# Maximum Norm Error Estimates of ADI Methods for a Two-Dimensional Fractional Subdiffusion Equation

Authors: Yuan-Ming Wang
Journal: Advances in Mathematical Physics
 (2013)

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## Abstract

This paper is concerned with two alternating direction implicit (ADI) finite difference methods for solving a two-dimensional fractional subdiffusion equation. An explicit error estimate for each of the two methods is provided in the discrete maximum norm. It is shown that the methods have the same order as their truncation errors with respect to the discrete maximum norm. Numerical results are given to confirm the
theoretical analysis results.


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## Body

## 1. Introduction

Fractional differential equations and fractional calculus arise in various application problems in science and engineering [1–16]. Various numerical methods have been developed for the computation of fractional differential equations [17–34]. Fractional subdiffusion equations describe a special type of anomalous diffusion [35], and it is a more difficult task to solve this kind of equation numerically.Numerical works for fractional subdiffusion equations are mostly focused on one-dimensional problems due to the memory effect in fractional derivatives; see, for example, [19, 20, 20–26, 31–33, 36–42]. A two-dimensional anomalous subdiffusion equation was numerically treated in [43, 44], where explicit and implicit finite difference schemes were proposed. Chen et al. [28] extended their work in [43] to a variable-order subdiffusion equation. Liu et al. [45] developed an implicit meshless approach based on the radial basis function for the numerical simulation of a two-dimensional subdiffusion problem. Chen and Liu [18] considered an implicit difference scheme for a three-dimensional fractional advection-diffusion equation, and a Richardson extrapolation was applied to improve the accuracy.The complexity of the fractional differential equations comes from the involving fractional derivatives that are nonlocal and have the character of history dependence and universal mutuality. This means that the computations would be costly if the implicit schemes were applied, especially for solving multidimensional problems [43, 44]. Some researchers have explored some techniques for reducing this cost. These techniques include the adaptive technique [46] and the matrix transfer technique [47, 48]. It is well known that alternating direction implicit (ADI) methods are unconditionally stable as the traditional implicit methods. On the other hand, they reduce a multidimensional problem to a series of independent one-dimensional problems, and thus the computational complexities and the computational cost can be greatly reduced. Therefore, ADI methods for fractional differential equations have the potential to significantly reduce the computational cost, while maintaining the stability of the numerical methods. The works in [29, 49–53] treated ADI finite difference methods for space fractional diffusion equations, and the work in [54] discussed ADI finite difference methods for fractional diffusion wave equations. Recently, Cui [55] derived an ADI compact finite difference scheme for a two-dimensional fractional subdiffusion equation, where the Grünwald formula is used to approximate the temporal Riemann-Liouville fractional derivative, and the spatial derivatives are approximated by a compact finite difference scheme. Another way to treat fractional subdiffusion problem is to transform the original subdiffusion equation into an equivalent equation by replacing the temporal Riemann-Liouville fractional derivative by the temporal Caputo fractional derivative; see, for example, [40, 42, 56]. An advantage of this approach is that the L1 approximation (see [1, 24, 25, 40, 57, 58]) can be used to deal with the temporal fractional derivative, and so the resulting scheme has the better temporal accuracy than the first order without the Crank-Nicolson technique (see [25, 40–42]). Based on the previous approach, Zhang and Sun [59] constructed two ADI finite difference schemes, called L1-ADI and BD-ADI schemes, for the following two-dimensional problem of subdiffusion equation with the temporal Caputo fractional derivative:
(1)𝒟0Ctαu(x,y,t)=Δu(x,y,t)+f(x,y,t),(x,y)∈Ω,t∈(0,T],u(x,y,t)=φ(x,y,t),(x,y)∈∂Ω,t∈(0,T],u(x,y,0)=ψ(x,y),(x,y)∈Ω¯,
where Ω=(0,l1)×(0,l2), Ω¯=Ω∪∂Ω, ∂Ω is the boundary of Ω, T is a positive constant, Δ is the two-dimensional Laplacian, and 0C𝒟tα denotes the temporal Caputo fractional derivative operator defined as
(2)𝒟0Ctαu(x,y,t)=1Γ(1-α)∫0t‍∂u(x,y,s)∂s(t-s)-αds,0<α<1.The main concern in that paper is the construction of the schemes and error estimates in the discreteH1-norm. Since H1-norm error estimates do not provide immediate insight on the phase error occurring during the time evolution, it is more preferable to give error estimates in the discrete maximum norm when we measure computation errors in practice. In this paper, we continue the investigation of the paper [59], by establishing a maximum norm error estimate for the ADI discretizations. It is known that an H1-norm error estimate does not imply a maximum norm error estimate for two-dimensional problems. We here present a technique of H2 discrete energy analysis in order to obtain an explicit maximum norm error estimate.The outline of the paper is as follows. In Section2, we derive ADI finite difference schemes for (1) and present our main results of the maximum norm error estimates. The proof of the main error results is given in Section 3. In Section 4, we give some numerical results demonstrating the accuracy of the schemes in the discrete maximum norm. Section 5 contains some concluding remarks.


## 2. ADI Schemes and Maximum Norm Error Estimates

We partitionΩ with nonisotropic uniform mesh sizes h1 and h2 in the x and y directions, respectively. The integers M1=l1/h1 and M2=l2/h2. The mesh points (xi,yj)=(ih1,jh2) (0≤i≤M1,0≤j≤M2). Let Ωh and ∂Ωh be the sets of mesh points lying in Ω and on ∂Ω, respectively, and let Ω¯h=Ωh∪∂Ωh. For any grid function v={vi,j∣0≤i≤M1,0≤j≤M2}, we denote
(3)δxvi,j=(vi+1,j-vi,j)h1,δyvi,j=(vi,j+1-vi,j)h2,δx2vi,j=(δxvi,j-δxvi-1,j)h1,δy2vi,j=(δyvi,j-δyvi,j-1)h2,Δhvi,j=δx2vi,j+δy2vi,j.For a positive integerN, we let τ=T/N be the time step. Define tn=nτ and
(4)Ui,jn=u(xi,yj,tn),fi,jn=f(xi,yj,tn),φi,jn=φ(xi,yj,tn),ψi,j=ψ(xi,yj).For the temporal approximation, we introduce the operators(5)δtUi,jn=1τ(Ui,jn+1-Ui,jn),DtαUi,jn=1μ(Ui,jn-∑k=1n-1‍(an-k-1-an-k)Ui,jk-an-1Ui,j0),
where μ=ταΓ(2-α) and ak=(k+1)1-α-k1-α.Using the Taylor expansion and theL1 approximation of 0C𝒟tαu(x,y,t) (see [25, 58, 59]), we have
(6)𝒟0Ctαu(xi,yj,tn)=DtαUi,jn+(Rt)i,jn,Δu(xi,yj,tn)=ΔhUi,jn-(Rx)i,jn-(Ry)i,jn,(xi,yj)∈Ωh,1≤n≤N,
where
(7)(Rx)i,jn=h126∫01‍(∂4u∂x4(xi+sh1,yj,tn)+∂4u∂x4(xi-sh1,yj,tn))(1-s)3ds,(Ry)i,jn=h226∫01‍(∂4u∂y4(xi,yj+sh2,tn)+∂4u∂y4(xi,yj-sh2,tn))(1-s)3ds.
There exists a positive constant C(α) independent of τ, h1, h2 and the time level n such that
(8)|(Rt)i,jn|≤C(α)max0≤t≤tn|∂2u∂t2(xi,yj,t)|τ2-α.
Substituting (6) into (1), we obtain
(9)DtαUi,jn=ΔhUi,jn+fi,jn-(Rt)i,jn-(Rx)i,jn-(Ry)i,jn,(xi,yj)∈Ωh,1≤n≤N.


### 2.1. Construction ofL1-ADI and BD-ADI Schemes

In order to construct an ADI scheme, we add the termμ2Dtαδx2δy2Ui,jn to (9). This yields
(10)Dtα(Ui,jn+μ2δx2δy2Ui,jn)=ΔhUi,jn+fi,jn+(R1)i,jn,(xi,yj)∈Ωh,1≤n≤N,
where
(11)(R1)i,jn=μ2Dtαδx2δy2Ui,jn-(Rt)i,jn-(Rx)i,jn-(Ry)i,jn.It was shown in [59] that |μ2Dtαδx2δy2Ui,jn|≤CL(α)τ2α for a positive constant CL(α) independent of τ, h1, h2 and the time level n. Thus, there exists a positive constant C1 independent of τ, h1, h2 and the time level n such that
(12)|(R1)i,jn|≤C1(τmin{2α,2-α}+h12+h22).Denote byui,jn the finite difference approximation to Ui,jn, and let bn,k=an-k-1-an-k. After multiplying (10) by μ and then dropping the term μ(R1)i,jn, we derive a finite difference scheme as follows:
(13)(I-μδx2)(I-μδy2)ui,jn=∑k=1n-1‍bn,k(ui,jk+μ2δx2δy2ui,jk)+an-1(ui,j0+μ2δx2δy2ui,j0)+μfi,jn,(xi,yj)∈Ωh,1≤n≤N,ui,jn=φi,jn,(xi,yj)∈∂Ωh,1≤n≤N,ui,j0=ψi,j,(xi,yj)∈Ω¯h.By introducing the intermediate variable vi,jn, we obtain the following L1-ADI scheme (see [59]):
(14)(I-μδx2)vi,jn=∑k=1n-1‍bn,k(ui,jk+μ2δx2δy2ui,jk)+an-1(ui,j0+μ2δx2δy2ui,j0)+μfi,jn,(xi,yj)∈Ωh,1≤n≤N,v0,jn=(I-μδy2)φ0,jn,vM1,jn=(I-μδy2)φM1,jn,0≤j≤M2,1≤n≤N,ui,j0=ψi,j,(xi,yj)∈Ω¯h,(I-μδy2)ui,jn=vi,jn,(xi,yj)∈Ωh,1≤n≤N,ui,0n=φi,0n,ui,M2n=φi,M2n,0≤i≤M1,1≤n≤N.Adding the termτμδx2δy2δtUi,jn to (9), we have
(15)DtαUi,jn+τμδx2δy2δtUi,jn=ΔhUi,jn+fi,jn+(R2)i,jn,
where
(16)(R2)i,jn=τμδx2δy2δtUi,jn-(Rt)i,jn-(Rx)i,jn-(Ry)i,jn.Since|τμδx2δy2δtUi,jn|≤CBD(α)τ1+α for a positive constant CBD(α) independent of τ, h1, h2 and the time level n (see [59]), there exists a positive constant C2 independent of τ, h1, h2 and the time level n such that
(17)|(R2)i,jn|≤C2(τmin{1+α,2-α}+h12+h22).By (15), we obtain the following finite difference scheme:
(18)(I-μδx2)(I-μδy2)ui,jn=μ2δx2δy2ui,jn-1+∑k=1n-1‍bn,kui,jk(I-μδx2)(I-μδy2)ui,jn=+an-1ui,j0+μfi,jn,(xi,yj)∈Ωh,1≤n≤N,ui,jn=φi,jn,(xi,yj)∈∂Ωh,1≤n≤N,ui,j0=ψi,j,(xi,yj)∈Ω¯h.It is equivalent to the following BD-ADI scheme (see [59]):
(19)(I-μδx2)vi,jn=μ2δx2δy2ui,jn-1+∑k=1n-1‍bn,kui,jk+an-1ui,j0+μfi,jn,(xi,yj)∈Ωh,1≤n≤N,v0,jn=(I-μδy2)φ0,jn,vM1,jn=(I-μδy2)φM1,jn,0≤j≤M2,1≤n≤N,ui,j0=ψi,j,(xi,yj)∈Ω¯h,(I-μδy2)ui,jn=vi,jn,(xi,yj)∈Ωh,1≤n≤N,ui,0n=φi,0n,ui,M2n=φi,M2n,0≤i≤M1,1≤n≤N.


### 2.2. Maximum Norm Error Estimates

For any grid functionv={vi,j∣0≤i≤M1,0≤j≤M2}, we define its maximum norm ∥v∥∞ by
(20)∥v∥∞=max0≤i≤M1,0≤j≤M2|vi,j|.LetUi,jn be the value of the solution of (1) at the mesh point (xi,yj,tn), and let ui,jn be the solution of the L1-ADI scheme (14) or the BD-ADI scheme (19). We now present our main results of the maximum norm estimate for the error ei,jn=Ui,jn-ui,jn in the following two theorems. Their proofs will be given in the next section.Theorem 1.
Assume that the solutionu(x,y,t) of (1) is sufficiently smooth, and let ui,jn be the solution of the L1-ADI scheme (14). Then
(21)(τ∑l=1n‍∥Ul-ul∥∞)1/2≤C1*(τmin{2α,2-α}+h12+h22),1≤n≤N,
where C1*=(l02/2)TC1 and l0=max{l1,l2}.Theorem 2.
Assume that the solutionu(x,y,t) of (1) is sufficiently smooth, and let ui,jn be the solution of the BD-ADI scheme (19). Then
(22)(τ∑l=1n‍∥Ul-ul∥∞)1/2≤C2*(τmin{1+α,2-α}+h12+h22),1≤n≤N,
where C2*=(l02/2)TC2 and l0=max{l1,l2}.Theorems1 and 2 show that the ADI difference solution ui,jn from (14) or (19) converges to the analytical solution Ui,jn of (1) in the discrete maximum norm. We also see from the estimates (12), (17), (21), and (22) that the L1-ADI scheme (14) and the BD-ADI scheme (19) have the same order as their truncation error with respect to the discrete maximum norm.


## 2.1. Construction ofL1-ADI and BD-ADI Schemes

In order to construct an ADI scheme, we add the termμ2Dtαδx2δy2Ui,jn to (9). This yields
(10)Dtα(Ui,jn+μ2δx2δy2Ui,jn)=ΔhUi,jn+fi,jn+(R1)i,jn,(xi,yj)∈Ωh,1≤n≤N,
where
(11)(R1)i,jn=μ2Dtαδx2δy2Ui,jn-(Rt)i,jn-(Rx)i,jn-(Ry)i,jn.It was shown in [59] that |μ2Dtαδx2δy2Ui,jn|≤CL(α)τ2α for a positive constant CL(α) independent of τ, h1, h2 and the time level n. Thus, there exists a positive constant C1 independent of τ, h1, h2 and the time level n such that
(12)|(R1)i,jn|≤C1(τmin{2α,2-α}+h12+h22).Denote byui,jn the finite difference approximation to Ui,jn, and let bn,k=an-k-1-an-k. After multiplying (10) by μ and then dropping the term μ(R1)i,jn, we derive a finite difference scheme as follows:
(13)(I-μδx2)(I-μδy2)ui,jn=∑k=1n-1‍bn,k(ui,jk+μ2δx2δy2ui,jk)+an-1(ui,j0+μ2δx2δy2ui,j0)+μfi,jn,(xi,yj)∈Ωh,1≤n≤N,ui,jn=φi,jn,(xi,yj)∈∂Ωh,1≤n≤N,ui,j0=ψi,j,(xi,yj)∈Ω¯h.By introducing the intermediate variable vi,jn, we obtain the following L1-ADI scheme (see [59]):
(14)(I-μδx2)vi,jn=∑k=1n-1‍bn,k(ui,jk+μ2δx2δy2ui,jk)+an-1(ui,j0+μ2δx2δy2ui,j0)+μfi,jn,(xi,yj)∈Ωh,1≤n≤N,v0,jn=(I-μδy2)φ0,jn,vM1,jn=(I-μδy2)φM1,jn,0≤j≤M2,1≤n≤N,ui,j0=ψi,j,(xi,yj)∈Ω¯h,(I-μδy2)ui,jn=vi,jn,(xi,yj)∈Ωh,1≤n≤N,ui,0n=φi,0n,ui,M2n=φi,M2n,0≤i≤M1,1≤n≤N.Adding the termτμδx2δy2δtUi,jn to (9), we have
(15)DtαUi,jn+τμδx2δy2δtUi,jn=ΔhUi,jn+fi,jn+(R2)i,jn,
where
(16)(R2)i,jn=τμδx2δy2δtUi,jn-(Rt)i,jn-(Rx)i,jn-(Ry)i,jn.Since|τμδx2δy2δtUi,jn|≤CBD(α)τ1+α for a positive constant CBD(α) independent of τ, h1, h2 and the time level n (see [59]), there exists a positive constant C2 independent of τ, h1, h2 and the time level n such that
(17)|(R2)i,jn|≤C2(τmin{1+α,2-α}+h12+h22).By (15), we obtain the following finite difference scheme:
(18)(I-μδx2)(I-μδy2)ui,jn=μ2δx2δy2ui,jn-1+∑k=1n-1‍bn,kui,jk(I-μδx2)(I-μδy2)ui,jn=+an-1ui,j0+μfi,jn,(xi,yj)∈Ωh,1≤n≤N,ui,jn=φi,jn,(xi,yj)∈∂Ωh,1≤n≤N,ui,j0=ψi,j,(xi,yj)∈Ω¯h.It is equivalent to the following BD-ADI scheme (see [59]):
(19)(I-μδx2)vi,jn=μ2δx2δy2ui,jn-1+∑k=1n-1‍bn,kui,jk+an-1ui,j0+μfi,jn,(xi,yj)∈Ωh,1≤n≤N,v0,jn=(I-μδy2)φ0,jn,vM1,jn=(I-μδy2)φM1,jn,0≤j≤M2,1≤n≤N,ui,j0=ψi,j,(xi,yj)∈Ω¯h,(I-μδy2)ui,jn=vi,jn,(xi,yj)∈Ωh,1≤n≤N,ui,0n=φi,0n,ui,M2n=φi,M2n,0≤i≤M1,1≤n≤N.


## 2.2. Maximum Norm Error Estimates

For any grid functionv={vi,j∣0≤i≤M1,0≤j≤M2}, we define its maximum norm ∥v∥∞ by
(20)∥v∥∞=max0≤i≤M1,0≤j≤M2|vi,j|.LetUi,jn be the value of the solution of (1) at the mesh point (xi,yj,tn), and let ui,jn be the solution of the L1-ADI scheme (14) or the BD-ADI scheme (19). We now present our main results of the maximum norm estimate for the error ei,jn=Ui,jn-ui,jn in the following two theorems. Their proofs will be given in the next section.Theorem 1.
Assume that the solutionu(x,y,t) of (1) is sufficiently smooth, and let ui,jn be the solution of the L1-ADI scheme (14). Then
(21)(τ∑l=1n‍∥Ul-ul∥∞)1/2≤C1*(τmin{2α,2-α}+h12+h22),1≤n≤N,
where C1*=(l02/2)TC1 and l0=max{l1,l2}.Theorem 2.
Assume that the solutionu(x,y,t) of (1) is sufficiently smooth, and let ui,jn be the solution of the BD-ADI scheme (19). Then
(22)(τ∑l=1n‍∥Ul-ul∥∞)1/2≤C2*(τmin{1+α,2-α}+h12+h22),1≤n≤N,
where C2*=(l02/2)TC2 and l0=max{l1,l2}.Theorems1 and 2 show that the ADI difference solution ui,jn from (14) or (19) converges to the analytical solution Ui,jn of (1) in the discrete maximum norm. We also see from the estimates (12), (17), (21), and (22) that the L1-ADI scheme (14) and the BD-ADI scheme (19) have the same order as their truncation error with respect to the discrete maximum norm.


## 3. Proof of the Main Results

Let𝒱h be the set of all grid functions v={vi,j∣0≤i≤M1,0≤j≤M2} defined in Ω¯h and vanishing on ∂Ωh. For arbitrary v,w∈𝒱h, we define the following inner products:
(23)(v,w)=h1h2∑i=1M1-1‍∑j=1M2-1‍vi,jwi,j,(δxv,δxw)x=h1h2∑i=0M1-1‍∑j=1M2-1‍(δxvi,j)(δxwi,j),(δyv,δyw)y=h1h2∑i=1M1-1‍∑j=0M2-1‍(δyvi,j)(δywi,j).For anyv∈𝒱h, we introduce the following norms:
(24)∥v∥=(v,v)1/2,∥δνv∥ν=(δνv,δνv)ν1/2(ν=x,y),|v|1=(∥δxv∥x2+∥δyv∥y2)1/2,∥δxδy2v∥x=(δxδy2v,δxδy2v)x1/2,∥δyδx2v∥y=(δyδx2v,δyδx2v)y1/2.Using a simple calculation, we have that for arbitraryv,w∈𝒱h,
(25)(δν2v,w)=-(δνv,δνw)ν,ν=x,y.Before proving Theorems1 and 2, we first introduce the following embedding theorem from [60, page 281].Lemma 3.
For anyv∈𝒱h, one has
(26)∥v∥∞≤l02(2l1l2)-1∥Δhv∥∞,l0=max{l1,l2}.Proof of Theorem1.
Letei,jn=Ui,jn-ui,jn. Then by (10) and (13),
(27)ei,jn-μΔhei,jn+μ2δx2δy2ei,jn=wi,jn-1+μ2δx2δy2wi,jn-1+μ(R1)i,jn,(xi,yj)∈Ωh,1≤n≤N,ei,jn=0,(xi,yj)∈∂Ωh,1≤n≤N,ei,j0=0,(xi,yj)∈Ω¯h,
where
(28)wi,jn-1=∑k=1n-1‍bn,kei,jk+an-1ei,j0.
This implies thaten={ei,jn∣0≤i≤M1,0≤j≤M2}∈𝒱h for each n. Taking the inner product of the first equation in (27) and -Δhen, we get
(29)(en+μ2δx2δy2en,-Δhen)+(-μΔhen,-Δhen)=(wn-1+μ2δx2δy2wn-1,-Δhen)+(μ(R1)n,-Δhen),1≤n≤N.
It follows from (25) that
(30)(en+μ2δx2δy2en,-Δhen)=|en|12+μ2(∥δyδx2en∥y2+∥δxδy2en∥x2).
Similarly,(31)(wn-1+μ2δx2δy2wn-1,-Δhen)=∑k=1n-1‍bn,k((δxek,δxen)x+(δyek,δyen)y+μ2(δyδx2ek,δyδx2en)y+μ2(δxδy2ek,δxδy2en)x).
Since(32)bn,k=an-k-1-an-k>0,∑k=1n-1‍bn,k≤1,
we have from Cauchy-Schwarz inequality that
(33)(wn-1+μ2δx2δy2wn-1,-Δhen)≤12∑k=1n-1‍bn,k(|ek|12+μ2(∥δyδx2ek∥y2+∥δxδy2ek∥x2))+12(|en|12+μ2(∥δyδx2en∥y2+∥δxδy2en∥x2)),(μ(R1)n,-Δhen)≤μ2(∥(R1)n∥2+∥Δhen∥2).
Substituting (30) and (33) into (29) gives the following:
(34)μ∥Δhen∥2+En≤En-1+μ∥(R1)n∥2,1≤n≤N,
where
(35)En=∑k=1n‍an-k(|ek|12+μ2(∥δyδx2ek∥y2+∥δxδy2ek∥x2)),E0=0.
This implies that(36)μ∑l=1n‍∥Δhel∥2+En≤μ∑l=1n‍∥(R1)l∥2,1≤n≤N.
SinceEn≥0 and by (12), ∥(R1)l∥2≤l1l2C12(τmin{2α,2-α}+h12+h22)2, the estimate (21) follows from (36) and (26) immediately.Proof of Theorem2.
The proof follows from the similar argument as that in the proof of Theorem1 and we give a sketch. Let ei,jn=Ui,jn-ui,jn. By (15) and (18),
(37)ei,jn-μΔhei,jn+μ2δx2δy2ei,jn=μ2δx2δy2ei,jn-1+wi,jn-1+μ(R2)i,jn,(xi,yj)∈Ωh,1≤n≤N,ei,jn=0,(xi,yj)∈∂Ωh,1≤n≤N,ei,j0=0,(xi,yj)∈Ω¯h,
where wi,jn-1 is defined by (28). Taking the inner product of the first equation in (37) and -Δhen, we get
(38)(en,-Δhen)+(-μΔhen,-Δhen)+(μ2δx2δy2(en-en-1),-Δhen)=(wn-1,-Δhen)+(μ(R2)n,-Δhen),1≤n≤N.
Since(39)(en,-Δhen)=|en|12,(-μΔhen,-Δhen)=μ∥Δhen∥2,(μ2δx2δy2(en-en-1),-Δhen)≥μ22(∥δyδx2en∥y2+∥δxδy2en∥x2-∥δyδx2en-1∥y2-∥δxδy2en-1∥x2),(wn-1,-Δhen)≤12(∑k=1n-1‍bn,k|ek|12+|en|12),(μ(R2)n,-Δhen)≤μ2(∥(R2)n∥2+∥Δhen∥2),
we obtain that
(40)μ∥Δhen∥2+Fn≤Fn-1+μ∥(R2)n∥2,1≤n≤N,
where
(41)Fn=∑k=1n‍an-k|ek|12+μ2(∥δyδx2en∥y2+∥δxδy2en∥x2),F0=0.
Therefore(42)μ∑l=1n‍∥Δhel∥2+Fn≤μ∑l=1n‍∥(R2)l∥2,1≤n≤N.
The estimate (22) follows from the previous inequality, (17), and (26) immediately.


## 4. Numerical Results

In this section, we give some numerical results to demonstrate the accuracy of theL1-ADI scheme (14) and the BD-ADI scheme (19) with respect to the discrete maximum norm. Some detailed numerical comparisons of these two ADI schemes with the implicit scheme proposed in [43] can be found in [59]. The dependence of the solution of (1) on the anomalous diffusion exponent α was also exhibited in [59] through some numerical results.In our numerical computations, we take an equal mesh size in each of the space directions; that is,h1=h2=h. We compute the discrete maximum norm error of the numerical solution ui,jn by
(43)error(τ,h)=(τ∑l=1N‍∥Ul-ul∥∞)1/2,
and its convergence orders by
(44)order(τ)=log2(error(2τ,h)error(τ,h)),order(h)=log2(error(τ,2h)error(τ,h)),
where Ui,jn represents the value of the exact analytic solution of (1) at (xi,yj,tn).Example 4.
We consider the problem (1) in the domain Ω=(0,π)×(0,π), and let T=1/2. Assume that the solution u(x,y,t) of this problem is u(x,y,t)=t2sin(x)sin(y). It can be checked that the corresponding known functions are given by
(45)f(x,y,t)=2sin(x)sin(y)(t2-αΓ(3-α)+t2),φ(x,y,t)=t2sin(x)sin(y),ψ(x,y)=0.In Table1, we present the maximum norm  error error(τ,h) and the temporal convergence   order   order(τ) of the numerical solution ui,jn by the L1-ADI scheme (14) and the BD-ADI scheme (19). We see that the L1-ADI scheme has the temporal accuracy of min{2α,2-α} in the discrete maximum norm, and the best temporal accuracy is attained at α=2/3. It is also seen that the BD-ADI scheme generates the temporal accuracy of min{1+α,2-α} in the discrete maximum norm, and it gets the best temporal accuracy when α=1/2. These observations coincide well with the theoretical analysis.Table 1
The maximum norm error and the temporal accuracy ofui,jn for Example 4.
τ
L
1-ADI scheme (h=π/200)
BD-ADI scheme (h=π/200)
α
error
(
τ
,
h
)
order
(
τ
)
α
error
(
τ
,
h
)
order
(
τ
)
1/20
1/2
1.1393833280
e
-
03
0.860979
1/3
1.5093973350
e
-
03
1.266265
1/40
6.2732035469
e
-
04
0.882911
6.2750871863
e
-
04
1.285881
1/80
3.4017847773
e
-
04
0.910189
2.5735409495
e
-
04
1.301046
1/160
1.8101416127
e
-
04
0.936057
1.0444249965
e
-
04
1.319085
1/320
9.4608749632
e
-
05
0.959731
4.1859417673
e
-
05
1.351071
1/640
4.8643355059
e
-
05
1.6408936533
e
-
05
1/20
2/3
1.2585768427
e
-
03
1.260363
1/2
3.6622305737
e
-
04
1.463622
1/40
5.2537885652
e
-
04
1.288661
1.3278581143
e
-
04
1.474131
1/80
2.1505367805
e
-
04
1.303399
4.7796268443
e
-
05
1.488745
1/160
8.7133397922
e
-
05
1.307319
1.7030874694
e
-
05
1.514438
1/320
3.5208078966
e
-
05
1.297645
5.9613630191
e
-
06
1.535827
1/640
1.4322284746
e
-
05
2.0559641134
e
-
06
1/20
3/4
2.3004703569
e
-
03
1.174836
2/3
1.4653603051
e
-
03
1.207510
1/40
1.0189566568
e
-
03
1.197125
6.3452327488
e
-
04
1.242740
1/80
4.4441136189
e
-
04
1.210140
2.6813016026
e
-
04
1.264770
1/160
1.9208646006
e
-
04
1.216775
1.1158658090
e
-
04
1.276115
1/320
8.2643912130
e
-
05
1.217031
4.6074754492
e
-
05
1.274583
1/640
3.5550691871
e
-
05
1.9044750580
e
-
05Table2 gives the maximum norm error   error(τ,h) and the spatial convergence order   order(h) of the numerical solution ui,jn by the L1-ADI scheme (14) and the BD-ADI scheme (19). As expected from the theoretical analysis, these two schemes have the second-order spatial accuracy.Table 2
The maximum norm error and the spatial accuracy ofui,jn for Example 4.
h
L
1-ADI scheme (α=2/3,τ=1/4000)
BD-ADI scheme (α=1/2,τ=1/4000)
error
(
τ
,
h
)
order
(
h
)
error
(
τ
,
h
)
order
(
h
)
π
/
4
1.4977394377
e
-
03
1.994343
1.8149573747
e
-
03
2.001730
π
/
8
3.7590600110
e
-
04
1.989441
4.5319556650
e
-
04
2.000950
π
/
16
9.4666844874
e
-
05
1.960585
1.1322433260
e
-
04
2.002088
π
/
32
2.4322216030
e
-
05
1.844870
2.8265146732
e
-
05
2.007753
π
/
64
6.7708311670
e
-
06
7.0284169450
e
-
06Example 5.
We consider the subdiffusion equation(46)ut(x,y,t)=𝒟t1-α0(Δu(x,y,t))+ex+y((1+α)tα-2Γ(2+α)Γ(1+2α)t2α),(x,y)∈Ω,t∈[0,1],
with boundary and initial conditions
(47)u(x,y,t)=ex+yt1+α,(x,y)∈∂Ω,t∈(0,1],u(x,y,0)=0,(x,y)∈Ω¯,
where Ω=(0,1)×(0,1) and 0𝒟t1-α denotes the Riemann-Liouville fractional derivative operator defined as
(48)𝒟t1-α0y(t)=1Γ(α)ddt∫0t‍y(s)(t-s)1-αds,0<α<1.Operating Riemann-Liouville fractional derivative operator0𝒟tα-1 on both sides of (46), we obtain the equivalent problem of the subdiffusion equation with the temporal Caputo fractional derivative [40, 42, 59] as follows:
(49)𝒟tα0Cu(x,y,t)=Δu(x,y,t)+ex+y(Γ(2+α)t-2t1+α),(x,y)∈Ω,t∈[0,1],u(x,y,t)=ex+yt1+α,(x,y)∈∂Ω,t∈(0,1],u(x,y,0)=0,(x,y)∈Ω¯.We now solve the aforementioned problem by theL1-ADI scheme (14) and the BD-ADI scheme (19). Tables 3 and 4 give the maximum norm error error(τ,h) and the convergence orders order(τ) and order(h) of the numerical solution ui,jn. It is easily seen that the numerical results confirm the theoretical analysis results.Table 3
The maximum norm error and the temporal accuracy ofui,jn for Example 5.
τ
L
1-ADI scheme (h=1/200)
BD-ADI scheme (h=1/200)
α
error
(
τ
,
h
)
order
(
τ
)
α
error
(
τ
,
h
)
order
(
τ
)
1/20
1/2
4.6386801902
e
-
03
0.835782
1/3
3.0682418814
e
-
03
1.247710
1/40
2.5989567998
e
-
03
0.906370
1.2920863692
e
-
03
1.295329
1/80
1.3866102907
e
-
03
0.947443
5.2645185538
e
-
04
1.315075
1/160
7.1902789341
e
-
04
0.970534
2.1158337290
e
-
04
1.325682
1/320
3.6693233527
e
-
04
0.983962
8.4413446753
e
-
05
1.335982
1/640
1.8551714096
e
-
04
3.3438043901
e
-
05
1/20
2/3
1.5621553331
e
-
03
1.192172
1/2
1.5491678500
e
-
03
1.423830
1/40
6.8366714097
e
-
04
1.262779
5.7740819917
e
-
04
1.463914
1/80
2.8491177407
e
-
04
1.297640
2.0931522211
e
-
04
1.481554
1/160
1.1589959988
e
-
04
1.317329
7.4956359178
e
-
05
1.496663
1/320
4.6507867889
e
-
05
1.333601
2.6562452368
e
-
05
1.524750
1/640
1.8453234155
e
-
05
9.2315064167
e
-
06
1/20
3/4
1.7962917770
e
-
03
1.063599
2/3
1.1129679333
e
-
03
1.112230
1/40
8.5941268352
e
-
04
1.128142
5.1483494465
e
-
04
1.171085
1/80
3.9318529135
e
-
04
1.161548
2.2863166690
e
-
04
1.211600
1/160
1.7576660520
e
-
04
1.180856
9.8720772742
e
-
05
1.239224
1/320
7.7528818903
e
-
05
1.191836
4.1818156321
e
-
05
1.255314
1/640
3.3937881480
e
-
05
1.7517727276
e
-
05Table 4
The maximum norm error and the spatial accuracy ofui,jn for Example 5.
h
L
1-ADI scheme (α=2/3,τ=1/4000)
BD-ADI scheme (α=1/2,τ=1/4000)
error
(
τ
,
h
)
order
(
h
)
error
(
τ
,
h
)
order
(
h
)
1
/
4
8.9083790169
e
-
04
1.901804
9.4164046481
e
-
04
1.901896
1
/
8
2.3839587645
e
-
04
1.985610
2.5197489313
e
-
04
1.974325
1
/
16
6.0196414976
e
-
05
2.072000
6.4124816806
e
-
05
2.032967
1
/
32
1.4316490746
e
-
05
2.263270
1.5669032476
e
-
05
2.158045
1
/
64
2.9821150680
e
-
06
3.5107982244
e
-
06


## 5. Conclusions

We have studied two ADI finite difference methods for a two-dimensional fractional subdiffusion equation. An explicit error estimate for each of the two methods has been provided in the discrete maximum norm. It has been shown that the methods have the same order as their truncation errors with respect to the discrete maximum norm. The maximum norm error estimates presented here are more preferable for measuring computation errors in practice, compared to theH1-norm error estimates in [59]. Numerical results have confirmed the theoretical analysis results.



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*Source: 293706-2013-07-31.xml*