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Research article Special Issues

Nonlocal scalar conservation laws with discontinuous flux

  • Received: 04 October 2022 Revised: 14 December 2022 Accepted: 19 December 2022 Published: 29 December 2022
  • We prove the well-posedness of entropy weak solutions for a class of space-discontinuous scalar conservation laws with nonlocal flux. We approximate the problem adding a viscosity term and we provide L and BV estimates for the approximate solutions. We use the doubling of variable technique to prove the stability with respect to the initial data from the entropy condition.

    Citation: Felisia Angela Chiarello, Giuseppe Maria Coclite. Nonlocal scalar conservation laws with discontinuous flux[J]. Networks and Heterogeneous Media, 2023, 18(1): 380-398. doi: 10.3934/nhm.2023015

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  • We prove the well-posedness of entropy weak solutions for a class of space-discontinuous scalar conservation laws with nonlocal flux. We approximate the problem adding a viscosity term and we provide L and BV estimates for the approximate solutions. We use the doubling of variable technique to prove the stability with respect to the initial data from the entropy condition.



    Fractional partial differential equations (FPDEs) have attracted considerable attention in various fields. Though research shows that many phenomena can be described by FPDEs such as physics [1], engineering [2], and other sciences [3,4]. However, finding the exact solutions of FPDEs by using current analytical methods such as Laplace transform, Green's function, and Fourier-Laplace transform (see [5,6] for examples) are often difficult to achieve[7]. Thus, proposing numerical methods to find approximate solutions of these equations has practical importance. Due to this fact, in recent years a large number of numerical methods have been proposed for solving FPDEs, for instances see [8,9,10,11,12] and the references therein.

    The time fractional diffusion-wave equation is obtained from the classical diffusion-wave equation by replacing the second order time derivative term with a fractional derivative of order α, 1<α<2, and it can describe the intermediate process between parabolic diffusion equations and hyperbolic wave equations. Many of the universal mechanical, acoustic and electromagnetic responses can be accurately described by the time fractional diffusion-wave equation, see [13,14] for examples. The fourth order space derivative arises in the wave propagation in beams and modeling formation of grooves on a flat surface, thus considerable attention has been devoted to fourth order fractional diffusion-wave equation and its applications, see [15]. In this paper, the following nonlinear time fractional diffusion-wave equation with fourth order derivative in space and homogeneous initial boundary conditions will be considered

    2u(x,t)t2+C0Dαtu(x,t)+Kc4u(x,t)x4=2u(x,t)x2+g(u)+f(x,t), (1.1)

    where 1<α<2, f(x,t) is a known function, g(u) is a nonlinear function of u with g(0)=0 and satisfies the Lipschitz condition, and C0Dαtu(x,t) denotes the temporal Caputo derivative with order α defined as

    C0Dαtu(x,t)=1Γ(2α)t0(ts)1α2u(x,s)s2ds.

    Recently, there exist many works on numerical methods for time fractional diffusion-wave equations (TFDWEs), see [16,17,18,19,20,21,22] and the references therein. Chen et al. [17] proposed the method of separation of variables with constructing the implicit difference scheme for fractional diffusion-wave equation with damping. Heydari et al. [19] have proposed Legendre wavelets (LWs) for solving TFDWEs where fractional operational matrix of integration for LWs was derived. Bhrawy et al. [16] have proposed Jacobi tau spectral procedure combined with the Jacobi operational matrix for solving TFDWEs. Ebadian et al. [18] have proposed triangular function (TFs) methods for solving a class of nonlinear TFDWEs where fractional operational matrix of integration for the TFs was derived. Mohammed et al. [21] have proposed shifted Legengre collocation scheme and sinc function for solving TFDWEs with variable coefficients. Zhou et al. [22] have applied Chebyshev wavelets collocation for solving a class of TFDWEs where fractional integral formula of a single Chebyshev wavelets in the Riemann-Liouville sense was derived. Khalid et al. [20] have proposed the third degree modified extended B-spline functions for solving TFDWEs with reaction and damping terms. Some other numerical methods were presented for solving time fractional diffusion equations, one can see [23,24,25,26] and the references therein.

    To the best of our knowledge, there is no existing numerical method which can be used to solve Eq (1.1) neither directly nor by transferring Eq (1.1) into an equivalent integro-differential equation. Thus, the aim of this study is devoted to constructing the high order numerical schemes to solve Eq (1.1), and carrying out the corresponding numerical analysis for the proposed schemes. Herein, we firstly transform Eq (1.1) into the equivalent partial integro-differential equations by using the integral operator. Secondly, the Crank-Nicolson technique is applied to deal with the temporal direction. Then, we use the midpoint formula to discretize the first order derivative, use the weighted and shifted Gr¨unwald difference formula to discretize the Caputo derivative, and apply the second order convolution quadrature formula to approximate the first order integral. The classical central difference formula, the fourth order Stephenson scheme, and the fourth order compact difference formula are applied for spatial approximations.

    The rest of this paper is organized as follows. In Section 2, some preparations and useful lemmas are provided and discussed. In Section 3, the finite difference scheme is constructed and analyzed. In Section 4, the compact finite difference scheme is deduced, and the convergence and the unconditional stability are strictly proved. Numerical experiments are provided to support the theoretical results in Section 5. Finally, some concluding remarks are given.

    Lemma 2.1. (see Lemma 6.2 in [27]) Eq (1.1) is equivalent to the following partial integro-differential equation,

    u(x,t)t+C0Dα1tu(x,t)+Kc0Jt4u(x,t)x4=0Jt2u(x,t)x2+0Jtg(u)+F(x,t), (2.1)

    where F(x,t)=0Jtf(x,t) and 0Jt is first order integral operator, i.e., 0Jtu(,t)=t0u(,s)ds.

    To discretize Eq (2.1), we introduce the temporal step size τ=T/N with a positive integer N, tn=nτ, and tn+1/2=(n+1/2)τ. Similarly, define the spatial step size h=L/M with a positive integer M, and denote xi=ih. Then, define a grid function space Θh={vni| 0nN,0iM,vn0=vnM=0}, and introduce the following notations, inner product, and norm, i.e., for un,vnΘh, we define

    Δxuni=12h(uni+1uni1),δ2xuni=1h2(uni12uni+uni+1),un,vn=hM1i=1univni,||un||2=un,un,
    Huni={(1+h212δ2x)uni=112(uni1+10uni+uni+1), 1iM1,uni, i=0 or M.

    Lemma 2.2. (see Lemmas 2.2 and 2.3 in [28]) If u(,t)C2([0,T]) and 0<γ<1, then it holds

    0Jtu(,tn+1/2)=12[0Jtu(,tn+1)+0Jtu(,tn)]+O(τ2).

    Furthermore, if u(,t)C3([0,T]), then we have

    ut(,tn+1/2)=u(,tn+1)u(,tn)τ+O(τ2)=δtu(,tn+1/2)+O(τ2)

    and

    C0Dγtu(,tn+1/2)=12(C0Dγtu(,tn+1)+C0Dγtu(,tn))+O(τ2).

    Lemma 2.3. (see Theorem 4.1 in [29]) Let {ωk} be the weights from generating function (3/22z+z2/2)1, i.e., ωk=13(k+1). If u(,t)C2([0,T]) and u(,0)=ut(,0)=0, then we have

    0Jtn+1u(,t)τn+1k=0ωn+1ku(,tk)=O(τ2).

    Lemma 2.4. (see Theorem 2.4 in [30]) For u(,t)L1(R), RLDγ+2tu(,t) and its Fourier transform belong to L1(R), if we use the weighted and shifted Gr¨unwald difference operator to approximate the Riemann-Liouville derivative, then it holds

    RL0Dγ0u(,tk+1)=τγk+1j=0σ(γ)ju(,tk+1j)+O(τ2),0<γ<1,

    where

    σ(γ)0=2+γ2c(γ)0,σ(γ)j=2+γ2c(γ)jγ2c(γ)j1,j1,

    and c(γ)j=(1)j(γj) for j0.

    Lemma 2.5. (see Lemma 1.2 in [31]) Suppose u(x,)C4([xi1,xi+1]), let ζ(s)=u(4)(xi+sh,)+u(4)(xish,), then

    δ2xu(xi,)=u(xi1,)2u(xi,)+u(xi+1,)h2=uxx(xi,)+h22410ζ(s)(1s)3ds.

    Lemma 2.6. (see Page 6 of [32]) Assume that u(x,)C8([0,L]) with u(0,)=u(L,)=ux(0,)=ux(L,)=0, and define the operator δ4x by

    δ4xuni=12h2(Δxvniδ2xuni),

    where vni is a compact approximation of ux(xi,tn), i.e.,

    16vni1+23vni+16vni+1=Δxuni.

    Then, we have the following approximation

    δ4xuni=4u(xi,tn)x4+O(h4).

    Furthermore, let un=(un1,un2,,unM1)T, then the matrix representation of the operator δ4x is

    Sun=6h4(3KP1K+2D)un,

    where

    K=(0110110110)(M1)×(M1)P=(4114114114)(M1)×(M1),

    and D=6IP with the identity matrix I.

    Lemma 2.7. (see Lemma 3.3 in [32]) The matrix S defined in Lemma 2.6 is symmetric positive definite.

    It follows from Lemma 2.7, there is an invertible matrix B such that, S=BTB. Then for wn,vnΘh, we have

    Swn,vn=BTBwn,vn=Bwn,Bvn. (2.2)

    The following lemma is required when we use compact operator H to increase the spatial accuracy.

    Lemma 2.8. (see Lemma 1.2 in [31]) Suppose u(x,)C6([xi1,xi+1]), 1iM1, and ζ(s)=5(1s)33(1s)5. Then it holds that

    112[uxx(xi1,)+10uxx(xi,)+uxx(xi+1,)]1h2[u(xi1,)2u(xi,)+u(xi+1,)]=h436010[u(6)(xish,)+u(6)(xi+sh,)]ζ(s)ds.

    In order to linearize the nonlinear function g(u), we can easily get the following lemma by Taylor expansions.

    Lemma 2.9. Assume that u(,t)C1([0,T])C2((0,T]), then the following approximation holds

    u(,tn+1)=2u(,tn)u(,tn1)+O(τ2).

    In this subsection, a finite difference scheme with the accuracy O(τ2+h2) for nonlinear Problem (2.1) is constructed.

    Assume that u(x,t)C8,3x,t([0,L]×[0,T]), and u(,0)=ut(,0)=0. Consider Eq (2.1) at the point u(xi,tn+1/2), we have

    u(xi,t)t|t=tn+1/2=C0Dα1tn+1/2u(xi,t)Kc0Jtn+1/24u(xi,t)x4+0Jtn+1/22u(xi,t)x2+0Jtn+1/2g(u(xi,t))+F(xi,tn+1/2).

    The Crank-Nicolson technique and Lemma 2.2 for the above equation yield

    u(xi,tn+1)u(xi,tn)τ=12[C0Dα1tn+1u(xi,t)+C0Dα1tnu(xi,t)]Kc2[0Jtn+14u(xi,t)x4+0Jtn4u(xi,t)x4]+12[0Jtn+12u(xi,t)x2+0Jtn2u(xi,t)x2]+12[0Jtn+1g(xi,t)+0Jtng(xi,t)]+F(xi,tn+1/2)+O(τ2). (3.1)

    Let u(xi,tn)=uni. Since the initial values are 0, thus the Riemannliouville derivative is equivalent to Caputo derivative. We apply Lemmas 2.3 and 2.4 to discretize the first order integral operator and Caputo derivative in Eq (3.1) respectively, apply Lemma 2.6 to discretize 4u(xi,t)x4, and Lemma 2.5 to discretize 2u(xi,t)x2, then we get

    un+1iuniτ=τ1α2[n+1k=0σ(α1)kun+1ki+nk=0σ(α1)kunki]Kcτ2[n+1k=0ωkδ4xun+1ki+nk=0ωkδ4xunki]+τ2[n+1k=0ωkδ2xun+1ki+nk=0ωkδ2xunki]+τ2[n+1k=0ωkg(un+1ki)+nk=0ωkg(unki)]+Fn+12i+(R1)n+1i, (3.2)

    where (R1)n+1i=O(τ2+h2+h4)=O(τ2+h2).

    It is clear that Eq (3.2) is a nonlinear system with respect to the unknown un+1i. To linearly solve Eq (3.2), we use u1i=u0i+τ(ut)0i+O(τ2) and Lemma 2.9 to linearize Eq (3.2) for n=0 and 1nN1, respectively, and then multiply Eq (3.2) by τ, i.e.,

    u1iu0i=τ2α2[1k=0σ(α1)ku1ki+σ(α1)0u0i]Kcτ22[1k=0ωkδ4xu1ki+ω0δ4xu0i]+τ22[1k=0ωkδ2xu1ki+ω0δ2xu0i]+τ22[ω0g(u0i+τ(ut)0i)+ω1g(u0i)+ω0g(u0i)]+τFn+12i+O(τ3+τh2) (3.3)

    and

    un+1iuni=τ2α2[n+1k=0σ(α1)kun+1ki+nk=0σ(α1)kunki]Kcτ22[n+1k=0ωkδ4xun+1ki+nk=0ωkδ4xunki]+τ22[n+1k=0ωkδ2xun+1ki+nk=0ωkδ2xunki]+τ22[n+1k=1ωkg(un+1ki)+nk=0ωkg(unki)]+τ2ω02g(2uniun1i)+τFn+12i+O(τ3+τh2), for 1nN1. (3.4)

    Noting (ut)0i=0, neglecting the truncation error term O(τ3+τh2) in both above equations, and replacing the uni with its numerical solution Uni, we deduce the following finite difference scheme for Problem (2.1)

    U1iU0i=τ2α2[1k=0σ(α1)kU1ki+σ(α1)0U0i]Kcτ22[1k=0ωkδ4xU1ki+ω0δ4xU0i]+τ22[1k=0ωkδ2xU1ki+ω0δ2xU0i]+τ22[ω0g(U0i)+ω1g(U0i)+ω0g(U0i)]+τFn+12i (3.5)

    and

    Un+1iUni=τ2α2[n+1k=0σ(α1)kUn+1ki+nk=0σ(α1)kUnki]Kcτ22[n+1k=0ωkδ4xUn+1ki+nk=0ωkδ4xUnki]+τ22[n+1k=0ωkδ2xUn+1ki+nk=0ωkδ2xUnki]+τ22[n+1k=1ωkg(Un+1ki)+nk=0ωkg(Unki)]+τ2ω02g(2UniUn1i)+τFn+12i, for 1nN1. (3.6)

    Remark 3.1. In case of g(u)=f(x,t)=0, the only solution of the finite difference Scheme (3.5) and (3.6) is zero solution.

    In this subsection, the convergence and stability of the finite difference Scheme (3.5) and (3.6) will be discussed. For convenience, let C be a generic constant, whose value is independent of discretization parameters and may be different from one line to another. To begin, we provide two lemmas that will be used in our convergence and stability analysis.

    Lemma 3.2. (see Proposition 5.2 in [33] and Lemma 3.2 in [34]) Let {ωk} and {σ(α1)k} be the weights defined in Lemmas 2.3 and 2.4, respectively. Then for any positive integer K and real vector (V1,V2,,VK)T, the inequalities

    K1n=0(nj=0ωjVn+1j)Vn+10

    and

    K1n=0(nj=0σ(α1)jVn+1j)Vn+10

    hold.

    Lemma 3.3. (see Lemma 4.2.2 in [35]) For any grid function wn,vnΘh, it holds

    δ2xwn,vn=δxwn,δxvn.

    Theorem 3.4. Assume u(x,t)C8,3x,t([0,L]×[0,T]) and u(,0)=ut(,0)=0, and let u(x,t) be the exact solution of Eq (2.1) and {Uni|0iM,1nN} be the numerical solution for Scheme (3.7) and (3.8). Then, for 1nN, it holds that

    unUnC(τ2+h2).

    Proof. Let us start by analyzing the error of (3.6). Subtracting Eq (3.6) from Eq (3.4), we have

    en+1ieni=τ2α2[n+1k=0σ(α1)ken+1ki+nk=0σ(α1)kenki]Kcτ22[n+1k=0ωkδ4xen+1ki+nk=0ωkδ4xenki]+τ22[n+1k=0ωkδ2xen+1ki+nk=0ωkδ2xenki]+τ22nk=0(ωk+1+ωk)[g(unki)g(Unki)]+τ2ω02[g(2uniun1i)g(2UniUn1i)]+O(τ3+τh2),

    where eni=uniUni. Since e0i=0, the above equation becomes

    en+1ieni=τ2α2[nk=0σ(α1)k(en+1ki+enki)]Kcτ22[nk=0ωkδ4x(en+1ki+enki)]+τ22[nk=0ωkδ2x(en+1ki+enki)]+τ22nk=0(ωk+1+ωk)[g(unki)g(Unki)]+τ2ω02[g(2uniun1i)g(2UniUn1i)]+O(τ3+τh2).

    Multiplying the both sides of the above equation by h(en+1i+eni) and summing over 1iM1. Then using Lemmas 3.3, 2.6, and Eq (2.2), we have

    en+12en2=τ2α2nk=0σ(α1)ken+1k+enk,en+1+enKcτ22nk=0ωkB(en+1k+enk),B(en+1+en)τ22nk=0ωkδx(en+1k+enk),δx(en+1+en)+τ22nk=0(ωk+1+ωk)g(unk)g(Unk),en+1+en+τ2ω02g(2unun1)g(2UnUn1),en+1+en+O(τ3+τh2),en+1+en.

    Summing the above equation over n from 1 to J1 leads to

    eJ2e12=τ2α2J1n=1nk=0σ(α1)ken+1k+enk,en+1+enKcτ22J1n=1nk=0ωkB(en+1k+enk),B(en+1+en)τ22J1n=1nk=0ωkδx(en+1k+enk),δx(en+1+en)+τ22J1n=1nk=0(ωk+1+ωk)g(unk)g(Unk),en+1+en+τ2ω02J1n=1g(2unun1)g(2UnUn1),en+1+en+J1n=1O(τ3+τh2),en+1+en. (3.7)

    Now, we turn to analyze e1. Subtracting Eq (3.5) from Eq (3.3), and by the similar deductions as above, we can derive that

    e12=τ2α2σ(α1)0e1+e0,e1+e0Kcτ22ω0B(e1+e0),B(e1+e0)τ22ω0δx(e1+e0),δx(e1+e0)+τ2ω0g(u0)g(U0),e1+e0+τ2ω12g(u0)g(U0),e1+e0+O(τ3+τh2),e1+e0. (3.8)

    Sum up Eq (3.7) and Eq (3.8), and apply Lemma 3.2, it deduces that

    eJ2τ22J1n=1nk=0(ωk+1+ωk)g(unk)g(Unk),en+1+en+τ2ω02J1n=1g(2unun1)g(2UnUn1),en+1+en+τ2ω0g(u0)g(U0),e1+e0+τ2ω12g(u0)g(U0),e1+e0+CJ1n=1τ3+τh2,en+1+en. (3.9)

    Using the Lipschitz condition of g and exchanging the order of two summations in the above inequality, we have

    eJ2Cτ2J1k=0J1n=k(ωn+1k+ωnk)eken+1+en+Cτ2J1n=1enen+1+en+CJ1n=1(τ3+τh2)en+1+en. (3.10)

    Assuming eP=max0pNep. Since τNn=k(ωn+1k+ωnk) is bounded (see [29]), then the above inequality yields

    ePCτP1k=0ek+C(τ2+h2). (3.11)

    Once the discrete Gronwall inequality has been applied to Inequality (3.11), we arrive at the estimate

    ePC(τ2+h2),

    thus the proof is completed.

    Theorem 3.5. Let {Uni|0iM,0nN} be the numerical solution of Scheme (3.5) and (3.6) for Problem (2.1). Then for 1KN, it holds

    UKC(max0nNg(Un)+max0nN1Fn+12). (3.12)

    Proof. Multiplying (3.6) by h(Un+1i+Uni) and summing up for i from 1 to M1, we have

    Un+12Un2=τ2α2nk=0σ(α1)kUn+1k+Unk,Un+1+UnKcτ22nk=0ωkδ4x(Un+1k+Unk),Un+1+Un+τ22nk=0ωkδ2x(Un+1k+Unk),Un+1+Un+τ22nk=0(ωk+1+ωk)g(Unk),Un+1+Un+τ2ω02g(2UnUn1),Un+1+UnKcτ22ωn+1δ4xU0,Un+1+Unτ2α2σ(α1)n+1U0,Un+1+Un+τ22ωn+1δ2xU0,Un+1+Un+τFn+12,Un+1+Un.

    Note that Eq (1.1) is equipped with the homogeneous initial conditions, thus it deduces

    Un+12Un2=τ2α2nk=0σ(α1)kUn+1k+Unk,Un+1+UnKcτ22nk=0ωkδ4x(Un+1k+Unk),Un+1+Un+τ22nk=0ωkδ2x(Un+1k+Unk),Un+1+Un+τ22nk=0(ωk+1+ωk)g(Unk),Un+1+Un+τ2ω02g(2UnUn1),Un+1+Un+τFn+12,Un+1+Un.

    Applying the similar deductions to get Eq (3.9), it achieves that

    UJ2CτJ1k=0g(Uk)(Un+1+Un)+τ22ω0J1n=1g(2UnUn1)(Un+1+Un)+CτJ1n=1Fn+12(Un+1+Un). (3.13)

    One can estimate g(2UnUn1) as the following

    g(2UnUn1)=g(2UnUn1)g(Un)+g(Un),g(2UnUn1)g(Un)+g(Un),C(Un+Un1)+g(Un). (3.14)

    Substituting Eq (3.14) into Eq (3.13) and using Young's inequality, then we have

    UJ2CτJ1n=0Un2+Cmax0nNg(Un)2+Cmax0nN1Fn+122. (3.15)

    By applying the Gronwall inequality to (3.15), it becomes

    UJ2C(max0nNg(Un)2+max0nN1Fn+122),

    and this completes the proof.

    In this subsection, a compact finite difference scheme with accuracy O(τ2+h4) for nonlinear Problem (2.1) is presented.

    Now let us act on both sides of Eq (3.1) with the compact operator H. Then, by using Lemma 2.8, we obtain

    H[u(xi,tn+1)u(xi,tn)τ]=12H[C0Dα1tn+1u(xi,t)+C0Dα1tnu(xi,t)]Kc2H[0Jtn+14u(xi,t)x4+0Jtn4u(xi,t)x4]+12[0Jtn+1δ2xu(xi,t)+0Jtnδ2xu(xi,t)]+12H[0Jtn+1g(xi,t)+0Jtng(xi,t)]+HFn+12i+O(τ2+h4). (4.1)

    Apply the similar deductions to get Eqs (3.3) and (3.4), it achieves

    H[u1iu0i]=τ2α2H[1k=0σ(α1)ku1ki+σ(α1)0u0i]Kcτ22H[1k=0ωkδ4xu1ki+ω0δ4xu0i]+τ22[1k=0ωkδ2xu1ki+ω0δ2xu0i]+τ22H[ω0g(u0i)+ω1g(u0i)+ω0g(u0i)]+τHFn+12i+O(τ3+τh4) (4.2)

    and

    H[un+1iuni]=τ2α2H[n+1k=0σ(α1)kun+1ki+nk=0σ(α1)kunki]Kcτ22H[n+1k=0ωkδ4xun+1ki+nk=0ωkδ4xunki]+τ22[n+1k=0ωkδ2xun+1ki+nk=0ωkδ2xunki]+τ22H[n+1k=1ωkg(un+1ki)+nk=0ωkg(unki)]+τ2ω02Hg(2uniun1i)+τHFn+12i+O(τ3+τh4), for 1nN1. (4.3)

    Neglecting the truncation error term O(τ3+τh4) in both above equations, and replacing the uni with its numerical solution Uni, we deduce the following compact finite difference scheme for Problem (2.1)

    H[U1iU0i]=τ2α2H[1k=0σ(α1)kU1ki+σ(α1)0U0i]Kcτ22H[1k=0ωkδ4xU1ki+ω0δ4xU0i]+τ22[1k=0ωkδ2xU1ki+ω0δ2xU0i]+τ22H[ω0g(U0i)+ω1g(U0i)+ω0g(U0i)]+τHFn+12i (4.4)

    and

    H[Un+1iUni]=τ2α2H[n+1k=0σ(α1)kUn+1ki+nk=0σ(α1)kUnki]Kcτ22H[n+1k=0ωkδ4xUn+1ki+nk=0ωkδ4xUnki]+τ22[n+1k=0ωkδ2xUn+1ki+nk=0ωkδ2xUnki]+τ22H[n+1k=1ωkg(Un+1ki)+nk=0ωkg(Unki)]+τ2ω02Hg(2UniUn1i)+τHFn+12i, for 1nN1. (4.5)

    Remark 4.1. In case of g(u)=f(x,t)=0, the only solution of the compact finite difference Scheme (4.4) and (4.5) is zero solution.

    In this subsection, we turn to analyze the convergence and stability of the compact finite difference Scheme (4.4) and (4.5). Firstly, we provide the following lemmas, which will be used in our convergence and stability analysis.

    Lemma 4.2. (see Lemma 5 in [36]) Let {σ(α1)k} be the weighted coefficients defined in Lemma 2.4, then for any positive integer n and wnΘh, it holds that

    nm=0mk=0σ(α1)kHwmk,wm0.

    Lemma 4.3. (see Lemma 4.2 in [37]) For any grid function wnΘh, we have

    23wn2Hwn,wnwn2.

    Theorem 4.4. Assume u(x,t)C8,3x,t([0,L]×[0,T]) and u(,0)=ut(,0)=0, and let u(x,t) be the exact solution of Eq (2.1) and {Uni|0iM,1nN} be the numerical solution for Scheme (4.4) and (4.5). Then, for 1nN, it holds that

    unUnC(τ2+h4).

    Proof. Let us start by analyzing the error of (4.5). Subtracting Eq (3.5) from Eq (4.3), we have

    H[en+1ieni]=τ2α2H[n+1k=0σ(α1)ken+1ki+nk=0σ(α1)kenki]Kcτ22H[n+1k=0ωkδ4xen+1ki+nk=0ωkδ4xenki]+τ22[n+1k=0ωkδ2xen+1ki+nk=0ωkδ2xenki]+τ22Hnk=0(ωk+1+ωk)[g(unki)g(Unki)]+τ2ω02H[g(2uniun1i)g(2UniUn1i)]+O(τ3+τh4),

    where eni=uniUni. Since e0i=0, the above equation becomes

    H[en+1ieni]=τ2α2[nk=0σ(α1)kH(en+1ki+enki)]Kcτ22[nk=0ωkHδ4x(en+1ki+enki)]+τ22[nk=0ωkδ2x(en+1ki+enki)]+τ22nk=0(ωk+1+ωk)H[g(unki)g(Unki)]+τ2ω02H[g(2uniun1i)g(2UniUn1i)]+O(τ3+τh4).

    Multiplying the both sides of the above equation by h(en+1i+eni) and summing over 1iM1. Then using Lemmas 2.6, 3.2, 4.2, and Eq (2.2), we have

    en+12en2τ2α2nk=0σ(α1)kH(en+1k+enk),en+1+enKcτ22nk=0ωkHB(en+1k+enk),B(en+1+en)τ22nk=0ωkδx(en+1k+enk),δx(en+1+en)+τ22nk=0(ωk+1+ωk)H(g(unk)g(Unk)),en+1+en+τ2ω02H(g(2unun1)g(2UnUn1)),en+1+en+Cτ3+τh4,en+1+en.

    Summing the above inequality over n from 1 to J1 leads to

    eJ2e12τ2α2J1n=1nk=0σ(α1)kH(en+1k+enk),en+1+enKcτ22J1n=1nk=0ωkHB(en+1k+enk),B(en+1+en)τ22J1n=1nk=0ωkδx(en+1k+enk),δx(en+1+en)+τ22J1n=1nk=0(ωk+1+ωk)H(g(unk)g(Unk)),en+1+en+τ2ω02J1n=1H(g(2unun1)g(2UnUn1)),en+1+en+CJ1n=1τ3+τh4,en+1+en. (4.6)

    Now, we turn to analyze e1. From Eqs (4.4), (4.2), and by the similar deductions as above, we can derive that

    e12τ2α2σ(α1)0H(e1+e0),e1+e0Kcτ22ω0HB(e1+e0),B(e1+e0)τ22ω0δx(e1+e0),δx(e1+e0)+τ2ω12H(g(u0)g(U0)),e1+e0+τ2ω0H(g(u0)g(U0)),e1+e0+Cτ3+τh4,e1+e0. (4.7)

    Sum up Eqs (4.6) and (4.7), and apply Lemmas 3.2 and 4.2, it deduces that

    eJ2τ22J1n=1nk=0(ωk+1+ωk)H(g(unk)g(Unk)),en+1+en+τ2ω02J1n=1H(g(2unun1)g(2UnUn1)),en+1+en+τ2ω12H(g(u0)g(U0)),en+1+en+τ2ω0H(g(u0)g(U0)),en+1+en+CJ1n=1τ3+τh4,en+1+en.

    According to the same technique as for dealing with (3.9), we can achieve

    ePC(τ2+h4),

    thus completes the proof.

    Theorem 4.5. Let {Uni|0iM,0nN} be the numerical solution of Scheme (4.4) and (4.5) for Problem (2.1). Then for 1KN, it holds

    UKC(max0nNg(Un)+max0nN1Fn+12).

    In this section, we carry out numerical experiments to verify the theoretical results and demonstrate the performance of our new schemes. All of the computations are performed by using a MATLAB on a computer with Intel(R) Core(TM) i5-8265U CPU 1.60GHz 1.80GHz and 8G RAM.

    Example 5.1. Consider the following problem with exact solution u(x,t)=t2+αsin2(πx)

    2u(x,t)t2+C0Dαtu(x,t)+4u(x,t)x4=2u(x,t)x2+f(x,t)+g(u),

    where T=1, 0<x<1, 0<tT, and 1<α<2. The nonlinear function g(u)=u2 and f(x,t) is

    f(x,t)=(2+α)(1+α)tαsin2(πx)+Γ(3+α)2t2sin2(πx)8π4t2+αcos(2πx)2π2t2+αcos(2πx)t2(2+α)sin4(πx).

    It is clear that u(x,t) satisfies all smoothness conditions required by Theorems 3.4 and 4.4, so that both of our schemes can be applied in this example. In Figures 1 and 2, we compare the exact solution with the numerical solution of finite difference Scheme (3.5) and (3.6) and compact finite difference Scheme (4.4) and (4.5). We easily see that the exact solution can be well approximated by the numerical solutions of our schemes.

    Figure 1.  The comparison of numerical solution of Scheme (3.5) and (3.6) with the exact solution for τ=h=0.01 and α=1.6.
    Figure 2.  The comparison of numerical solution of the compact finite difference Scheme (4.4) and (4.5) with the exact solution for τ=h=0.01 and α=1.6.

    First, we in Tables 1, 2 and 3 show that the errors, time and space convergence order 2 and CPU times (second) of the finite difference Scheme (3.5) and (3.6) for α=1.25,1.5,1.75. The average CPU time, expressed as the mean time (mean) for α=1.25,1.5,1.75. Specifically, Table 1 tests the case that when τ=h. In Table 2, we set h=0.001, a value small enough such that the spatial discretization errors are negligible as compared with the temporal errors, and choose different time step size. In Table 3, we set τ=0.001, a value small enough such that the temporal discretization errors are negligible as compared with the spatial errors, and choose different space step size. From all scenarios above, we conclude that the temporal and spatial convergence order is 2. It verifies Theorem 3.4.

    Table 1.  The errors, CPU times (second) for different α, and numerical convergence orders of Scheme (3.5) and (3.6) for different τ=h.
    τ=h α=1.25 α=1.5 α=1.75 CPU time
    error order error order error order mean
    1/5 6.6627×102 7.8031×102 8.9815×102 0.0896
    1/10 1.8412×102 1.8555 2.1839×102 1.8371 2.5456×102 1.8190 0.0973
    1/20 4.8132×103 1.9355 5.7273×103 1.9310 6.6917×103 1.9275 0.0994
    1/40 1.2137×103 1.9876 1.4621×103 1.9698 1.7210×103 1.9591 0.1359

     | Show Table
    DownLoad: CSV
    Table 2.  The errors, CPU times (second) for different α, and temporal numerical convergence orders of Scheme (3.5) and (3.6) for h=0.001 and different τ.
    τ α=1.25 α=1.5 α=1.75 CPU time
    error order error order error order mean
    1/5 7.0844×102 8.2130×102 9.3783×102 0.5852
    1/10 1.9012×102 1.8977 2.2432×102 1.8724 2.6040×102 1.8486 1.0501
    1/20 4.9405×103 1.9442 5.8537×103 1.9381 6.8169×103 1.9335 2.4071
    1/40 1.2435×103 1.9903 1.4917×103 1.9724 1.7504×103 1.9615 6.7799

     | Show Table
    DownLoad: CSV
    Table 3.  The errors, CPU times (second) for different α, and spatial numerical convergence orders of Scheme (3.5) and (3.6) for τ=0.001 and different h.
    h α=1.25 α=1.5 α=1.75 CPU time
    error order error order error order mean
    1/5 4.7813×103 4.7510×103 4.7111×103 2.2382
    1/10 6.1943×104 2.9484 6.1518×104 2.9492 6.0963×104 2.9501 2.2952
    1/20 1.2773×104 2.2778 1.2654×104 2.2815 1.2503×104 2.2857 2.5410
    1/40 2.8950×105 2.1415 2.8363×105 2.1575 2.7665×105 2.1761 3.4293

     | Show Table
    DownLoad: CSV

    On the other hand, we check the numerical convergence orders and CPU times (second) in time and space of the compact finite difference Scheme (4.4) and (4.5) for α=1.25,1.5,1.75 in Tables 4 and 5, respectively. The average CPU time, expressed as the mean time (mean) for α=1.25,1.5,1.75. As expected, the numerical results reflect that the compact finite difference has a convergence order of 2 and 4 in time and space, respectively, which verifies our Theorem 4.4.

    Table 4.  The errors, CPU times (second) for different α, and temporal convergence orders of Scheme (4.4) and (4.5) for h=0.001 and different τ.
    τ α=1.25 α=1.5 α=1.75 CPU time
    error order error order error order mean
    1/5 7.0844×102 8.2129×102 9.3783×102 0.9501
    1/10 1.9012×102 1.8978 2.2432×102 1.8724 2.6040×102 1.8486 2.3622
    1/20 4.9407×103 1.9441 5.8538×103 1.9381 6.8169×103 1.9335 7.6793
    1/40 1.2436×103 1.9901 1.4919×103 1.9723 1.7506×103 1.9612 28.9326

     | Show Table
    DownLoad: CSV
    Table 5.  The errors, CPU times (second) for different α, and spatial convergence orders of Scheme (4.4) and (4.5) for τ=0.0005 and different h.
    h α=1.25 α=1.5 α=1.75 CPU time
    error order error order error order mean
    1/5 3.8110×103 3.7871×103 3.7555×103 12.5566
    1/10 2.5308×104 3.9125 2.5141×104 3.9130 2.4922×104 3.9135 14.0490
    1/20 2.2087×105 3.5183 2.1851×105 3.5243 2.1557×105 3.5312 18.1726
    1/40 1.8261×106 3.5964 1.7163×106 3.6703 1.5904×106 3.7607 43.9104

     | Show Table
    DownLoad: CSV

    We in this paper constructed two linearized finite difference schemes for time fractional nonlinear diffusion-wave equations with the space fourth-order derivative. The equations were transformed into equivalent partial integro-differential equations. Then, the Crank-Nicolson technique, the midpoint formula, the weighted and shifted Gr¨unwald difference formula, the second order convolution formula, the classical central difference formula, the fourth-order approximation and the compact difference technique were applied to construct the two proposed schemes. The finite difference Scheme (3.5) and (3.6) has the accuracy O(τ2+h2). The compact finite difference Scheme (4.4) and (4.5) has the accuracy O(τ2+h4). It should be mentioned that our schemes require the exact solution u(,t)C3([0,T]), while it requires u(,t)C4([0,T]) if one discretizes Eq (1.1) directly to get the second order accuracy in time. Theoretically, the convergence and the unconditional stability of the two proposed schemes are proved and discussed. All of the numerical experiments can support our theoretical results.

    This research is supported by Natural Science Foundation of Jiangsu Province of China (Grant No. BK20201427), and by National Natural Science Foundation of China (Grant Nos. 11701502 and 11871065).

    The authors declare that they have no competing interests.



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