) . Furthermore,

∥U n ∥L∞ ≤ ∥un ∥L∞ + C3 ∥en ∥2 ≤ ∥un ∥L∞ + C3 τ ≤ η1 , 2τ 1−α ∥Dτ U n ∥2 ≤ ∥Dτ un ∥2 + ∥Dτ en ∥2 ≤ ∥Dτ un ∥2 + ≤ η1 , Γ (2 − α ) 1 ( Γ (2−α) ) 1−α for 1 ≤ n ≤ K . When τ ≤ τ7 = min {(C3 )−1 , }. Taking τ1∗ = min {τi }, 1 ≤ i ≤ 7, the proof is end. □ 2

(49)

3.2. Spatial error analysis 1,0

1 ,0

For any v ∈ H 2 (Ω ), it has ∥ΠN v∥L∞ ≤ C ∥v∥2 . Therefore, we obtain the boundedness of ∥ΠN U n ∥L∞ for n = 1, 2, . . . , K . Define the following constant

η2 = max ∥ΠN1,0 U n ∥L∞ + 1.

(50)

1≤n≤K

Denote

ξNn = ΠN1,0 U n − UNn ,

n = 0, 1, . . . , K .

(51)

Based on (12) and (21), we have (Dτ ξNn , v ) = − k1 (∇ξNn , v ) + k2 (∆ξNn , v ) + (F2n−1 , v )

(52)

− (Dτ (U n − ΠN1,0 U n ), v ), n ≥ 2, here F2n−1 = 2G(U n−1 ) − G(U n−2 ) − (2G(UNn−1 ) − G(UNn−2 )).

(53)

Combining (13) and (22) presents

ξN1 ,v µ

(

)

( = −k1 (∇ξN1 , v ) + k2 (∆ξN1 , v ) + (F20 , v ) −

1,0

U 1 − ΠN U 1

µ

) ,v ,

(54)

where F20 = G(U 0 ) − G(UN0 ).

(55)

Next, the following theorem is shown to prove the error estimate of ξ

n N.

Theorem 3.2. Suppose that Eqs. (1) has a unique solution u ∈ H01 (Ω ) ∩ H r (Ω ) satisfying (3). Then, the Legendre–Galerkin spectral method (12)–(13) has a unique solution UNn , n = 1, 2, . . . , K . There exists τ2∗ > 0, N1∗ > 0 satisfying, 3

∥ξNn ∥ ≤ N − 2 ,

(56)

≤ η2 ,

(57)

∥

UNn L∞

∥

when τ ≤ τ2∗ , N ≥ N1∗ . Proof. We take v = ξN1 in (54),

( ) ∥ξN1 ∥2 + µk2 ∥∇ξN1 ∥2 = µk1 (ξN1 , ∇ξN1 ) + µ(F20 , ξN1 ) − U 1 − ΠN1,0 U 1 , ξN1 .

(58)

1,0

Since UN0 = ΠN u0 , the following estimate holds

∥UN0 ∥L∞ ≤ ∥ΠN1,0 U 0 ∥L∞ + ∥ΠN1,0 U 0 − UN0 ∥L∞ ≤ η2 .

(59)

It is easy to obtain

µk1 (ξN1 , ∇ξN1 ) ≤

(

1,0

µk21 4k2

U 1 − ΠN U 1 , ξN1

)

∥ξN1 ∥2 + µk2 ∥∇ξN1 ∥2 , ≤ˆ C3 N −4 +

∥ξN1 ∥2 2

(60)

,

and

µ(F20 , ξN1 ) ≤ ˆ C1 µ∥ξN1 ∥2 + ˆ C2 N − 4 ,

(61)

Please cite this article as: H. Zhang, J. Jia and X. Jiang, An optimal error estimate for the two-dimensional nonlinear time fractional advection–diffusion equation with smooth and non-smooth solutions, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.12.013.

8

H. Zhang, J. Jia and X. Jiang / Computers and Mathematics with Applications xxx (xxxx) xxx

where τ ≤ τ7 . We can see that C2 + ˆ C3 )N −4 . C4 µ∥ξN1 ∥2 + 2(ˆ ∥ξN1 ∥2 ≤ ˆ

(62)

C5 N −2 , ∥ξN1 ∥ ≤ ˆ

(63)

Then,

when τ ≤ τ8 . It implies 3

∥ξN1 ∥ ≤ N − 2 ,

(64)

in which N ≥ N1 = (ˆ C5 )2 . The estimate (56) holds for n = 1. We suppose (56) holds for n ≤ k − 1. Then, from Lemma 3.1

∥UNn ∥L∞ ≤ ∥ΠN1,0 U n ∥L∞ + ∥ΠN1,0 U n − UNn ∥L∞ ≤ ∥ΠN1,0 U n ∥L∞ + C2 N ∥ξNn ∥ ≤ ∥ΠN1,0 U n ∥L∞ + C2 N − 2 1

ΠN1,0 U n L∞

≤∥ ≤ η2 ,

∥

(65)

+1

when N ≥ N2 = (C2 )2 . Now, we verify (56) for n = k. Let v = ξNk in (52), (Dτ ξNk , ξNk ) + k2 ∥∇ξNk ∥2

(66)

( ) =k1 (ξNk , ∇ξNk ) + (F2k−1 , ξNk ) − Dτ (U k − ΠN1,0 U k ), ξNk . Similar to the above process k1 (ξNk , ∇ξNk ) ≤

k21 4k2

∥ξNk ∥2 + k2 ∥∇ξNk ∥2 ,

(F2k−1 , ξNk ) ≤ ˆ C6 N −4 + ˆ C7 (∥ξNk−1 ∥2 + ∥ξNk−2 ∥2 ) +

(

1,0

Dτ (U k − ΠN U k ), ξNk

)

1 2

∥ξNk ∥2 ,

(67)

1

≤ˆ C8 N −4 + ∥ξNk ∥2 . 2

It is easy to see that (Dτ ξNk , ξNk ) ≤(ˆ C6 + ˆ C8 )N −4 +

( 1+

k21

)

4k2

∥ξNk ∥2

(68)

+ˆ C7 (∥ξNk−1 ∥2 + ∥ξNk−2 ∥2 ). From Lemma 3.3, there exists a positive constant τ9

∥ξNk ∥ ≤ ˆ C9 N −2 ,

(69)

when τ ≤ τ9 . It further indicates that 3

∥ξNk ∥ ≤ N − 2 ,

(70)

∥UNk ∥L∞ ≤ ∥ΠN1,0 U k ∥L∞ + ∥ξNk ∥L∞ ≤ ∥ΠN1,0 U k ∥L∞ + C2 N − 2 ≤ η2 , 1

when N ≥ N3 = max {(ˆ C9 )2 , (C2 )2 }. Let τ2∗ = min {τ7 , τ8 , τ9 } and N1∗ = max {N1 , N2 , N3 }. The proof is completed.

□

3.3. Error analysis for the fully discrete scheme Denote

θNn = ΠN1,0 un − UNn ,

n = 0, 1, . . . , K .

(71)

The error equation of θNn is derived, (Dτ θNn , v ) = − k1 (∇θNn , v ) + k2 (∆θNn , v ) + (F3n−1 , v )

− (Dτ (un − ΠN1,0 un ), v ) + (Rn , v ), n ≥ 2,

(72)

here F3n−1 = 2G(un−1 ) − G(un−2 ) − (2G(UNn−1 ) − G(UNn−2 )).

(73)

Please cite this article as: H. Zhang, J. Jia and X. Jiang, An optimal error estimate for the two-dimensional nonlinear time fractional advection–diffusion equation with smooth and non-smooth solutions, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.12.013.

H. Zhang, J. Jia and X. Jiang / Computers and Mathematics with Applications xxx (xxxx) xxx

9

Together (8) with (13),

(

θN1 ,v µ

)

= − k1 (∇θN1 , v ) + k2 (∆θN1 , v ) + (F30 , v ) ( ) 1,0 u1 − ΠN u1 − , v + (R1 , v ), µ

(74)

where F30 = G(u0 ) − G(UN0 ).

(75)

The following optimal error estimate of the numerical method (12)–(13) can be shown. Theorem 3.3. Suppose that r ≥ 1, Eq. (1) has the exact solution u ∈ H01 (Ω ) ∩ H r (Ω ) satisfying (3), and the fully discrete scheme (12)–(13) has a unique solution UNn . The following estimates are obtained

∥u1 − UN1 ∥ ≤ C ∗ (τ + N −r ),

(76)

∥un − UNn ∥ ≤ C (τ 2−α + N −r ), 2 ≤ n ≤ K ,

(77)

where C , C ∗ are two positive constants which are independent of n, τ , and N. Proof. Taking v = θN1 in (74), it is obvious to obtain the following estimate,

∥θN1 ∥ ≤ ˜ C1 (τ + N −r ).

(78)

For 2 ≤ n ≤ K , let v = θNn in (72), (Dτ θNn , θNn ) + k2 ∥∇θNn ∥2

(79)

= k1 (θNn , ∇θNn ) + (F3n−1 , θNn ) − (Dτ (un − ΠN1,0 un ), θNn ) + (Rn , θNn ). We get (Dτ θNn , θNn ) ≤˜ C2 N −2r + ˜ C3 (∥θNn−1 ∥2 + ∥θNn−2 ∥2 )

(80)

+˜ C4 ∥θNn ∥2 + Cη2 τ 4−2α . There exists a positive constant τ10 such that

∥θNn ∥ ≤ ˜ C5 (τ 2−α + N −r ),

(81)

where τ ≤ τ10 . The above equation indicates

∥un − UNn ∥ ≤ ∥un − ΠN1,0 un ∥ + ∥θNn ∥ ≤ (˜ C1 η + ˜ C5 )(τ 2−α + N −r )

(82)

for 2 ≤ n ≤ K . Taking τ ≤ τ0 = min {τ1 , τ2 , τ10 }, and N ≥ N0 = N1 , (76) holds when τ ≤ τ0 , N ≥N0 , and C = ˜ C1 , (77) ∗

∗

holds when τ ≤ τ0 , N ≥ N0 , and C = ˜ C1 η + ˜ C5 .

∗

∗

□

4. Correction method

ˆ σk + ˜ Suppose that the solution u satisfies u(t) − u(0) = u(t)σm+1 , 0 ≤ t ≤ T , where 0 < σk < σk+1 and ∥˜ u(t)∥∞ k=1 uk t is bounded for 0 ≤ t ≤ T . Then we develop the correction method to deal with the non-smooth solutions, ∑m

C α n 0 Dt u

= D τ un +

m 1 ∑

τα

n ˜n Wnα,j (uj − u0 )+˜ Rn = Dm τ u +R ,

(83)

j=1

σ −2−2α where the truncation error ˜ Rn satisfies ˜ Rn = O(τ 2−α tn m+1 ) + O(τ σm+1 +1 tn−α−1 ) and Wnα,j are the starting weights satisfying

D τ un +

m 1 ∑

τα

j=1

Wnα,j (uj − u0 ) =

Γ (1 + σj ) σj −α t , Γ (1 + σj − α )

(84)

in which u = t σj , 0 < σj < σj+1 . We can get Wnα,j (1 ≤ j ≤ m) from (84) and Wnα,j is independent of τ for n > 0. The above result yields the following fully form, when n = 1, then 1 1 1 0 1 (Dm τ UN , v ) = −k1 (∇ UN , v ) + k2 (∆UN , v ) + (G(UN ), v ) + (f , v ),

(85)

Please cite this article as: H. Zhang, J. Jia and X. Jiang, An optimal error estimate for the two-dimensional nonlinear time fractional advection–diffusion equation with smooth and non-smooth solutions, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.12.013.

10

H. Zhang, J. Jia and X. Jiang / Computers and Mathematics with Applications xxx (xxxx) xxx

Fig. 1. The exact solution and the numerical solution for Example 5.1.

when n ≥ 2, we have n−2 n−1 n n n (Dm τ UN , v ) = − k1 (∇ UN , v ) + k2 (∆UN , v ) + (2G(UN ) − G(UN ), v )

+ (f n , v ),

(86)

in which v ∈ VN0 . Since the correction terms are bounded, we can move these terms to the right side of Eqs. (85)–(86), the correction terms do not have a influence on the stability [36] for the scheme (85)–(86). 5. Numerical results In this section, we give some numerical examples to illustrate the accuracy of the numerical scheme. We define the convergence orders of time and space in the L2 -norm sense as follows,

{ log(∥error(τ1 )∥/∥error(τ2 )∥) order =

where error = ∥uexact − Example 5.1.

in time,

log(τ1 /τ2 )

in space,

log(∥error(N1 )∥/∥error(N2 )∥) log(N2 /N1 )

ukN

∥ is the error equation, τ1 ̸= τ2 , and N1 ̸= N2 .

The following time fractional equation with a source term for Ω = [0, 1]2 is provided,

⎧ C α 2 ⎪ ⎨0 Dt u = −∇ u + ∆u + u + f , (x, y, t) ∈ Ω × I , u(x, y, 0) = x2 (1 − x)2 y2 (1 − y)2 , (x, y) ∈ Ω , ⎪ ⎩u(· , t) = 0, (x, y) ∈ ∂ Ω , t ∈ I ,

(87)

and f (x, y, t) =

2t 2−α

x2 (1 − x)2 y2 (1 − y)2 + (t 2 + 1)(4x3 − 6x2 + 2x)y2 (1 − y)2 Γ (3 − α ) + (t 2 + 1)x2 (1 − x)2 (4y3 − 6y2 + 2y) − 2(t 2 + 1)x2 (1 − x)2 2

2

2

2

(6y − 6y + 1) − 2(t + 1)(6x − 6x + 1)y (1 − y)

(88)

2

− (t 2 + 1)2 x4 (1 − x)4 y4 (1 − y)4 . The exact solution is u = (t 2 + 1)x2 (1 − x)2 y2 (1 − y)2 . For this model, we set T = 1. In the example, we take τ = 0.01 and N = 32. Fig. 1 shows the numerical solution and the exact solution of Example 5.1 when α = 0.8. We can see that the numerical solution is in good agreement with the exact solution. Table 1 exhibits the L2 -errors, L∞ -errors, and convergence order versus τ for different α when N = 32. It illustrates that our method has (2 −α ) order accuracy in time. Fig. 2 presents the L2 -errors and L∞ -errors versus N for different α when taking τ = 0.001. It is obvious that our method achieves the spectral accuracy in space. The above numerical results are consistent with the theoretical analysis. Example 5.2. The following equation with non-smooth solution is presented to verify the effectiveness of the correction method,

⎧C 0.1 3 ⎨0 Dt u = −∇ u + ∆u + u − u + f , (x, y, t) ∈ Ω × I , u(x, y, 0) = 0, (x, y) ∈ Ω , ⎩ u(· , t) = 0, (x, y) ∈ ∂ Ω , t ∈ I ,

(89)

Please cite this article as: H. Zhang, J. Jia and X. Jiang, An optimal error estimate for the two-dimensional nonlinear time fractional advection–diffusion equation with smooth and non-smooth solutions, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.12.013.

H. Zhang, J. Jia and X. Jiang / Computers and Mathematics with Applications xxx (xxxx) xxx

11

Table 1 The L2 -error, L∞ -error, and convergence order versus τ for different α in Example 5.1.

α

τ

L2 -error

Order

L∞ -error

Order

0.4

1/10 1/20 1/40 1/80 1/160

4.5305e−07 1.5554e−07 5.2902e−08 1.7859e−08 5.9946e−09

– 1.5424 1.5559 1.5667 1.5749

1.3121e−06 4.5089e−07 1.5348e−07 5.1848e−08 1.7412e−08

– 1.5410 1.5547 1.5657 1.5742

0.8

1/10 1/20 1/40 1/80 1/160

2.6295e−06 1.1519e−06 5.0334e−07 2.1959e−07 9.5706e−08

– 1.1908 1.1944 1.1967 1.1981

7.6371e−06 3.3465e−06 1.4626e−06 6.3813e−07 2.7815e−07

– 1.1903 1.1942 1.1966 1.1980

Fig. 2. The errors of L2 −norm and L∞ −norm versus N for different α in Example 5.1.

and f (x, y, t) =

6 ( ∑ 2π 2 t 0.1k

k+1

k=1

+

Γ (0.1k + 1)t 0.1k−0.1 (k + 1)Γ (0.1k + 1 − 0.1)

)

sin(π x) sin(π y)

6 ∑ t 0.1k (cos(π x) sin(π y) + sin(π x) cos(π y)) k+1 k=1 ( 6 )3 k 6 ∑ ∑ t 0.1k t 0.1 − sin(π x) sin(π y) + sin(π x) sin(π y) . k+1 k+1

+π

k=1

(90)

k=1

∑6

k t 0.1

2 The exact solution is u = k=1 k+1 sin(π x) sin(π y) and it has Ω = (0, 2) , I = (0, 1]. The numerical solution and the exact solution of Example 5.2 are shown in Fig. 3 when N = 32, τ = 0.01, it demonstrates that the numerical solution is well in line with the exact solution. In terms of the regularity of the solution, we choose σj = 0.1j. Tables 2 and 3 compare the results of the two schemes (12) and (85)–(86). From Tables 2 and 3, we can find that schemes with correction terms have higher order accuracy than the scheme with no correction terms (m = 0), which shows the effectiveness of the correction method.

6. Conclusions In this paper, we develop a linearized L1 Legendre–Galerkin spectral method for the two-dimensional nonlinear time fractional advection–diffusion equation. We use L1 method and Legendre–Galerkin spectral method to discrete the time direction and the space direction, respectively. The numerical method is stable without the CFL conditions by using the error splitting argument technique and the discrete fractional Gronwall type inequality. For the strong singularity of the time fractional advection–diffusion equation at t = 0, we propose the correction method. Finally, numerical experiments are shown to verify the theoretical analysis and the effectiveness of the correction method. Please cite this article as: H. Zhang, J. Jia and X. Jiang, An optimal error estimate for the two-dimensional nonlinear time fractional advection–diffusion equation with smooth and non-smooth solutions, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.12.013.

12

H. Zhang, J. Jia and X. Jiang / Computers and Mathematics with Applications xxx (xxxx) xxx

Fig. 3. The exact solution and the numerical solution for Example 5.2. Table 2 The errors and order of L2 -norm versus τ for Example 5.2.

τ 1/10 1/20 1/40 1/80 1/160

m = 0

m = 1

m = 3

Error

Order

Error

Order

Error

Order

4.0211e−05 3.3439e−05 1.7270e−05 8.2951e−06 3.8696e−06

– 0.2660 0.9533 1.0580 1.1001

8.9573e−05 2.1094e−05 6.2146e−06 1.9736e−06 6.6954e−07

– 2.0862 1.7631 1.6548 1.5596

6.6355e−05 1.2076e−05 2.6610e−06 5.6403e−07 1.0771e−07

– 2.4581 2.1821 2.2381 2.3886

Table 3 The errors and order of L∞ -norm versus τ for Example 5.2.

τ 1/10 1/20 1/40 1/80 1/160

m = 0

m = 1

m = 3

Error

Order

Error

Order

Error

Order

5.7605e−05 4.1738e−05 2.0796e−05 9.8325e−06 4.5501e−06

– 0.4648 1.0051 1.0807 1.1117

1.1296e−04 2.4006e−05 7.2182e−06 2.3385e−06 8.0748e−07

– 2.2343 1.7337 1.6261 1.5341

8.3635e−05 1.3201e−05 2.8832e−06 5.9935e−07 1.1145e−07

– 2.6635 2.1949 2.2662 2.4270

CRediT authorship contribution statement Hui Zhang: Conceptualization, Methodology, Writing - original draft, Software. Junqing Jia: Software, Methodology, Writing - original draft, Writing - review & editing. Xiaoyun Jiang: Conceptualization, Methodology, Writing - review & editing, Supervision. Acknowledgments This work has been supported by the National Natural Science Foundation of China (Grants Nos. 11672163, 11771254), the Natural Science Foundation of Shandong Province (Grant No. ZR2017MA030), the Fundamental Research Funds for the Central Universities (Grant No. 2019ZRJC002). References [1] P. Becker-Kern, M.M. Meerschaert, H.P. Scheffler, Limit theorems for coupled continuous time random walks, Ann. Probab. 32 (2004) 730–756. [2] M.M. Meerschaert, H.P. Scheffler, Limit theorems for continuous time random walks with infinite mean waiting times, J. Appl. Probab. 41 (2004) 623–638. [3] J. Zhang, X. Zhang, B. Yang, An approximation scheme for the time fractional convection–diffusion equation, Appl. Math. Comput. 335 (2018) 305–312. [4] Y. Zhang, D.A. Benson, D.M. Reeves, Time and space nonlocalities underlying fractional-derivative models: Distinction and literature review of field applications, Adv. Water. Resour. 32 (2009) 561–581. [5] A. Jannelli, M. Ruggieri, M.P. Speciale, Exact and numerical solutions of time-fractional advection-diffusion equation with a nonlinear source term by means of the lie symmetries, Nonlinear Dynam. 92 (2018) 543–555. [6] G.M. Zaslavsky, Fractional kinetic equation for hamiltonian chaos: chaotic advection, tracer dynamics ad turbulent dispersion, Physica D 76 (1994) 110–122. [7] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999. [8] Y. Liu, Y. Du, H. Li, S. He, W. Gao, Finite difference/finite element method for a nonlinear time-fractional fourth-order reaction–diffusion problem, Comput. Math. Appl. 70 (2015) 573–591.

Please cite this article as: H. Zhang, J. Jia and X. Jiang, An optimal error estimate for the two-dimensional nonlinear time fractional advection–diffusion equation with smooth and non-smooth solutions, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.12.013.

H. Zhang, J. Jia and X. Jiang / Computers and Mathematics with Applications xxx (xxxx) xxx

13

[9] Y. Liu, Y. Du, H. Li, J. Wang, A two-grid finite element approximation for a nonlinear time-fractional Cable equation, Nonlinear Dynam. 4 (2016) 2535–2548. [10] Y. Liu, Z. Yu, H. Li, F. Liu, J. Wang, Time two-mesh algorithm combined with finite element method for time fractional water wave model, Int. J. Heat Mass Transfer 120 (2018) 1132–1145. [11] H. Chen, S. Lv, W. Chen, A unified numerical scheme for the multi-term time fractional diffusion and diffusion-wave equations with variable coefficients, J. Comput. Appl. Math. 330 (2018) 380–397. [12] W. Jiang, Z. Chen, A collocation method based on reproducing kernel for a modified anomalous subdiffusion equation, Numer. Methods Partial Differential Equations 30 (2014) 289–300. [13] S. Yi, H. Sun, A hybrided trapezoidal-difference scheme for nonlinear time-fractional fourth-order advection-dispersion equation based on Chebyshev spectral collocation method, Adv. Appl. Math. Mech. 11 (2019) 197–215. [14] C. Li, Q. Yi, Finite difference method for two-dimensional nonlinear time-fractional subdiffusion equation, Fract. Calc. Appl. Anal. 21 (2018) 1046–1072. [15] Q. Liu, Y. Gu, P. Zhuang, F. Liu, Y. Nie, An implicit RBF meshless approach for time fractional diffusion equations, Comput. Mech. 48 (2011) 1–12. [16] M.M. Meerschaert, C. Tadjeran, Finite difference approximations for fractional advection–dispersion flow equations, J. Comput. Appl. Math. 172 (2004) 65–77. [17] A. Yildirim, H. Koçak, Homotopy perturbation method for solving the space–time fractional advection–dispersion equation, Adv. Water Resour. 32 (2009) 1711–1716. [18] F. Liu, P. Zhuang, V. Anh, I. Turner, K. Burrage, Stability and convergence of the difference methods for the space–time fractional advection-diffusion equation, Appl. Math. Comput. 191 (2007) 12–20. [19] B. Yu, X. Jiang, H. Qi, Numerical method for the estimation of the fractional parameters in the fractional mobile/immobile advection-diffusion model, Int. J. Comput. Math. 95 (2018) 1131–1150. [20] J. Huang, N. Nie, Y. Tang, A second order finite difference-spectral method for space fractional diffusion equations, Sci. Chin. Math. 57 (2014) 1303–1317. [21] M. Zheng, F. Liu, V. Anh, I. Turner, A high-order spectral method for the multi-term time-fractional diffusion equations, Appl. Math. Model. 40 (2016) 4970–4985. [22] R. Ren, H. Li, W. Jiang, M. Song, An efficient Chebyshev-Tau method for solving the space fractional diffusion equations, Appl. Math. Comput. 224 (2013) 259–267. [23] Z. Liu, S. Lv, F. Liu, Fully discrete spectral method for Sine-Gordon equation with smooth and non-smooth solutions, Appl. Math. Comput. 333 (2018) 212–224. [24] Z. Liu, F. Liu, F. Zeng, An alternating direction implicit spectral method for solving two dimensional multi-term time fractional mixed diffusion and diffusion-wave equations, Appl. Numer. Math. 136 (2019) 139–151. [25] Y. Lin, C. Xu, Finite difference/spectral approximations for the time-fractional diffusion equation, J. Comput. Phys. 225 (2007) 1533–1552. [26] H. Zhang, X. Jiang, Spectral method and Bayesian parameter estimation for the space fractional coupled nonlinear Schrödinger equations, Nonlinear Dynam. 95 (2019) 1599–1614. [27] R.E. Ewing, M.F. Wheeler, Galerkin methods for miscible displacement problems in porous media, SIAM J. Numer. Anal. 17 (1980) 351–365. [28] M. Luskin, A Galerkin method for nonlinear parabolic equations with nonlinear boundary conditions, SIAM J. Numer. Anal. 16 (1979) 284–299. [29] F. Zeng, F. Liu, C. Li, K. Burrage, I. Turner, V. Anh, Crank–Nicolson ADI spectral method for the two-dimensional riesz space fractional nonlinear reaction–diffusion equation, SIAM J. Numer. Anal. 52 (2014) 2599–2622. [30] J. Cannon, Y. Lin, Nonclassical H1 projection and Galerkin methods for non-linear parabolic integro-differential equations, SIAM J. Numer. Anal. 25 (1988) 187–201. [31] N.J. Ford, M.M. Rodrigues, N. Vieira, A numerical method for the fractional Schrödinger type equation of spatial dimension two, Frac. Cacl. Appl. Anal. 16 (2013) 1–15. [32] B. Li, Mathematical Modeling Analysis and Computation for Some Complex and Nonlinear Flow Problems (Ph.D. Thesis), City University of Hong Kong, Hong Kong, 2012. [33] C. Li, Q. Yi, A. Chen, Finite difference methods with non-uniform meshes for nonlinear fractional differential equations, J. Comput. Phys. 316 (2016) 614–631. [34] N. Liu, W. Jiang, A numerical method for solving the time fractional Schrödinger equation, Adv. Comput. Math. 44 (2018) 1235–1248. [35] F. Zeng, I. Turner, K. Burrage, A stable fast time-stepping method for fractional integral and derivative operators, J. Sci. Comput. 77 (2018) 283–307. [36] F. Zeng, Z. Zhang, G. Karniadakis, Second-order numerical methods for multi-term fractional differential equations: Smooth and non-smooth solutions, Comput. Methods Appl. Mech. Engrg. 327 (2017) 478–502. [37] H. Liao, D. Li, J. Zhang, Sharp error estimate of the nonuniform l1 formula for linear reaction-subdiffusion equations, SIAM J. Numer. Anal. 56 (2018) 1112–1133. [38] J. Shen, T. Tang, L. Wang, Spectral Methods: Algorithms, Analysis and Applications, Springer, Heidelberg, 2011. [39] S. Guo, L. Mei, Y. Li, An efficient Galerkin spectral method for two-dimensional fractional nonlinear reaction–diffusion-wave equation, Comput. Math. Appl. 74 (2017) 2449–2465. [40] R.A. Adams, J.J.F. Fournier, Sobolev Spaces, Academic Pres, New York, 2003.

Please cite this article as: H. Zhang, J. Jia and X. Jiang, An optimal error estimate for the two-dimensional nonlinear time fractional advection–diffusion equation with smooth and non-smooth solutions, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.12.013.