# L1-Multiscale Galerkin’s Scheme with Multilevel Augmentation Algorithm for Solving Time Fractional Burgers’ Equation

Authors: Jian Chen; Yong Huang; Taishan Zeng
Journal: Journal of Function Spaces
 (2021)

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## Abstract

In this paper, we consider the initial boundary value problem of the time fractional Burgers equation. A fully discrete scheme is proposed for the time fractional nonlinear Burgers equation with time discretized byL1-type formula and space discretized by the multiscale Galerkin method. The optimal convergence orders reach Oτ2−α+hr in the L2 norm and Oτ2−α+hr−1 in the H1 norm, respectively, in which τ is the time step size, h is the space step size, and r is the order of piecewise polynomial space. Then, a fast multilevel augmentation method (MAM) is developed for solving the nonlinear algebraic equations resulting from the fully discrete scheme at each time step. We show that the MAM preserves the optimal convergence orders, and the computational cost is greatly reduced. Numerical experiments are presented to verify the theoretical analysis, and comparisons between MAM and Newton’s method show the efficiency of our algorithm.


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## Body

## 1. Introduction

In this paper, we consider the following time fractional Burgers equation [1–7]:
(1)D0ctαux,t+ux,tuxx,t−uxxx,t=fx,t,x,t∈Ω,with the initial and boundary conditions, given by
(2)ux,0=u0x,0≤x≤1,u0,t=u1,t=0,0<t≤T,where 0<α<1,Ω=x,t∣0≤x≤1,0<t≤T,u0x and fx,t are given functions, and the notation 0cDtα denotes the Caputo fractional partial derivative of order α, defined by
(3)D0ctαux,t=1Γ1−α∫0t1t−sα∂ux,s∂sds,α∈0,1,in which Γ· represents the Gamma fuction.The time fractional Burgers equation is a kind of nonlinear subdiffusion convection equation occurring in several physical problems such as unidirectional propagation of weakly nonlinear acoustic waves through a gas-filled pipe, propagation of weak shock, compressible turbulence shallow-water waves, shock waves in a viscous medium, waves in bubbly liquids, and electromagnetic waves [1, 4, 8]. Till now, there have been several analytical techniques developed to solve the time fractional Burgers equation. These methods include the Cole-Hopf transformation, Laplace transform, variable separation method [8], Adomian decomposition method [4], homotopy analysis method [6], and so on.However, even if some fractional differential equations can be solved, the expressions of their exact solutions are often expressed by special functions, which are difficult to apply in practice. Moreover, due to the nonlocality of the fractional operators, analytical methods do not always work well on most fractional differential equations in real applications. Hence, it is of great importance to develop reliable and efficient numerical methods for solving fractional differential equations. Nowadays, the numerical methods cover the quadratic B-spline Galerkin method [2], cubic B-spline finite element method [3], finite difference methods [1, 9–13], and Fourier pseudospectral schemes [14]. There are also some other numerical methods (see, for example, [8, 15–17]).In this paper, we first present a fully discrete scheme for solving the time fractional Burgers equation with the time approximated by theL1-type formula and the space discretization based on the multiscale Galerkin method. We give rigorous convergence analysis for the fully discrete scheme, which shows that the scheme enjoys the optimal convergence order Oτ2−α+hr in the L2 norm and Oτ2−α+hr−1 in the H1 norm, respectively, where τ,h, and r are the time step size, space step size, and the order of piecewise polynomial space, respectively. Since the time fractional Burgers equation is a nonlinear differential equation, the fully discrete scheme results in a system of nonlinear algebraic equation at each time step. Iteration methods such as the Newton iteration method and the quasi-Newton iteration method are often employed to solve these nonlinear equations. In this case, a large amount of computational effort is demanded to compute and update the Jacobian matrix in each iteration process. The higher accuracy of the approximate solution is required, the larger dimension of the subspace is needed, and the longer computational time is consumed. To overcome this problem, we develop the multilevel augmentation method for solving the fully discrete scheme. The MAM solves a nonlinear equation at a high level consisting of two parts: solving the nonlinear equation only in a fixed initial subspace with the dimension much lower than that of the whole approximate subspace; compensating the error by matrix-vector multiplications at the high level. The MAM reduces the computational costs significantly and leads to a fast solution for the fully discrete scheme. We prove that the MAM preserves the same optimal convergence order as the original fully discrete scheme. The idea of MAM was first introduced in [18] for solving the linear Fredholm integral equations of the second kind. The theoretical setting of MAM was established by Chen et al. in [19] for solving operator equations covering both first kind and second kind equations; they further develop MAM for solving the nonlinear Hammerstein integral equation in [20]. We modified the framework and extended the idea of MAM to solve general nonlinear operator equations of the second kind and applied it to the Sine-Gordon equation in [21]. Readers are referred to [22–27] and the references therein for more applications of MAM.This paper is organized in seven sections. In “Preliminaries,” some necessary notations, multiscale orthonormal bases in Sobolev space, and useful lemmas are introduced. In “L1 Scheme for Discretization of Caputo Derivative in Time,” we introduce the L1-formula for time discretization. In “Fully Discrete Scheme and Convergence,” a fully discrete scheme for time fractional Burgers equation is established, and the convergence analysis are given. The MAM and its convergence analysis are developed in “Multilevel Augmentation Method for Solving the Fully Discrete Scheme.” The numerical experiments are provided in “Numerical Experiments” to verify the theoretical estimates. Finally, a conclusion is included in “Conclusion.”


## 2. Preliminaries

DenoteI=0,1. Let ·,· stand for the inner product on the space L2I with the L2 norm ·2. We denote by H01I the Sobolev space of elements u satisfying the homogeneous boundary conditions that u0=u1=0. The inner product and norm of H01I are defined by
(4)u,v≔u′,v′=∫01u′xv′xdx,u,v∈H01I,u1≔u,u,u∈H01I,respectively. Let n be a positive integer, we denote by Xn the subspace of H01I whose elements are the piecewise polynomials of order r with knots j/2n,j−1∈ℤ2n−1, where the notation ℤn≔0,1,2⋯,n−1. Obviously, the sequence of Xn is nested, that is
(5)Xn⊂Xn+1,n∈ℕ0≔0,1,2⋯,which yields the following decomposition:
(6)Xn=Xn−1⊕⊥Wn=X0⊕⊥W1⊕⊥W2⊕⊥⋯⊕⊥Wn,where Wn is the orthogonal complement of Xn−1 in Xn.It is easily concluded from the definition ofXn and Wn that the dimensions of Xn and Wn are given byxn≔dimXn=r−12n−1 and wn≔dimWn=xn−xn−1=r−12n−1,respectively.Define two affine mappings on the intervalI by φ0x=x/2 and φ1x=x+1/2,x∈I, which map the interval 0,1 into 0,1/2 and 1/2,1, respectively. Associated with the two mappings, we introduce two linear operators as follows:
(7)T0ux≔22uφ0−1x=22u2x,T1ux≔22uφ1−1x=22u2x−1.Lemma 1 (see [26]).
Letwij,j∈ℤwi, be an orthonormal basis of Wi,i≥1. Then the functions T0wij,T1wij:j∈ℤwi form an orthonormal basis for Wi+1.Lemma1 shows that the space Wn can be recursively constructed by the linear operators T0 and T1 once W1 has been given. Therefore, the basis of the space Xn can be constructed by Lemma 1 step by step. For the details of the construction and more, the readers can refer to [26].LetPn be an orthogonal projection operator from H01I into Xn with respect to the inner product ·,·, that is, for all u∈H01I,(8)Pnu,v=u,v,v∈Xn,or
(9)∂xPnu,∂xv=∂xu,∂xv,v∈Xn.The following approximation results on the operatorPn will be used later. Throughout this paper, unless stated otherwise, c denotes a generic positive constant whose value may differ in different occurrences.Lemma 2 (see [28]).
Ifu∈HrI∩H01I, then
(10)u−Pnu2≤chrur,u−Pnu1≤chr−1ur,where h≔2−n.


## 3.L1 Scheme for Discretization of Caputo Derivative in Time

For a positive integerN, let τ=T/N be the time step size and ti=iτ for i=0,1,⋯,N. Let ui be the solution of ux,t on t=ti.Define(11)ui+12≔ui+1+ui2andδtui+12≔ui+1−uiτ.For the approximation of fractional derivativeD0ctαgti, we use the following L1 scheme [29, 30]:
(12)D0ctαgti≃Dταgti≔μa0gti−∑k=1i−1ai−k−1−ai−kgtk−ai−1gt0,where 0<α<1,μ=τ−α/Γ2−α and ak=k+11−α−k1−α.Lemma 3 (see [30]).
If0<α<1 and ak=k+11−α−k1−α,k∈ℕ0, then
(13)1=a0>a1>a2>⋯>ai>⋯⟶0,ask⟶∞,1−αk+1−α<ak<1−αk−α.Lemma 4 (see [30]).
Suppose0<α<1,gt∈C20,ti. Let
(14)Rgti≔D0ctαgti−Dταgti.Then(15)∣Rgti∣≤1Γ2−α1−α12+22−α2−α−1+2−αmax0≤t≤tig′′tτ2−α.


## 4. Fully Discrete Scheme and Convergence

In this section, we present a fully discrete scheme for the time fractional Burgers equation (1), and we derive the error estimates and convergence of the proposed fully discrete scheme. The Galerkin method associated with the multiscale basis introduced in “Preliminaries” is employed to discretize the spatial variable. The fully discrete scheme in weak formulation for (1) reads as follows: for each t=ti,i=1,2⋯n, find uni∈Xn such that
(16)Dταuni,vn+unxi,vnx+uniunxi,vn=fi,vn,vn∈Xn,un0=Πnu0,where fi=fx,tt=ti and Πn denotes the interpolation operator.We present an optimal error estimate of the fully discrete scheme (16) in the following theorem.Theorem 5.
Suppose that the problem (1)–(2) has a unique solution u∈C20,T;L2Ω∩C10,T;HrΩ. Then
(17)ui−uni2≤cτ2−α+hr,where ui=ux,tt=ti.Proof.
Denoteeni≔Pnui−uni,i=0,1,2⋯N. We conclude from (1) and (16) that eni satisfies
(18)Dταeni,vn+enxi,vnx=DταPnui−ui,vn+Dταui−0cDtiαu,vn+uiuxi−uniunxi,vn.
Takingvn=eni in (18), we have
(19)Dταeni,eni+eni12=Pn−IDταui,eni+Dταui−0cDtiαu,eni+uiuxi−uniunxi,eni.
We estimate the terms of the right-hand side of (19) one by one. For the first term in right-hand side of (19), using the Cauchy-Schwarz inequality and Lemma 2, we have
(20)Pn−IDταui,eni≤chrDταui2eni2≤ch2rDταui22+12eni22.
To estimate the second term in the right-hand side of (19), we conclude from Lemma 4 that
(21)Dταui−0cDtiαu,eni≤Dταui−0cDtiαu2eni2≤cτ4−2r+12eni22.
For the last term in the right-hand side of (19), using integration by parts and the Cauchy-Schwartz inequality, we have
(22)uiuxi−uniunxi,eni=12∫Iui−uniui+unienxidx≤Mui−uni2eni1≤M2ui−uni22+eni12≤M2ui−Pnui22+M2eni22+eni12≤ch2r+M2eni22+eni12,where ui≤M and uni≤M, due to the smoothness of u and the approximation un∈Xn⊂H01I.
On the other hand(23)Dταeni,eni=τ−αΓ2−αa0eni−∑k=1i−1ai−k−1−ai−kenk−ai−1en0,eni,=a0τ−αΓ2−αeni22−τ−αΓ2−α∑k=1i−1ai−k−1−ai−kenk+ai−1en0,eni.
Substituting (20)–(23) into (19) and noting that 1=a0>a1>⋯>ai>⋯, we deduce that
(24)τ−αΓ2−αeni22≤τ−αΓ2−α∑k=1i−1ai−k−1−ai−k∥enk∥22+∥eni∥222+ai−1∥en0∥22+∥eni∥222+M2+1eni22+cτ2−α+hr2=τ−α2Γ2−αeni22+∑k=1i−1enk22+en022+M2+1eni22+cτ2−α+hr2.
Chooseτ such that τ−α/2Γ2−α>M/2+1, and denote σ=τ−α/2Γ2−α,λ=τ−α/2Γ2−α−M/2+1; then, we have
(25)eni22≤σλ∑k=1i−1enk22+σλen022+cτ2−α+hr2.
By Gronwall’s inequality, we have(26)eni2≤cen022+cτ2−α+hr,which, together with Lemma 2 and the initial error estimate, yields that
(27)ui−uni2≤ui−Pnui2+eni2≤cτ2−α+hr.
This completes the proof.Remark 6.
If we choosevn=Dταeni in (19) and make a similar analysis as the above Theorem 5, we can obtain the optimal convergence order in H1 norm
(28)ui−uni1≤cτ2−α+hr−1.


## 5. Multilevel Augmentation Method for Solving the Fully Discrete Scheme

At each time step, the fully discrete scheme (16) leads to a nonlinear system, which makes the computational cost expensive. We present a fast multilevel augmentation method in this section to solve these nonlinear systems. To this end, we rewrite (16) into
(29)unxi,vnx+μuni+uniunxi,vn=fi+μrni,vn,where rni=∑k=1i−1ai−k−1−ai−kunk+ai−1un0 and μ=τ−α/Γ2−α.Define a nonlinear operatorK:X⟶X as follows:
(30)Ku,v1≔μu+uux,v,v∈X.Similar to the proof of Lemma3 in [23], we applied the Riesz representation theorem to the right-hand side of (29); there exists a element f~i∈X, such that
(31)f~i,v1=fi+μrni,v,∀v∈Xn.Then, Equation (29) can be reformulated as
(32)uni,vn1+Kuni,vn1=f~i,vn,vn∈Xn,or equivalently
(33)I+PnKuni=Pnf~i.Since Equation (16) has been reformulated as a nonlinear operator equation of the second kind (33), and K has the properties (P1) and (P2) described in [23], then MAM developed in [21] is applicable.We now briefly describe the MAM for solving (33). As we presented in “Preliminaries,” the approximation subspace sequence is nested, for a fixed positive integer k,n≔k+m,m is any nonnegative integer, and we have the following decomposition:
(34)Xk+m≔Xk⊕⊥Wk,m,withWk,m≔Wk+1⊕⊥Wk+2⊕⊥⋯⊕⊥Wk+m.Now, we are in a position to solve (33) with n≔k+m, and k is fixed and smaller than n. Firstly, we solve (33) with n≔k exactly and obtain uki. Next, we obtain an approximation of uk+1i of (33) with n≔k+1. To this end, we decomposeuk+1i=uk+1i,L+uk+1i,H, with uk+1i,L∈Xk,uk+1i,H∈Wk+1.With the help of (34), Equation (33) with n≔k+1 can be rewritten as an equivalent form as
(35)Pk+1−Pkuk+1i,L+uk+1i,H−Pk+1−PkKuk+1i=Pk+1−Pkf~i,(36)PkI−Kuk+1i,L+uk+1i,H=Pkf~i.Note that(37)Pk+1−Pkuk+1i,L+uk+1i,H=uk+1i,H.Equation (35) becomes
(38)uk+1i,H=Pk+1−Pkf~i+Kuk+1i.Theuk+1i in the right-hand side can be approximated by the previous level solution uk,0i≔uki. We compute
(39)uk,1i,H=Pk+1−Pkf~i+Kuk,0i∈Wk+1.Replaceuk+1i,H in (36) by uk,1i,H, and solve uk,1i,L∈Xk from
(40)PkI−Kuk,1i,L+uk,1i,H=Pkf~i.Let(41)uk,1i≔uk,1i,L+uk,1i,H,which is an approximation to the solution uk+1i.This procedure is repeatedm times to obtain an approximation uk,mi of the solution uk+mi of (33) with n=k+m. The solution uk,mi is called a multilevel augmentation solution. Since at any step l=0,1,⋯m, we only need to invert the same nonlinear operator PkI−K with a fixed small k instead of the nonlinear operator Pk+lI−K. This means the algorithm has a high computational efficiency. At every time step, the fully discrete scheme (33) is solved by the MAM, and the whole process can be summarized as the following algorithm:Algorithm 1: (MAM for time fractional Burgers equation).
Letk,m be two fixed positive integers and n≔k+m.Step 1: obtain the approximation of initial value functionun0≔Pnu0x=∑i,j∈Jncijwijx,wherecij=u0x,wij1,i,j∈Jn.Step 2: fori=1:N (T=Nτ), do the following:(I): Solveuki∈Xk from (33) with n≔k. Set uk,0i≔uki,l=1(II): Computeuk,li,H≔Pk+l−Pkf~i−Kuk,l−1i, that isuk,li,H=∑i,j∈Jk,mcijwijx,whereJk,m=Jk+m\Jk,cij=f~i−Kuk,l−1i,wij1 withf~i,wij1≔fi,wij+μrk,mi,wij.(III): solveuk,li,L∈Xk from the following equationPkI+Kuk,li,L+uk,li,H=Pkf~i,that isI+Kuk,li,L+uk,li,H,wij1=f~i,wij1,foralli,j∈Jk.(IV): Letuk,li≔uk,li,L+uk,li,H. Set l←l+1 and go back to (III) until l=m..⋯Theorem 7.
Letu be the exact solution of (1) and uk,mi be the approximation solution obtained by Algorithm 1. Suppose that the solution of Equation (33) belongs to HrI for i=1,2,⋯T/τ. Then, there exist a positive integer N such that for all k≥N and m∈ℕ(42)ui−uk,mi2≤Cτ2−α+hr.Proof.
As stated in [20, 21, 23], uk,mi is the solution of the equation:
(43)I+PkKuk,mi=Pk+mf~i−Pk+m−PkKuk,l−1i,which is equivalent to the following discrete form:
(44)uk,m,xi,vx+μuk,mi+uk,miuk,m,xi,v=μuk,mi−uk,m−1i,v2+fi,v+μrk,mi,v+uk,miuk,m,xi−uk,m−1iuk,m−1,xi,v2,where v∈Xk+m,v2=Pk+m−Pkv. Rearranging the terms, we have
(45)Dταuk,mi,v+uk,m,xi,vx=−uk,miuk,m,xi,v+μuk,mi−uk,m−1i,v2+uk,miuk,m,xi−uk,m−1iuk,m−1,xi,v2+fi,v.
Noting that the exact solutionu at t=ti satisfies
(46)Dταui,v+uxi,vx=−uiuxi,v+fi,v+Dταui−0cDtiαu,v.
Subtracting (45) from (46), we obtain that for all v∈Xk+m,v2=Pk+m−Pkv(47)Dταui−uk,mi,v+uxi−uk,m,xi,vx=−uiuxi−uk,miuk,m,xi,v+Dταui−0cDtiαu,v−μuk,mi−uk,m−1i,v2−uk,miuk,m,xi−uk,m−1iuk,m−1,xi,v2.
Denoteρi≔ui−Pk+mui and ei≔Pk+mui−uk,mi, then ui−uk,mi=ρi+ei. Using these notations and noting that ρxi,vx=0, we derive the error equation as follows:
(48)Dταei,v+exi,vx=Dταρi,v+Dταui−0cDtiαu,v−uiuxi−uk,miuk,m,xi,v−μuk,mi−uk,m−1i,v2−uk,miuk,m,xi−uk,m−1iuk,m−1,xi,v2.
LetM1=maxuk,mi∞,i∈ℤN and M′=maxM1,M, where M is the positive constant appearing in (22); then, ui∞≤M′,uk,mi∞≤M′. We take v=ei in (48) and estimate the terms in the right-hand side of (48).
For the first three terms in the right-hand side of (48), similar to the analysis of (20)–(22), we have
(49)Dταρi,ei=Pk+m−IDταui,ei≤ch2rDταui22+12ei22,(50)Dταui−0cDtiαu,ei≤cτ4−2r+12ei22,(51)uiuxi−uk,miuk,m,xi,ei≤ch2r+M′ei22+12ei12.
By the Cauchy-Schwartz inequality, Young’s inequality, and noting thatv22≤v2, we have
(52)−μuk,mi−uk,m−1i,v2≤μ2σ1uk,mi−uk,m−1i22+σ14ei22.
For the last term in the right-hand side of (48), it follows from integration by parts, the Cauchy-Schwartz inequality, and the Young inequality that
(53)uk,miuk,m,xi−uk,m−1iuk,m−1,xi,v2≤2M′uk,mi−uk,m−1i2ei1≤M′2uk,mi−uk,m−1i22+12ei12.
On the other hand side, as presented in (23), we have
(54)Dταei,ei≤ταΓ2−αei22−ταΓ2−α∑k=1i−1ai−k−1−ai−kek+ai−1e0,ei.
Combining (49)–(54) and exi,exi=ei12, we have
(55)τ−αΓ2−αei22≤τ−αΓ2−α∑k=1i−1ai−k−1−ai−k∥ek∥22+∥ei∥222+ai−1∥e0∥22+∥ei∥222+1+M′+σ14ei22+μ2σ1uk,mi−uk,m−1i22+cτ2−α+hr2=τ−α2Γ2−αei22+∑k=1i−1ek22+e022+1+M′+σ14ei22+μ2σ1uk,mi−uk,m−1i22+cτ2−α+hr2.
Chooseτ such that τ−α/2Γ2−α>1+M′+σ1/4, and denote σ¯=τ−α/2Γ2−α,μ¯≔σ¯−1+M′+σ1/4. Then, we have
(56)ei22≤σ¯μ¯∑k=1i−1ek22+e022+τ4−2α4μ¯Γ22−α+2M′2μ¯uk,mi−uk,m−1i22+cτ2−α+hr2.
When the exact solution of (33) belongs to HrI, there exists a positive integer N, for all k≥N and any m∈ℕ (see [21, 23]):
(57)uk,mi−uk,m−1i≤chr.
Combining (56) and (57), we conclude from Gronwall’s inequality that
(58)ei2≤e02+cτ2−αhr+hr+cτ2−α+hr≤∥e0∥2+cτ2−α+hr.
Noting thate02≤chr, then
(59)ui−uk,mi2≤ui−Pk+mui2+ei2≤Cτ2−α+hr.
This completes the proof.Remark 8.
If we choosev=Dταei in (48) and make a similar analysis as the above Theorem 7, we can obtain the optimal convergence order in H1 norm:
(60)ui−uk,mi1≤cτ2−α+hr−1.


## 6. Numerical Experiments

We present in this section numerical examples to illustrate the efficiency and accuracy of our proposed method. The computer programs are run on a personal computer with 2.5G CPU and 8G memory.Example 1.
We consider the time fractional Burgers equation (1) with the exact solution:
(61)ux,t=4t2−4t+1x2x−12.The corresponding initial condition and forcing term are(62)ux,0=u0x=x2x−12fx,t=8t2−αΓ3−α−4t1−αΓ2−αx2x−12+24t2−4t+12x3x−132x−1−24t2−4t+16x2−6x+1.Both piecewise linear (r=2) and quadratic (r=3) multiscale orthonormal bases introduced in “Preliminaries” are employed in our numerical approximation. The numerical results are reported in Tables 1–5. k and m stand for the numbers of initial level and augmentation level used in the MAM, respectively. xn denotes the dimension of approximation subspace Xn with n=k+m. Table 1 shows the L2 errors and temporal convergence rates for different α using the MAM with k,m=3,7 for linear basis and k,m=2,5 for quadratic basis. It is seen that our numerical scheme has an accuracy of 2−α, which is in agreement with our theoretical analysis. In the spatial direction, we illustrate the accuracy, convergence order, and computational efficiency of the MAM, with a comparison to those of the direct Newton’s method (DNM) for solving the fully discrete scheme (16). The numerical results listed in Tables 2 and 3 are linear basis cases, and Tables 4 and 5 are quadratic basis cases. We can easily see from these tables that both MAM and DNM have the optimal convergence orders in the H1 norm (1 for the linear case and 2 for the quadratic case) and in the L2 norm (2 for the linear case and 3 for the quadratic case). We also observe that MAM and DNM have nearly the same accuracy, while MAM takes significantly less time than DNM. To intuitively show the approximation effect, we plot in Figure 1 the absolute error surface of the approximation solution u2,5 obtained by MAM.Example 2.
We consider the time fractional Burgers equation (1) with initial condition:
(63)u0x=0,fx,t=2t2−αsin2πxΓ3−α+πt4sin4πx+4π2t2sin2πx.Table 1
Errors and convergent orders of MAM in temporal direction for Example1.
ατLinear basis (k=3,m=7)Quadratic basis (k=2,m=5)u∗−u3,72Rateu∗−u2,52Rate1/31/43.8885e-43.8871e-41/81.3023e-41.57811.3012e-41.57891/164.2962e-51.59994.2893e-51.60101/321.3946e-51.62331.3985e-51.61681/644.2759e-61.70554.5237e-61.62831/1281.3118e-61.70471.4543e-61.63721/21/47.8755e-47.8742e-41/82.8852e-41.44872.8850e-41.44861/161.0421e-41.46921.0448e-41.46541/323.7337e-51.48083.7550e-51.47631/641.3340e-51.48481.3427e-51.48371/1284.5749e-61.54394.7858e-61.48833/41/41.8737e-31.8737e-31/87.9862e-41.23037.9879e-41.23001/163.3825e-41.23943.3833e-41.23941/321.4287e-41.24341.4285e-41.24401/646.0343e-51.24346.0206e-51.24651/1282.5613e-51.23632.5352e-51.2478Table 2
Errors and convergent orders of MAM and DNM via linear basis in spatial direction withk=3,τ=0.001, and α=0.25 for Example 1.
mxnMAMDNMu∗−u3,m1Rateu∗−u3,m2Rateu∗−un1Rateu∗−un2Rate073.1264e-21.2161e-33.1264e-21.2161e-31151.7726e-20.81863.1564e-41.94611.6011e-20.96543.1282e-41.95902318.9080e-30.99277.7686e-52.02258.0530e-30.99157.8743e-51.99013634.4596e-30.99821.9377e-52.00334.0324e-30.99791.9714e-51.997941272.2305e-30.99964.8450e-61.99982.0169e-30.99954.9263e-62.000652551.1153e-30.99991.2132e-61.99761.0086e-30.99991.2342e-61.997065115.5768e-41.00002.8891e-72.07025.0429e-41.00003.3678e-71.8737Table 3
Comparison of CPU time between MAM and DNM via linear basis withk=3,τ=0.001, and α=0.25 for Example 1.
n=k+mxnTMAMTDNM3=3+071.331.284=3+1153.213.705=3+2317.0912.226=3+36315.4541.967=3+412741.47152.718=3+5255112.52570.639=3+6511340.874581.64Table 4
Errors and convergent orders of MAM and DNM via quadratic basis in spatial direction withk=2,τ=0.001, and α=0.1 for Example 1.
mxnMAMDNMu∗−u2,m1Rateu∗−u2,m2Rateu∗−un1Rateu∗−un2Rate071.5670e-26.0316e-41.5670e-26.0316e-41154.0566e-31.94968.1280e-52.89164.0054e-31.96797.7216e-52.96562311.0196e-31.99221.0218e-52.99181.0068e-31.99229.7080e-62.99173632.5528e-41.99791.2796e-62.99732.5203e-41.99811.2151e-62.998041276.3843e-51.99951.5919e-73.00706.3030e-51.99951.5096e-73.008952551.5962e-51.99992.1784e-82.86941.5759e-51.99991.7198e-83.1339Table 5
Comparison of CPU time between MAM and DNM via quadratic basis withk=2,τ=0.001, and α=0.1 for Example 1.
n=k+mxnTMAMTDNM2=2+072.442.483=2+1155.188.914=2+2318.4931.605=2+36314.04128.976=2+412723.16518.637=2+525543.772506.75Figure 1
The graph of error functionux,t−u2,5x,t for Example 1.The exact solution of this problem is(64)ux,t=t2sin2πx.The numerical results are presented in Figure2 and Tables 6–9, where Figure 2 displays the convergence orders in temporal direction with different α, and Tables 6–9 show the accuracy, convergence order, and computing time for the spatial direction. All the numerical results verify our theoretical analysis and also show the efficiency of the proposed algorithm.Figure 2
The temporal convergence order of MAM for Example2.Table 6
Errors and convergent orders of MAM and DNM via linear basis in spatial direction withk=3,τ=0.005, and α=0.25 for Example 2.
mxnMAMDNMu∗−u3,m1Rateu∗−u3,m2Rateu∗−un1Rateu∗−un2Rate079.9718e-13.8317e-29.9718e-13.8317e-21155.1444e-10.95487.8333e-32.29035.0239e-10.98919.6573e-31.98832312.5795e-10.99591.9554e-32.00212.5167e-10.99732.4190e-31.99723631.2907e-10.99904.8547e-42.01001.2589e-10.99936.0479e-41.999941276.4546e-20.99981.2093e-42.00526.2955e-20.99981.5096e-42.002252553.2274e-20.99992.9983e-52.01203.1478e-21.00003.7490e-52.009665111.6137e-21.00007.2066e-62.05681.5739e-21.00009.1214e-62.0392710238.0687e-31.00001.5895e-62.18087.8696e-31.00002.0338e-62.1651Table 7
Comparison of CPU time between MAM and DNM via linear basis withk=3,τ=0.005, and α=0.25 for Example 2.
n=k+mxnTMAMTDNM3=3+070.290.514=3+1150.781.085=3+2311.673.756=3+3633.8613.187=3+41279.8248.168=3+525527.78179.339=3+651187.87716.0210=3+71023289.367004.77Table 8
Errors and convergent orders of MAM and DNM via quadratic basis in spatial direction withk=2,τ=0.005, and α=0.1 for Example 2.
mxnMAMDNMu∗−u2,m1Rateu∗−u2,m2Rateu∗−un1Rateu∗−un2Rate073.9445e-11.5119e-23.9445e-11.5119e-21151.0367e-11.92792.0779e-32.86311.0124e-11.96201.9499e-32.95492312.9991e-21.78933.8616e-42.42792.5478e-21.99052.4561e-42.98893637.5607e-31.98795.0534e-52.93396.3800e-31.99763.0758e-52.997341271.8938e-31.99726.3432e-62.99401.5957e-31.99943.8482e-62.998752554.7367e-41.99938.1030e-72.96873.9896e-41.99995.0488e-72.9301Table 9
Comparison of CPU time between MAM and DNM withk=2,τ=0.005, and α=0.1 for Example 2.
n=k+mxnTMAMTDNM2=2+070.570.573=2+1151.332.024=2+2312.167.645=2+3633.4332.326=2+41275.89129.807=2+525510.34543.35


## 7. Conclusion

In this article, theL1-discretization formula and the multiscale Galerkin method are adopted to discrete the Caputo fractional derivative and spatial variable, respectively, and the multilevel augmentation algorithm is proposed for solving the resulting fully discrete scheme which is a nonlinear system at each time step. The MAM only needs to solve nonlinear systems in a fixed subspace with much lower dimension than that for the whole approximation subspace and compensate the error by multiplications of matrices and vectors at the high level. Therefore, the computational cost is greatly reduced. Numerical experiments are presented to confirm our theoretical results. Compared with the DNM, the proposed MAM has substantial advantages in computing time and is suitable for solving large-scale and high-accuracy problems.



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*Source: 5581102-2021-03-02.xml*