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Efficient time second-order SCQ formula combined with a mixed element method for a nonlinear time fractional wave model

  • In this article, a kind of nonlinear wave model with the Caputo fractional derivative is solved by an efficient algorithm, which is formulated by combining a time second-order shifted convolution quadrature (SCQ) formula in time and a mixed element method in space. The stability of numerical scheme is derived, and an optimal error result for unknown functions which include an original function and two auxiliary functions are proven. Further, the numerical tests are conducted to confirm the theoretical results.

    Citation: Yining Yang, Yang Liu, Cao Wen, Hong Li, Jinfeng Wang. Efficient time second-order SCQ formula combined with a mixed element method for a nonlinear time fractional wave model[J]. Electronic Research Archive, 2022, 30(2): 440-458. doi: 10.3934/era.2022023

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  • In this article, a kind of nonlinear wave model with the Caputo fractional derivative is solved by an efficient algorithm, which is formulated by combining a time second-order shifted convolution quadrature (SCQ) formula in time and a mixed element method in space. The stability of numerical scheme is derived, and an optimal error result for unknown functions which include an original function and two auxiliary functions are proven. Further, the numerical tests are conducted to confirm the theoretical results.



    A large number of fractional partial differential equation (FPDE) models have been found in many fields of science and engineering, such as fractional wave model [1,2,3,4,5], fractional diffusion model [6,7,8,9,10], fractional FitzHugh-Nagumo monodomain model [11], fractional water wave model [12,13], fractional Maxwell model [14,15], fractional Allen-Cahn model [16], fractional constitutive model [17] and fractional Fokker-Planck model [18]. With the continuous developments of scholars' research on FPDE models, the important fractional wave equations studied by theoretical or numerical methods [1,2,3,4,5] have received a lot of attention. However, as scholars know that due to the existing of the fractional derivative, their exact solutions for fractional wave equations are hard to be found by some theoretical methods. So, numerical solutions of fractional wave models are studied by designing efficient numerical algorithms, such as finite element method [2,19,20,21,22], meshless method [3], finite difference method [23,24,25,26,27,28,29,30,31,32,33], spectral method [34,35] and collocation method [36,37]. In this article, we focus on the following wave model with a nonlinear term and a high-order Caputo time fractional derivative

    2ut2+βutβ3ux2t+f(u)=d(x,t),(x,t)Ω×J,u(a,t)=u(b,t)=0,t¯J,u(x,0)=0,ut(x,0)=0,x¯Ω, (1.1)

    where Ω=(a,b) is an open space domain and J=(0,T] with 0<T< is a time interval. The source term d(x,t) is a known smooth function, the nonlinear term satisfies f(u)C2(R) with f(0)=0, and the Caputo fractional derivative is defined by

    βu(x,t)tβ=1Γ(2β)t02u(x,s)s2ds(ts)β1, 1<β<2. (1.2)

    The fractional wave model (1.1), which describe many physical phenomena including nerve conduction and wave propagation, can be degenerated into the pseudo-hyperbolic equation for β=1 and the hyperbolic wave equation for β=2, respectively. In [38], Wang et al. developed a mixed element method with an L2-1σ formula for solving the fractional wave model (1.1) with the time Caputo fractional derivative, which was proposed by improving the H1-Galerkin mixed element method [39,40,41,42,43,44]. The improved mixed element method can approach three unknown functions simultaneously. However, in [38], the optimal theory error result for the auxiliary variable v depends on the parameter β12, from which the optimal estimate result of v(tn)vnh cannot abtained by choosing any fractional parameter β(1,2).

    In this article, we develop a fully discrete mixed finite element scheme, where the mixed element method is used to approximate the space direction and the generalized BDF2-θ [45] that is a shifted convolution quadrature (SCQ) method [46] is applied to the approxiamtion of the time direction at any time tnθ. Based on the formulated fully discrete mixed element method with a second-order SCQ formula, we prove the stability and derive optimal error estimates for three unknown functions. More importantly, with a comparison to the theory error results in [38], we can obtain the optimal error result in L2-norm for the auxiliary variable v at time tn by choosing the shifted parameter θ=0. Finally, we implement two numerical examples to verify our optimal theory results.

    The rest of the article is outlined as follows: In Section 2, the fully discrete scheme based on the combination between a mixed element method and an SCQ formula (generalized BDF2-θ) is derived; In Section 3, the stability is proven by using useful lemmas; The optimal error estimates for the fully discrete scheme are derived in Section 4. Two experiments, in Section 5, are conducted to further confirm our theoretical results. Finally, in the last section we give the conclusions and advancements.

    Throughout the article, we denote by C a positive generic constant which is free of time and space meshes, and may be changed at different occurrences.

    By setting the parameter α=β1 and an auxiliary variable v=ut as shown in [38], we get

    βu(x,t)tβ=1Γ(1α)t0v(x,s)sds(ts)α=αv(x,t)tα,0<α<1. (2.1)

    Further, by introducing the other auxiliary variable σ=vx, we can rewrite the model (1.1) as the following coupled system with the low order space-time derivatives

    v=ut,σ=vx,vt+αvtασx+f(u)=d(x,t),(x,t)Ω×J,u(a,t)=u(b,t)=v(a,t)=v(b,t)=0,tˉJ,u(x,0)=v(x,0)=0,xˉΩ. (2.2)

    For the fully discrete scheme, we first divide time interval [0,T] by the nodes tn=nτ (n=0,1,2,...,N) with the time step length size τ=T/N, where tn satisfy 0=t0<t1<t2<<tN=T, N is a positive integer. Setting ϕn=ϕ(,tn), the generalized BDF2-θ (See [45]) for the Caputo fractional differential operator with α(0,1] at time tnθ is

    αϕnθtα=ταnj=0ω(α)jϕnj+O(τ2)Ψα,nτϕ+O(τ2). (2.3)

    The convolution weights {ω(α)j}j=0 are the coefficients of the following generating function with the relation ω(α)(ξ)=j=0ω(α)jξj,

    ω(α)(ξ)=(3α2θ2α2α2θαξ+α2θ2αξ2)α,0θmin{α,12}. (2.4)

    For the convenience of application in calculation, we provide the relationship among these convolution weights {ω(α)j}j=0.

    Lemma 2.1. (See [45]) The convolution weights ω(α)k for the generalized BDF2-θ can be arrived at by the recursive formula

    {ω(α)0=(3α2θ2α)α,ω(α)1=2(θα)(2α3α2θ)1α,ω(α)k=2αk(3α2θ)[2(αθ)(k1α1)ω(α)k1+(α2θ)(1k22α)ω(α)k2],k2.

    For the term v(tnθ)x, we have the following formula

    ϕ(tnθ)x=(1θ)ϕnx+θϕn1x+O(τ2)ϕnθx+O(τ2). (2.5)

    Now, by combining Eqs (2.2), (2.3) with (2.5), we have

    (a)Ψ1,nτu=vnθ+Rnθ1,(b)σnθ=vnθx+Rnθ2,(c)Ψ1,nτv+Ψα,nτvσnθx+f(unθ)=d(x,tnθ)+Rnθ3, (2.6)

    where

    Rnθ1=Ψ1,nτuu(tnθ)t+v(tnθ)vnθ=O(τ2),Rnθ2=σnθσ(tnθ)+v(tnθ)xvnθx=O(τ2),Rnθ3=Ψ1,nτvv(tnθ)t+Ψα,nτvαv(tnθ)tα+σ(tnθ)xσnθx+f(u(tnθ))f(unθ)=O(τ2).

    Based on Eq (2.6), the mixed weak formulation is to find (un,vn,σn)L2×H10×H1, such that

    (a) (Ψ1,nτu,w)=(vnθ,w)+(Rnθ1,w),(b) (σnθ,ψx)=(vnθx,ψx)+(Rnθ2,ψx),(c) (Ψ1,nτσ,χ)+(ταnj=0ω(α)jσnj,χ)+(σnθx,χx)=(g(unθ)Inθ0σ,χ)(d(x,tnθ),χx)+(Rnθ3,χx)+(Rnθ4,χ), (2.7)

    where g(unθ)=f(unθ),Inθ0σ=τ(12σ0+n2k=1σk+(1θ2)σn1+12(1θ)σn),Rnθ4=g(unθ)(tnθ0σdtInθ0σ)=O(τ2).

    Setting (ˉun,ˉvn,ˉσn)L2×H10×H1 be the time approximate solutions of (un,vn,σn), we have

    (a) (Ψ1,nτˉu,w)=(ˉvnθ,w),(b) (ˉσnθ,ψx)=(ˉvnθx,ψx),(c) (Ψ1,nτˉσ,χ)+(ταnj=0ω(α)jˉσnj,χ)+(ˉσnθx,χx)=(g(ˉunθ)Inθ0ˉσ,χ)(d(x,tnθ),χx). (2.8)

    For formulating the fully discrete mixed element scheme, we provide the following mixed finite element spaces

    Lh={uh|uhPmoneachelement,mN},Vh={vh|vhPkoneachelement,vh(a)=vh(b)=0,vhxL2,kZ+},Hh={σh|σhProneachelement,σhxL2,rZ+},

    where Ps the set of polynomials of x with the degree of sN. Based on Eq (2.7), we obtain the mixed element scheme. That is to find (unh,vnh,σnh)Lh×Vh×HhL2×H10×H1, such that

    (a) (Ψ1,nτuh,wh)=(vnθh,wh),  whLh,(b) (σnθh,ψhx)=(vnθhx,ψhx),  ψhVh,(c) (Ψ1,nτσh,χh)+(ταnj=0ω(α)jσnjh,χh)+(σnθhx,χhx)=(g(unθh)Inθ0σh,χh)(d(x,tnθ),χhx),  χhHh. (2.9)

    Remark 2.2. 1) For implementing the computation based on the system (2.9), we need to consider the following case for n=1. For this case, we only need to take the semi-discrete approximation of the nonlinear term

    g(ˉu1θ)I1θ0ˉσ=g(ˉu0)I10ˉσ=g(ˉu0)τˉσ0,

    and the fully discrete approximation

    g(¯uh1θ)I1θ0¯σh=g(¯uh0)I10¯σh=g(¯uh0)τˉσ0h.

    2) Now, we illustrate how to derive the Eq (2.7)(c). We multiply Eq (2.6)(c) by χx, and then make the inner product on the space domain ˉΩ=[a,b]. Taking the first term as an example, we deduce it in detail. By the integration by part, we obtain for vH10(Ω)

    (Ψ1,nτv,χx)=(Ψ1,nτvx,χ)+[χΨ1,nτv]|ba=(Ψ1,nτσ,χ),

    which also shows that χ only needs to belong to H1(Ω). For this problem, readers can also see other references [39,40,41].

    Remark 2.3. 1) In Ref [45], one can see that the generalized BDF2-θ is given by

    αϕnθtα=ταnj=0ω(α)jϕnj+ταkj=1w(α)n,jϕj+O(τ2)Ψα,nτϕ+Sα,nτ,kϕ+O(τ2), (2.10)

    where Ψα,nτϕ and Sα,nτ,k are called the convolution part and the starting part, respectively. If we only consider the model with a sufficiently smooth exact solution, the starting part will disappear. For this problem, readers can see the detailed illustrations in [45]. Here, we just study the case without the starting part.

    2) Readers can know easily from many references that the following relationship between the Caputo fractional derivative and the Riemann-Liouville derivative holds

    RL0Dαtϕ(t)=C0Dαtϕ(t)+n1j=0ϕ(j)(0)Γ(1+jα)tjα,n1α<n, (2.11)

    which imply that if initial values ϕ(j)(0)=0, the equality RL0Dαtϕ(t)=C0Dαtϕ(t) holds. In this article, the Caputo fractional derivative C0Dαtϕ(t) is written as αϕ(t)tα.

    Now we need to introduce some useful lemmas for the next analysis.

    Lemma 3.1. (See [45]) For series {ϕm} m2, we have

    Ψ1,mτ(ϕ,ϕm)14τ(Hm(ϕ)Hm1(ϕ)), (3.1)

    with

    Hm(ϕ)=(32θ)ϕm2(12θ)ϕm12+2ϕmϕm12. (3.2)

    In addition, we have

    Hm(ϕ)ϕm2, (3.3)

    where θ[0,12].

    Proof. Here, we just need to take θ=α2, and then follow the derived process as in [47] to get the result.

    Lemma 3.2. (See [45])Let ω(α)k be the coefficients of generating function ω(α)(ξ) and the parameter θ satisfies 0θmin{α,12}, where α(0,1). Then we have for any vector (ϕ0,ϕ1,,ϕn)Rn+1

    nm=0mk=0(ω(α)mkϕk,ϕm)0,n1. (3.4)

    Lemma 3.3. (See [45]) With the shifted parameter θ12 and ϕ0=0, we have for any vector (ϕ1,ϕ2,,ϕn)Rn

    nm=1(ϕmθ,ϕm)0,n1, (3.5)

    where vmθ(1θ)vm+θvm1.

    Without loss of generality, we will analyze the stability of the numerical scheme Eq (2.9) for the case of the source term d(x,t)=0.

    Theorem 3.4. For the fully discrete system (2.9), the following stability holds

    unh2+σnh2+vnθh2+vnθhx2C(u0h2+σ0h2), (3.6)

    where C is a positive constant independent on mesh parameters τ and h.

    Proof. In Eq (2.9)(a), we take wh=unh, use Eq (3.1) and Cauchy-Schwarz inequality as well as Young inequality to get for n2

    14τ(Hn(uh)Hn1(uh))(vnθh,unh)1θ2vnh2+θ2vn1h2+12unh2. (3.7)

    Sum Eq (3.7) for j=2 to n, and use Eqs (3.2) and (3.3) to arrive at

    unh2Hn(uh)τnj=2((2θ)vjh2+2θvj1h2+2ujh2)+H1(uh)τnj=2((2θ)vjh2+2θvj1h2+2ujh2)+C(u1h2+u0h2). (3.8)

    Letting ψh=vnθh in Eq (2.9)(b), using Cauchy-Schwarz inequality as well as Young inequality, noting that vnθhVhH10 and making use of Poincaré inequality, we have

    vnθh2Cvnθhx2Cσnθh2. (3.9)

    In Eq (2.9)(c), we set χh=σnh, replace n with k and sum for k=2 to n to get

    nk=2(Ψ1,kτσh,σkh)+ταnk=2(kj=0ω(α)kjσjh,σkh)+nk=2(σkθhx,σkhx)=nk=2(τg(ukθh)(12σ0h+k2j=1σjh+(1θ2)σk1h+12(1θ)σkh),σkh), (3.10)

    Use Hölder inequality as well as Young inequality to arrive at

    14τnk=2(Hk(σh)Hk1(σh))+ταnk=2(kj=0ω(α)kjσjh,σkh)+nk=2(σkθhx,σkhx)τnk=2(g(ukθh)12σ0h+k2j=1σjh+(1θ2)σk1h+12(1θ)σkhσkh)Cτnk=2(σkhkj=0σjh)Cnk=2(τkj=0σjh2+τ(k+1)2σkh2). (3.11)

    Multiplying Eq (3.11) by 4τ and using Young inequality as well as triangle inequality, we have

    Hn(σh)+4τ1αnk=2(kj=0ω(α)kjσjh,σkh)+4τnk=2(σkθhx,σkhx)Cτnk=0σkh2+H1(σh)Cτnk=0σkh2+C(σ1h2+σ0h2). (3.12)

    Now we only need to estimate the case for n=1. Similar to the processes of Eqs (3.8) and (3.12), we easily derive

    u1h2Cτ(v1h2+τv0h2+u1h2)+u0h2, (3.13)

    and

    σ1h2+τ1α1j=0(ω(α)1jσjh,σ1h)+τ(σ1θhx,σ1hx)Cσ0h2. (3.14)

    Combining Eqs (3.8), (3.12), (3.13) with (3.14), we have

    unh2C(u0h2+τnj=0vjh2+τnj=1ujh2), (3.15)
    σnh2+τ1αnk=0(kj=0ω(α)kjσjh,σkh)+τnk=1(σkθhx,σkhx)C(σ0h2+τnk=0σkh2). (3.16)

    Combining Lemmas 3.2–3.3, Eqs (3.9), (3.15) with (3.16), we have

    unh2+σnh2+vnθh2+vnθhx2C(u0h2+σ0h2+τnk=0(ukh2+vkh2+σkh2)). (3.17)

    Use Gronwall lemma to finish the proof.

    In this section, we obtain an error estimate for the numerical scheme Eq (2.9). To facilitate the analysis, we first introduce three projection operators with the corresponding estimate inequalities.

    Lemma 4.1. Define an L2-projection operator Λh:L2(Ω)Lh by

    (ˉuΛhˉu,ωh)=0,  ωhLh, (4.1)

    with an estimate inequality

    ˉuΛhˉu+ˉutΛhˉutChm+1(ˉum+1+ˉutm+1),  ˉuHm+1(Ω). (4.2)

    Lemma 4.2. (See [41]).Define an elliptic projection operator Υh:H10(Ω)Vh, such that

    (ˉvxΥhˉvx,ϕhx)=0,  ϕhVh, (4.3)

    with an estimate inequality

    ˉvΥhˉv+hˉvΥhˉv1Chk+1ˉvk+1,  ˉvH10(Ω)Hk+1(Ω). (4.4)

    Lemma 4.3. (See [41]) Define a Rize projection operator Πh:H1(Ω)Hh by

    A(ˉσΠhˉσ,χh)=0,  χhHh, (4.5)

    where A(ˉσ,ϕ)(ˉσx,ϕx)+λ(ˉσ,ϕ) and A(ϕ,ϕ)μ0ϕ21,μ0>0is a constant. Further, the estimate inequality holds

    ˉσΠhˉσ+ˉσtΠhˉσt+hˉσΠhˉσ1Chr+1(ˉσr+1+ˉσtr+1),  ˉσHr+1(Ω). (4.6)

    Theorem 4.4. With Λhˉu(0)=ˉu0h, Υhˉv(0)=ˉv0h and Πhˉσ(0)=ˉσ0h, there exists a positive constant C independent of (h,τ) such that

    u(tn)unh+σ(tn)σnh+v(tnθ)vnθhC(hmin{m+1,r+1,k+1}+τ2). (4.7)

    Proof. For convenience, we write errors as

    ˉu(tn)unh=(ˉu(tn)Λhˉun)+(Λhˉununh)=ρn+ϑn,ˉv(tn)vnh=(ˉv(tn)Υhˉvn)+(Υhˉvnvnh)=ζn+ξn,ˉσ(tn)σnh=(ˉσ(tn)Πhˉσn)+(Πhˉσnσnh)=ηn+δn,

    Subtract Eq (2.9)(a) from Eq (2.8)(a), set ωh=ϑn, apply the projection Eq (4.1) and use Cauchy-Schwarz inequality and Young inequality to obtain

    (Ψ1,nτϑ,ϑn)=(Ψ1,nτρ,ϑn)+(ζnθ+ξnθ,ϑn)12(Ψ1,nτρ2+ξnθ2+ζnθ2)+32ϑn2. (4.8)

    Replace n by m, sum from m=2 to n, and use Lemmas 3.1 to have

    ϑn2Hn(ϑ)Cτnm=2(Ψ1,nτρ2+ξnθ2+ζnθ2)+Cτnm=2ϑm2+C(ϑ12+ϑ02). (4.9)

    Subtract Eq (2.9)(b) from Eq (2.8)(b), take ψh=ξnθ and apply projection Eq (4.3) to get

    (δnθ,ξnθx)=(ηnθ,ξnθx)+ξnθx2. (4.10)

    Use Cauchy-Schwarz inequality, Young inequality and Poincaré inequality to arrive at

    ξnθx2+ξnθ2C(ηnθ2+δn2+δn12). (4.11)

    Subtract Eq (2.9)(c) from Eq (2.8)(c), choose χh=δn, apply projection Eq (4.5) and use Hölder inequality as well as Young inequality to get

    (Ψ1,nτδ,δn)+(Ψα,nτδ,δn)+(δnθx,δnx)=(Ψ1,nτη,δn)(Ψα,nτη,δn)+λ(ηnθ,δn)(g(ˉunθ)Inθ0ˉσg(unθh)Inθ0σh,δn)=(Ψ1,nτη,δn)(Ψα,nτη,δn)+λ(ηnθ,δn)(g(ˉunθ)Inθ0ˉσg(unθh)Inθ0ˉσ+g(unθh)Inθ0ˉσ g(unθh)Inθ0σh,δn)=(Ψ1,nτη,δn)(Ψα,nτη,δn)+λ(ηnθ,δn)((g(ˉunθ)g(unθh))Inθ0ˉσ,δn)(g(unθh)Inθ0(ˉσσh),δn)12Ψ1,nτη2+12Ψα,nτη2+λ(1θ)2ηn2+λθ2ηn12+(32+λ2)δn2+((g(ˉunθ)g(unθh)Inθ0ˉσ+g(unθh)Inθ0(ˉσσh))δn12Ψ1,nτη2+12Ψα,nτη2+λ(1θ)2ηn2+λθ2ηn12+(32+λ2)δn2+(Cg(ς)(ρnθ+ϑnθ+Cτ(12(η0+δ0)+n1j=1(ηj+δj)+(1θ2)(ηn1+δn1)+12(1θ)(ηn+δn)))δn12Ψ1,nτη2+12Ψα,nτη2+C(ηn2+ηn12+ρn2+ρn12)+C(δn2+ϑn2+ϑn12)+Cτnk=0(ηk2+δk2). (4.12)

    Replace n by m and sum for m=2 to n to arrive at

    nm=2(Ψ1,mτδ,δm)+ταnm=2(mj=0ω(α)mjδj,δm)+nm=2(δmθx,δmx)Cnm=2(Ψα,mτη2+Ψ1,mτη2+δm2)+Cnm=1(ρm2+ηm2+ϑm2)+Cτnm=1mk=0(ηk2+δk2). (4.13)

    Noting that the similar method in [45,48], we have

    Ψα,nτηΠh(αˉσnθtα)αˉσnθtαChr+1, (4.14)

    and

    Ψ1,nτηChr+1. (4.15)

    Multiply Eq (4.13) by 4τ, and combine Lemmas 4.1 and 4.3 with Eqs (4.14) and (4.15) to arrive at

    δn2+τ1αnm=2(mj=0ω(α)mjδj,δm)+τnm=2(δmθx,δmx)C(h2k+2+h2r+2)+Cτnm=0δm2+Cτnm=1ϑm2+δ12+δ02. (4.16)

    Combining Eqs (4.16), (4.9) with (4.11), we have the estimate

    δn2+ϑn2+ξnθ2+ξnθx2+τ1αnm=2(mj=0ω(α)mjδj,δm)+τnm=2(δmθx,δmx)C(h2k+2+h2r+2+τnm=2(Ψ1,nτρ2+ζnθ2)+Cτnm=0δm2+Cτnm=1ϑm2+C(ϑ12+δ12+ϑ02+δ02). (4.17)

    For the case n=1, we use a similar derivation to Eq (4.17) to get

    δ12+ϑ12+ξ1θ2+ξ1θx2+τ1α(1j=0ω(α)1jδj,δm)+τ(δ1θx,δ1x)C(h2k+2+h2r+2+h2m+2+ϑ02+δ02). (4.18)

    Combining Eq (4.17) with (4.18) and using Gronwall inequality, we have

    δn2+ϑn2+ξnθ2+ξnθx2+τ1αnm=0(mj=0ω(α)mjδj,δm)+τnm=1(δmθx,δmx)C(h2k+2+h2r+2+h2m+2). (4.19)

    Apply Lemmas 3.2–3.3 and combine Eqs (4.2), (4.4), (4.6) with triangle inequality to arrive at

    ˉu(tn)unh+ˉσ(tn)σnh+ˉv(tnθ)vnθhChmin{m+1,r+1,k+1}. (4.20)

    Combining Eq (4.20) with triangle inequality, we have

    u(tn)unh+σ(tn)σnh+v(tnθ)vnθhu(tn)ˉu(tn)+ˉu(tn)unh+σ(tn)ˉσ(tn)+ˉσ(tn)σnh+v(tnθ)ˉv(tnθ)+ˉv(tnθ)vnθhˉu(tn)unh+ˉσ(tn)σnh+ˉv(tnθ)vnθh+Cτ2, (4.21)

    which implies that we finish the proof.

    In this section, we will consider two numerical examples based on the linear element to validate our optimal theory results. In numerical experiments, we need to use the recursive formula provided in Lemma 2.1.

    In this test, we calculate the convergence rate in time and space. By taking the space domain ˉΩ=[0,1] and the time domain [0,1], the nonlinear term f(u)=u2 and the source term

    d(x,t)=(6t+6Γ(4β)t3β+3π2t2)sin(πx)+(t3sin(πx))2,

    with given initial and boundary conditions in Eq (1.1), we can validate easily that the exact solution is u=t3sinπx.

    In Table 1, with fixed space step length size h=11000 and changed time step length parameters τ=110,114,118, we arrive at the approximating time second-order convergence rate in L2-norm for three functions based on different parameters β=1.3,1.5,1.7 and θ=0.1,0.2,0.3. Similarly, by choosing the same parameters β and θ as the ones in Table 1 with space-time step length parameters τ=12000 and h=110,130,150, we calculate the approximating a priori error results with the second-order convergence rate in Table 2. The data computed in Tables 1 and 2 show the optimal convergence results are achieved by using our method.

    Table 1.  The errors and convergence rates in time with h=11000.
    β θ τ uuh Rate vvh Rate σσh Rate
    1/10 1.3016E-02 - 5.8621E-03 - 1.8795E-02 -
    0.1 1/14 6.7031E-03 1.9722 3.1046E-03 1.8891 9.8770E-03 1.9122
    1/18 4.0738E-03 1.9816 1.9155E-03 1.9214 6.0722E-03 1.9357
    1/10 1.0864E-02 - 5.1423E-03 - 1.6540E-02 -
    1.3 0.2 1/14 5.5735E-03 1.9837 2.6895E-03 1.9263 8.5751E-03 1.9524
    1/18 3.3800E-03 1.9901 1.6452E-03 1.9558 5.2236E-03 1.9723
    1/10 8.7318E-03 - 4.2714E-03 - 1.3791E-02 -
    0.3 1/14 4.4618E-03 1.9954 2.1909E-03 1.9842 7.0052E-03 2.0131
    1/18 2.6999E-03 1.9989 1.3219E-03 2.0104 4.2068E-03 2.0292
    1/10 1.3694E-02 - 7.6728E-03 - 2.5054E-02 -
    0.1 1/14 7.0334E-03 1.9802 4.0506E-03 1.8986 1.3074E-02 1.9329
    1/18 4.2656E-03 1.9899 2.4853E-03 1.9438 7.9743E-03 1.9674
    1/10 1.1421E-02 - 6.9048E-03 - 2.2668E-02 -
    1.5 0.2 1/14 5.8412E-03 1.9927 3.6073E-03 1.9296 1.1693E-02 1.9674
    1/18 3.5340E-03 1.9995 2.1967E-03 1.9738 7.0731E-03 2.0001
    1/10 9.1253E-03 - 5.9755E-03 - 1.9778E-02 -
    0.3 1/14 4.6449E-03 2.0070 3.0749E-03 1.9746 1.0033E-02 2.0170
    1/18 2.8024E-03 2.0105 1.8517E-03 2.0180 5.9965E-03 2.0480
    1/10 1.4835E-02 - 1.0860E-02 - 3.6502E-02 -
    0.1 1/14 7.6140E-03 1.9822 5.8365E-03 1.8454 1.9333E-02 1.8888
    1/18 4.6125E-03 1.9944 3.6181E-03 1.9027 1.1890E-02 1.9345
    1/10 1.2430E-02 - 1.0006E-02 - 3.3861E-02 -
    1.7 0.2 1/14 6.3535E-03 1.9946 5.3398E-03 1.8665 1.7790E-02 1.9128
    1/18 3.8395E-03 2.0042 3.2932E-03 1.9232 1.0879E-02 1.9572
    1/10 9.9846E-03 - 8.9741E-03 - 3.0669E-02 -
    0.3 1/14 5.0785E-03 2.0092 4.7432E-03 1.8951 1.5940E-02 1.9450
    1/18 3.0599E-03 2.0159 2.9049E-03 1.9510 9.6719E-03 1.9879

     | Show Table
    DownLoad: CSV
    Table 2.  The errors and convergence rates in space with τ=12000.
    β θ h uuh Rate vvh Rate σσh Rate
    1/10 1.6537E-02 - 4.9369E-02 - 3.2169E-01 -
    0.1 1/30 1.9382E-03 1.9514 5.7725E-03 1.9536 3.6219E-02 1.9880
    1/50 7.0494E-04 1.9800 2.0990E-03 1.9804 1.3053E-02 1.9978
    1/10 1.6537E-02 - 4.9369E-02 - 3.2169E-01 -
    1.3 0.2 1/30 1.9383E-03 1.9514 5.7725E-03 1.9536 3.6219E-02 1.9880
    1/50 7.0499E-04 1.9799 2.0990E-03 1.9804 1.3053E-02 1.9978
    1/10 1.6537E-02 - 4.9369E-02 - 3.2169E-01 -
    0.3 1/30 1.9383E-03 1.9514 5.7725E-03 1.9536 3.6219E-02 1.9880
    1/50 7.0505E-04 1.9798 2.0990E-03 1.9804 1.3053E-02 1.9978
    1/10 1.6445E-02 - 4.9220E-02 - 3.0753E-01 -
    0.1 1/30 1.9279E-03 1.9512 5.7568E-03 1.9533 3.4563E-02 1.9896
    1/50 7.0124E-04 1.9799 2.0934E-03 1.9803 1.2451E-02 1.9986
    1/10 1.6445E-02 - 4.9220E-02 - 3.0753E-01 -
    1.5 0.2 1/30 1.9280E-03 1.9511 5.7568E-03 1.9533 3.4563E-02 1.9896
    1/50 7.0130E-04 1.9797 2.0934E-03 1.9803 1.2451E-02 1.9986
    1/10 1.6445E-02 - 4.9220E-02 - 3.0753E-01 -
    0.3 1/30 1.9281E-03 1.9511 5.7568E-03 1.9533 3.4563E-02 1.9896
    1/50 7.0136E-04 1.9796 2.0934E-03 1.9803 1.2451E-02 1.9986
    1/10 1.6339E-02 - 4.9077E-02 - 2.9293E-01 -
    0.1 1/30 1.9161E-03 1.9509 5.7420E-03 1.9530 3.2857E-02 1.9914
    1/50 6.9697E-04 1.9797 2.0883E-03 1.9801 1.1831E-02 1.9995
    1/10 1.6339E-02 - 4.9077E-02 - 2.9293E-01 -
    1.7 0.2 1/30 1.9161E-03 1.9508 5.7420E-03 1.9530 3.2857E-02 1.9914
    1/50 6.9704E-04 1.9796 2.0883E-03 1.9801 1.1831E-02 1.9995
    1/10 1.6339E-02 - 4.9077E-02 - 2.9293E-01 -
    0.3 1/30 1.9162E-03 1.9508 5.7420E-03 1.9530 3.2857E-02 1.9914
    1/50 6.9710E-04 1.9795 2.0883E-03 1.9801 1.1831E-02 1.9995

     | Show Table
    DownLoad: CSV

    In this numerical example, we consider the same space-time domain and the nonlinear term as shown in the first example, and choose the exact solution u=t3sin(3πx)x+1 with the corresponding source term. By choosing the space parameter h=1/30, time step length size τ=1/200, fractional parameter β=1.7 and shifted parameter θ=0.3, we obtain the comparison in Figures 13 between the figures of numerical solutions and the figures of exact solutions at t=0.25,0.5,0.75,1, from which one can visually see the approximation effect.

    Figure 1.  Comparison between u and uh at different time t.
    Figure 2.  Comparison between v and vh at different time t.
    Figure 3.  Comparison between σ and σh at different time t.

    Remark 5.1. 1) From these two examples, readers can see that the linear basis functions for three finite element spaces are used. In this article, the presented time second-order fully discrete mixed finite element scheme is derived by combining Pani's space H1-mixed element method with a time second-order SCQ formula, so it also does not need to meet the LBB condition. Further, the degrees with k, m and r of three polynomial basis functions can be freely selected.

    2) By introducing two auxiliary variables, the original problem is transformed into a low order coupled system in space-time directions. In this case, many efficient numerical approximation schemes in time for solving this system can be constructed. From the computational point of view, there are some small differences among them. However, the related technical difficulty of theoretical analysis by using different approximation technique is even a big difference, which will bring many challenges to researchers. For example, in this article, the positive definite property is used for analyzing the stability and error estimate, which differs from the iterative technique shown in [38].

    From the data computed by our fully discrete SCQ mixed element method, one can see clearly that the convergence orders for both space and time are optimal, which is in agreement with our theory result. With a comparison to the standard Galerkin finite element method for directly solving the studied fractional wave model, the advantage of this method is that three unknown functions can be approximated simultaneously. However, the computing time problem is its limitations, which urges us to further study the fast computing technology based on this method.

    In the future, we will extend this method to solve multidimensional fractional wave models and multi-term time fractional wave equations [49], and consider other SCQ formulas [46,50] and their numerical theories.

    Authors thank four anonymous reviewers and editors very much for their valuable comments and suggestions for improving our work. This work is supported by the National Natural Science Foundation of China (12061053, 12161063), Natural Science Foundation of Inner Mongolia (2020MS01003, 2021MS01018), Young Innovative Talents Project of Grassland Talents Project, Program for Innovative Research Team in Universities of Inner Mongolia Autonomous Region (NMGIRT2207) and Scientific Research Projection of Higher Schools of Inner Mongolia (NJZY21266).

    The authors declare there is no conflicts of interest.



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