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Research article

Pointwise-in-time α-robust error estimate of the ADI difference scheme for three-dimensional fractional subdiffusion equations with variable coefficients

  • Received: 30 October 2023 Revised: 12 December 2023 Accepted: 02 January 2024 Published: 10 January 2024
  • 65M15, 65M06

  • In this paper, a fully-discrete alternating direction implicit (ADI) difference method is proposed for solving three-dimensional (3D) fractional subdiffusion equations with variable coefficients, whose solution presents a weak singularity at t=0. The proposed method is established via the L1 scheme on graded mesh for the Caputo fractional derivative and central difference method for spatial derivative, and an ADI method is structured to change the 3D problem into three 1D problems. Using the modified Grönwall inequality we prove the stability and α-robust convergence. The results presented in numerical experiments are in accordance with the theoretical analysis.

    Citation: Wang Xiao, Xuehua Yang, Ziyi Zhou. Pointwise-in-time α-robust error estimate of the ADI difference scheme for three-dimensional fractional subdiffusion equations with variable coefficients[J]. Communications in Analysis and Mechanics, 2024, 16(1): 53-70. doi: 10.3934/cam.2024003

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  • In this paper, a fully-discrete alternating direction implicit (ADI) difference method is proposed for solving three-dimensional (3D) fractional subdiffusion equations with variable coefficients, whose solution presents a weak singularity at t=0. The proposed method is established via the L1 scheme on graded mesh for the Caputo fractional derivative and central difference method for spatial derivative, and an ADI method is structured to change the 3D problem into three 1D problems. Using the modified Grönwall inequality we prove the stability and α-robust convergence. The results presented in numerical experiments are in accordance with the theoretical analysis.



    This paper focuses on the 3D subdiffusion equations with variable coefficients

    Dαtua1uxxa2uyya3uzz+bu=f(x,y,z,t),(x,y,z)Λ,t(0,T], (1.1)
    u(x,y,z,0)=g(x,y,z),(x,y,z)Λ, (1.2)
    u(x,y,z,t)=φ(x,y,z,t),(x,y,z)Λ,t(0,T] (1.3)

    in which: a1, a2, a3, and b are positive constants; f, φ, and g are smooth functions; and Λ=(0,Υx)×(0,Υy)×(0,Υz) is a rectangular field bounded by Λ. The Caputo fractional derivative can be defined via

    Dαtu(,t)=1Γ(1α)t0(tξ)αu(,ξ)ξdξ,α(0,1).

    The fractional subdiffusion equation with variable coefficients is derived from the classical partial differential equation. Many scholars have studied the existence and uniqueness of the solution of this equation [1,2]. The high-dimensional subdiffusion problem can accurately describe the mechanical and physical processes with non-locality and spatial global correlation, it has become an important tool to describe a variety of complex mechanical and physical behaviors [3,4,5], and many numerical schemes have been proposed in [6,7,8,9,10,11,12,13,14,15]. Among them, the ADI method is favored by many scholars. Tian and Ge [16] offered the fourth-order ADI difference method for 2D unsteady convection diffusion problems. Wang et al. [17,18] considered the 2D fractional integro-differential equation by using an ADI compact difference method. Zhang et al. [19] provided the 2D semilinear multidelay parabolic equations by using Crank-Nicolson ADI compact difference schemes. Zhang et al. [20] considered the 2D space-fractional nonlinear Ginzburg-Landau equation via linearized ADI difference schemes. Wang et al. [21,22] presented an analysis of the convergence of the α-robust H1-norm for the ADI differece scheme for the 2D time fractional diffusion equation (TFDE). Chen et al. [23,24] considered the ADI difference scheme for a class of two-dimensional multi-term TFDEs and gave the point error estimation. Qiao et al. [25,26,27] presented the ADI difference scheme for 2D TFDE with a weakly singular kernel. Zhang et al. [28] presented an efficient ADI difference scheme for the nonlocal evolution problem in 3D space. Zhou et al. [29,30] solved the 3D TFDE via the ADI method.

    Along with the previous research, many scholars considered the subdiffusion equation with variable coefficients. Ngondiep [31] discussed a class of convection-diffusion-reaction equations with variable coefficients and gave a two-level fourth-order scheme. Wei et al. [32] gave the anisotropic finite element method (FEM) for 2D time fractional variable coefficient diffusion equations on graded meshes. Zeng and Tan [33] proposed a two-grid FEM for nonlinear TFDEs with variable coefficients. Ma et al. [34] presented the L1 robust analysis of the fourth-order block-centered FDM for 2D TFDEs with variable coefficient. As we all know, the study of the variable coefficient subdiffusion equation is almost based on one or two dimensions. It is not easy to solve three-dimensional fractional subdiffusion equations with variable coefficients due to the existence of the weak singularity near the initial time. In this paper, we put forward an α-robust L1 ADI difference scheme on graded mesh to overcome these difficulties. The L1 scheme on graded meshes is used to discretize the Caputo fractional derivative, and the difference method is used to approximate the spatial derivative. Using the modified Grönwall inequality, we prove the stability and α-robust convergence of the proposed method.

    In this article, the main achievements are as follows:

    ● We construct an ADI difference method for the 3D subdiffusion equation with variable coefficients. Most previous work has focused on 1D or 2D problems.

    ● We introduce a new norm to prove the stability and convergence of the proposed method.

    ● The α-robust convergence is proved strictly, which is able to avoid the blow-up phenomenon in the case of α1.

    The structure of this paper is as follows. In Section 2, the ADI scheme is developed for the 3D variable coefficient subdiffusion problems. In Section 3, we demonstrate the stability and convergence of the method. In Section 4, the three-dimensional numerical example is given to verify the effectiveness and accuracy of the proposed method. We conclude the paper in Section 5.

    Set hx=Υx1, hy=Υy2, hz=Υz3 for the positive integers 1, 2, and 3. Let N1 be a positive integer number and set tn=T(n/N)γ for n=0,1,,N, γ1. Denote time step ιn=tntn1 for 1nN.

    Set ¯Λh={(xi,yr,zk)|0i1,0r2,0k3}, and Λh=¯ΛhΛ, Λh=ΛhΛ. Let unirk be the approximation solution of (1.1)-(1.3).

    For n=1,2,,N, by the L1 scheme [35,36], denote

    DαNunirk:=1Γ(1α)n1q=0uq+1irkuqirkιq+1tq+1tq(tnξ)αdξ=pn,1unirkpn,nu0irkn1q=1(pn,qpn,q+1)unqirk (2.1)

    where

    pn,1=ιαnΓ(2α),pn,q=(tntnq)1α(tntnq+1)1αιnq+1Γ(2α),1qn.

    Thus, the problem (1.1)-(1.3) can be approximated by

    DαNunirka1δ2xunirka2δ2yunirka3δ2zunirk+bunirk=fnirk,(xi,yr,zk)Λh,1nN, (2.2)
    u0irk=g(xi,yr,zk),(xi,yr,zk)¯Λh, (2.3)
    unirk=φ(xi,yr,zk,tn),(xi,yr,zk)Λh,1nN (2.4)

    where

    δ2xunirk=uni+1,r,k2unirk+uni1,r,kh2x,
    δ2yunirk=uni,r+1,k2unirk+uni,r1,kh2y,
    δ2zunirk=uni,r,k+12unirk+uni,r,k1h2z.

    Set Sn=d1n,1=ιαnΓ(2α), and to establish the ADI scheme, add a small term

    (a1a2S2nδ2xδ2y1+bSn+a1a3S2nδ2xδ2z1+bSn+a2a3S2nδ2yδ2z1+bSna1a2a3S3nδ2xδ2yδ2z(1+bSn)2)DαNunijk (2.5)

    onto the left side of Equation (2.2), which is order O(N2α). Then, one has

    (I+a1a2S2nδ2xδ2y1+bSn+a1a3S2nδ2xδ2z1+bSn+a2a3S2nδ2yδ2z1+bSna1a2a3S3nδ2xδ2y(1+bSn)2)DαNunirka1δ2xunirka2δ2yunirka3δ2zunirk+bunirk=fnirk,(xi,yr,zk)Λh,1nN, (2.6)
    unirk=φ(xi,yr,zk,tn),(xi,yr,zk)Λh,1nN, (2.7)
    u0irk=g(xi,yr,zk),(xi,yr,zk)¯Λh. (2.8)

    Multiplying both sides of (2.6) by Sn, we get

    (I+a1a2S2nδ2xδ2y1+bSn+a1a3S2nδ2xδ2z1+bSn+a2a3S2nδ2yδ2z1+bSna1a2a3S3nδ2xδ2y(1+bSn)2)unirka1Snδ2xunirka2Snδ2yunirka3Snδ2zunirk+bSnunirk=Snfnirk+Sn(I+a1a2S2nδ2xδ2y1+bSn+a1a3S2nδ2xδ2z1+bSn+a2a3S2nδ2yδ2z1+bSna1a2a3S3nδ2xδ2yδ2z(1+bSn)2)(pn,nu0irk+n1q=1(pn,qpn,q+1)unqirk).

    Then, rewriting Eqs. (2.6)-(2.8) in an ADI form:

    (Ia1Sn1+bSnδ2x)(Ia2Sn1+bSnδ2y)(Ia3Sn1+bSnδ2z)unirk=Fnirk,(xi,yr,zk)Λh,1nN, (2.9)
    u0irk=g(xi,yr,zk),(xi,yr,zk)¯Λh, (2.10)
    unirk=φ(xi,yr,zk,tn),(xi,yr,zk)Λh,1nN (2.11)

    in which

    Fnirk=Sn1+bSn(I+a1a2S2nδ2xδ2y1+bSn+a1a3S2nδ2xδ2z1+bSn+a2a3S2nδ2yδ2z1+bSna1a2a3S3nδ2xδ2yδ2z(1+bSn)2)(pn,nu0irk+n1q=1(pn,qpn,q+1)unqirk)+Sn1+bSnfnirk.

    Set un13irk=(Ia3Sn1+bSnδ2z)unirk and un23irk=(Ia2Sn1+bSnδ2y)unirk. To get the solution {unirk} of problem (2.6)-(2.8), we just need to solve three of the one-dimensional equations.

    First, for fixed r{1,2,,21} and k{1,2,,31}, we can solve {un23irk} in the x-direction as follows:

    {(Ia1Sn1+bSnδ2x)un23irk=Fnirkfor1i11,un230rk=(Ia2Sn1+bSnδ2y)(Ia3Sn1+bSnδ2z)un0rk,un231,r,k=(Ia2Sn1+bSnδ2y)(Ia3Sn1+bSnδ2z)un1,r,k. (2.12)

    Then, for fixed i{1,2,,11} and k{1,2,,31}, we can solve {un13irk} in the y-direction as follows:

    {(Ia2Sn1+bSnδ2y)un13irk=un23irkfor1r21,un13i0k=(Ia3Sn1+bSnδ2z)uni0k,un13i,2,k=(Ia3Sn1+bSnδ2z)uni,2,k. (2.13)

    At last, for i{1,2,,11} and r{1,2,,21}, we utilize {un23irk,un13irk} in the z-direction to solve

    {(Ia3Sn1+bSnδ2z)unirk=un13irkfor1k31,unir0=φ(xi,yr,z0,tn),uni,r,3=φ(xi,yr,z3,tn) (2.14)

    where the solution {unijk} is attained.

    For grid function u={unirk|0i1,0r2,0k3,0nN}, we define

    un2=hx11i=1hy21r=1hz31k=1(unirk)2,δxun2=hx1i=1hy21r=1hz31k=1(δxuni12,r,k)2,δxδyun2=hx1i=1hy2r=1hz31k=1(δxδyuni12,r12,k)2,δxδyδzun2=hx1i=1hy2r=1hz3k=1(δxδyδzuni12,r12,k12)2,δxδyδ2zun2=hx1i=1hy2r=1hz31k=1(δxδyδ2zuni12,r12,k)2,Δhun2=hx11i=1hy21r=1hz31k=1(Δhunirk)2

    where δxui12,r,k=1hx(uirkui1,r,k). The norms δyun, δzun, δyδzun, δxδzun, δxδzδ2yun and δyδzδ2xun can be defined similarly.

    Set ¯U={uirk|uirk=0 if (xi,yr,zk)Λh and (xi,yr,zk)¯Λh}, for u, w¯U, we define

    L1=a1Sn1+bSn,L2=a2Sn1+bSn,L3=a3Sn1+bSn,hun2=δxun2+δyun2+δzun2,un2H1=un2+hun2,δ2xunωn2B=(L1L2δ2xδyunδ2xδyωn+L1L3δ2xδzunδ2xδzωn+L2L3δxδyδzunδxδyδzωn+L1L1L2L3δ2xδyδzunδ2xδyδzωn),δxun2B=L1L2δ2xδyun2+L1L3δ2xδzun2+L2L3δxδyδzun2+L1L2L3δ2xδyδzun2,un2A=L1δxun2+L2δyun2+L3δzun2+L1(1+bSn)δxun2B+L2(1+bSn)δyun2B+L3(1+bSn)δzun2B.

    This section mainly discusses the stability analysis of ADI difference scheme (2.6)-(2.8).

    Lemma 3.1. For grid functions u,ω¯U, one has

    hxhyhz11i=121r=131k=1(unirk+(1+bSn)(L1L2δ2xδ2yunirk+L1L3δ2xδ2zunirk+L2L3δ2yδ2zunirkL1L2L3δ2xδ2yδ2zunirk))(L1δ2x+L2δ2y+L3δ2z)ωnirkunAωnA (3.1)

    where the inequality is equal when ω=u.

    Proof. Using the Cauchy-Schwartz inequality, one has

    hxhyhz11i=121r=131k=1(unirk+(1+bSn)(L1L2δ2xδ2yunirk+L1L3δ2xδ2zunirk+L2L3δ2yδ2zunirkL1L2L3δ2xδ2yδ2zunirk))(L1δ2x+L2δ2y+L3δ2z)ωnirkL1δxunδxωn+L1(1+bSn)δ2xunωn2B+L2δyunδyωn+L2(1+bSn)δ2yunωn2B+L3δzunδzωn+L3(1+bSn)δ2zunωn2B[L1δxun2+L2δyun2+L3δzun2+L1(1+bSn)δxun2B+L2(1+bSn)δyun2B+L3(1+bSn)δzun2B]12[L1δxωn2+L2δyωn2+L3δzωn2+L1(1+bSn)δxωn2B+L2(1+bSn)δyωn2B+L3(1+bSn)δzωn2B]12unAωnA.

    The proof is completed.

    For n=1,2,,N and j=1,2,,n1, define a positive sequence

    ϱn,n=1,ϱn,j=njq=11pn,qpn,q+1ϱnq,j>0.

    Lemma 3.2. [37,38] Assume that the sequences {mn}n=1, {gn}n=1 are nonnegative, and assume that the grid function {un:n=0,1,,N} satisfies u00 and

    (DαNun)unmnun+(gn)2,n=1,2,,N.

    Then

    unu0+Snnj=1ϱn,j(mj+gj)+max

    Lemma 3.3. [39] When , one has

    Next, the stability theorem is presented.

    Theorem 3.4. For , the solution of ADI scheme (2.6)-(2.8) satisfies

    Proof. Multiplying both sides of , and summing over for , we have

    (3.2)

    Then, using the Cauchy-Schwartz inequality, Young's inequality, and Lemma 3.1, one has

    and

    Thus, we get

    which obtains

    (3.3)

    Finally, Lemma 3.2 and Lemma 3.3 are applied in (3.3), and the proof is completed.

    Lemma 3.5. [24] Assume that , . Then, for , we have

    (3.4)

    Lemma 3.6. [40] For grid function , one has

    Define

    (3.5)
    (3.6)

    and

    (3.7)

    According to the definition of the small term (2.5), one has

    (3.8)

    Using a Taylor expansion, one has

    Applying Lemma 3.5 and Eq. (3.8), one has

    (3.9)

    We now show the -norm convergence analysis.

    Theorem 3.7. Suppose that , , . one has

    (3.10)

    where is an -robust constant that does not blow up as .

    Proof. Subtracting Eqs. (1.1)-(1.3) from Eqs. (2.6)-(2.8), the error equations are presented via

    (3.11)
    (3.12)
    (3.13)

    Multiplying both sides of (3.11) by and summing over for , one has

    Since

    and

    Then, we get

    That is,

    (3.14)

    Thus

    (3.15)

    By the Poincaré inequality, Eq. (3.9), and Lemma 3.5, we have

    (3.16)

    From (3.6), (3.14), and (3.16), using Lemma 3.2 and Lemma 3.3, and noting that , one has

    (3.17)

    Finally, using Lemma 3.6 and the definition of , the proof is completed.

    Example 4.1. For problem (1.1)-(1.3), considering , , , , , and

    The exact solution is . Set , , , and . In Tables 13, we fix to test the error and convergence in time. In Table 1, choosing different , we give errors, convergence orders, and CPU time in time for . In Table 2, set , , , and , and choosing different we give errors, convergence orders and CPU time in time for . In Table 2, the convergence order goes down for and , which can be caused by the round-off error computing the coefficients of the L1 scheme (2.1), see Remark 2.1 in [41]. In Table 3, set , , , and , and choosing different we give errors, convergence orders, and CPU time in time for . Due to the influence of multiple variable coefficient values such as , , , , the degree of freedom of the equation fitting is higher, and there may be some differences for CPU times. From Tables 13, it is obtained that the order of convergence in time is , which is consistent with the theoretical analysis.

    Table 1.  The errors, convergence orders, and CPU time for with , , , and in time.
    N
    Error Order CPU(s) Error Order CPU(s)
    4 3.0652e-1 - 48.83 2.2682e-1 - 51.04
    8 2.5808e-1 0.2481 143.74 1.4655e-1 0.6301 145.86
    16 2.1683e-1 0.2513 458.23 9.3775e-2 0.6442 448.40
    32 1.7966e-1 0.2713 1613.40 5.8933e-2 0.6701 1563.70
    N
    Error Order CPU(s) Error Order CPU(s)
    4 1.5083e-1 - 51.43 9.5021e-2 - 51.82
    8 7.2715e-2 1.0526 146.19 3.8592e-2 1.2999 146.22
    16 3.4023e-2 1.1057 462.02 1.5557e-2 1.3108 452.52
    32 1.5515e-2 1.1329 1559.60 8.3539e-3 1.2918 1586.70

     | Show Table
    DownLoad: CSV
    Table 2.  The errors, convergence orders, and CPU time in time for with , , , and .
    N
    Error Order CPU(s) Error Order CPU(s)
    4 1.209e-3 - 51.55 1.164e-3 - 51.88
    8 5.127e-4 1.2371 152.91 4.590e-4 1.3427 150.91
    16 2.715e-4 0.9173 519.34 2.067e-4 1.1510 478.29
    32 1.809e-4 0.5859 1727.10 1.085e-4 0.9297 1657.20
    N
    Error Order CPU(s) Error Order CPU(s)
    4 9.694e-4 - 51.79 6.210e-4 - 52.53
    8 3.808e-4 1.3480 146.84 2.589e-4 1.2623 147.80
    16 1.597e-4 1.2538 476.11 1.114e-4 1.2269 474.38
    32 7.104e-5 1.1686 1654.50 4.799e-5 1.2144 1628.10

     | Show Table
    DownLoad: CSV
    Table 3.  The errors, convergence orders, and CPU time for with , , , and in time.
    N
    Error Order CPU(s) Error Order CPU(s)
    4 7.0416e-1 - 48.81 4.7668e-1 - 54.38
    8 6.0163e-1 0.2270 149.75 2.9985e-1 0.6688 147.75
    16 4.9789e-1 0.2731 466.34 1.8209e-1 0.7196 471.73
    32 4.0077e-1 0.3131 1692.90 1.0794e-1 0.7545 1693.90
    N
    Error Order CPU(s) Error Order CPU(s)
    4 2.8199e-1 - 58.09 1.6609e-1 - 53.54
    8 1.2866e-1 1.1320 148.79 6.5972e-2 1.3321 148.34
    16 5.6762e-2 1.1806 470.31 2.6820e-2 1.296 468.54
    32 2.4623e-2 1.2049 1678.6 1.1297e-2 1.2479 1646.70

     | Show Table
    DownLoad: CSV

    In Tables 46, we fix to present errors and convergence in space for . In Table 4, set , , , and , and by choosing different we give -norm errors, convergence orders and CPU time in space. In Table 5, choosing different we give -norm errors, convergence orders and CPU time in space for , , , and . In Table 6, choosing different we give -norm errors, convergence orders, and CPU time in space for the case where , , , and . In Tables 4-6, we can find that the spatial convergence order is order 2, which is consistent with our theory. From Tables 1-6 we can notice that our scheme is computationally efficient and consistent with the analysis.

    Table 4.  -norm errors, convergence orders, and CPU time for with , , , and in space.
    Error Order CPU(s) Error Order CPU(s)
    4 7.2337e-2 - 31.74 6.9950e-2 - 46.11
    8 1.9667e-3 1.8790 174.84 1.9154e-2 1.8604 290.63
    16 4.7608e-3 2.0465 1080.70 4.8763e-3 1.9738 1023.10
    32 7.8250e-4 2.6051 4364.90 1.0680e-3 2.1909 4603.40

     | Show Table
    DownLoad: CSV
    Table 5.  -norm errors, convergence orders, and CPU time in space for with , , , and .
    Error Order CPU(s) Error Order CPU(s)
    4 2.509e-3 - 31.23 2.511e-3 - 30.28
    8 7.034e-4 1.8347 170.04 7.064e-4 1.8296 168.96
    16 1.822e-4 1.9483 820.18 1.857e-4 1.9277 814.60
    32 4.262e-5 2.0963 3704.50 4.621e-5 2.0064 3738.70

     | Show Table
    DownLoad: CSV
    Table 6.  -norm errors, convergence orders, and CPU time for with , , , and in space.
    Error Order CPU(s) Error Order CPU(s)
    4 9.8959e-2 - 31.55 9.2094e-2 - 30.58
    8 2.6733e-2 1.8882 170.77 2.5135e-2 1.8734 170.73
    16 6.4312e-3 2.0556 825.40 6.2566e-3 2.0163 815.74
    32 1.0216e-4 2.6543 3759.60 1.2250e-3 2.3526 3760.00

     | Show Table
    DownLoad: CSV

    In this paper, an ADI scheme is proposed to solve the 3D variable coefficient subdiffusion problem, and our theoretical analysis shows that the method is unconditionally stable, its temporal convergence order is order , and its spatial convergence order is order 2. We present numerical results when taking different coefficients, which show that our ADI scheme is consistent with the theoretical analysis and is very efficient in solving such problems. The ADI technique proposed can reduce the computational cost from spatial discretization. In the future, we will consider some fast and parallel numerical methods [42,43,44] to improve computational efficiency in time.

    The work was supported by National Natural Science Foundation of China Mathematics Tianyuan Foundation (12226337, 12226340, 12126321, 12126307), Scientific Research Fund of Hunan Provincial Education Department (21B0550, 22C0323, 23C0193), Hunan Provincial Natural Science Foundation of China (2022JJ50083, 2023JJ50164).

    The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

    The authors declare there is no conflict of interest.



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