Local,Discontinuous,Galerkin,Scheme,for,Space,Fractional,Allen-Cahn,Equation

Can Li · Shuming Liu

Abstract This paper is concerned with the effi cient numerical solution for a space fractional Allen-Cahn (AC) equation. Based on the features of the fractional derivative, we design and analyze a semi-discrete local discontinuous Galerkin (LDG) scheme for the initial-boundary problem of the space fractional AC equation. We prove the optimal convergence rates of the semi-discrete LDG approximation for smooth solutions. Finally, we test the accuracy and effi ciency of the designed numerical scheme on a uniform grid by three examples.Numerical simulations show that the space fractional AC equation displays abundant dynamical behaviors.

Keywords Fractional Allen-Cahn equation · Local discontinuous Galerkin scheme · Error estimates

In this paper, we are concerned with the effi cient semi-discrete local discontinuous Galerkin (LDG) scheme for the space fractional Allen-Cahn (AC) equation. More precisely, we consider the following initial-boundary problem [ 18, 27]:

where Ω=(a,b) , f( u) is a nonlinear function, ε is a positive constant, and Lαdenotes the nonlocal operator

with left and right Riemann-Liouville fractional derivatives of order α , def ined by

and

Model ( 1) is the generalization of the classic AC equation which was originally introduced in 1979 by Allen and Cahn [ 2]. Usually, f(u)=F′(u) for a given energy F( u), such as the Ginzburg-Landau double well potential F(u)=(u2-1)2, is widely applied. To well understand the physical mechanisms of the phase transition in materials science, many eff orts have been made to develop effi cient numerical methods for the classic AC equation[ 14- 16, 23- 25].

Recently, many researchers are of particular interest to the numerical methods of model( 1) due to its new physical and mathematical properties [ 18]. With the help of the spectral decomposition, Bueno-Orovio et al. [ 5] discussed a Fourier spectral method for model( 1) with the periodic boundary condition. Using the classic Crank-Nicolson scheme for the time discretization and the second-order central difference for the space discretization,Hou et al. [ 18] presented a fully discretized scheme for the nonlocal model ( 1). Unlike the numerical discretization of the classic Allen-Cahn equation, the main diffi culty arises in the nonlocality of the fractional derivative operator. Recently, Li et al. [ 20] developed a fast f inite difference method for a space-time fractional AC equation based on the structure of the iterative matrix. Very recently, Du and Yang [ 12] designed an asymptotically compatible Fourier spectral method for a nonlocal AC model. And they proved that the nonlocal AC equation converges to the classic AC equation in the local limit from the view of theory and numerical methods. As one of the high-order numerical methods, the LDG method works well for the classic AC equation [ 14, 16]. The f inite difference and spectral methods have been used to solve the nonlinear fractional reaction-diff usion equations, see [ 3, 4,30, 31], but the LDG solver for this kind of equation is scarce. There are many theoretical investigations of the nonlocal AC equation, such as the traveling wave solutions [ 3, 6],dynamics of multiple fronts [ 27]. To investigate the microscopic dynamic behavior, there is urgent need to design high-order and robust numerical methods for the nonlocal AC equation. In view of the advantages of the LDG method [ 10, 26], we try to analyze a semidiscrete LDG scheme for model ( 1) in this work. Several attempts were made to develop the LDG methods for solving space fractional diff usion equations, which include [ 7, 8, 11,19, 21, 28]. Motivated by the previous works [ 11, 19], we construct the LDG scheme by splitting the fractional derivative into two classical f irst derivative and a weakly singular integral. For the potential function f( u), we shall suppose that there exists a positive constant L, such that

To design the LDG scheme, the original model is transformed into a system by setting auxiliary variables. Then, the semi-discrete LDG scheme is obtained by the LDG approximation to the space variable. The stability and convergence of the semi-discrete LDG scheme are rigorously analyzed. The numerical results indicate that the LDG scheme works well for our considered model. In addition, the dissipative inf luence of the fractional derivative term is investigated using the proposed numerical scheme. For very small diff usion coeff icient, the numerical results show that the inf luence of the nonlocal dispersive is out of action. For the same diff usion coeffi cient, the dissipative behavior increases as the order of the fractional derivative tends to 2.

The content of the paper is organized as follows. In Sect. 2, we construct a semi-discrete LDG scheme for the considered equation with the help of the fractional calculus.In Sect. 3, we discuss the stability and error estimates of the semi-discrete LDG scheme.Numerical examples and physical simulations are presented in Sect. 4. This paper ends with some conclusions in Sect. 5.

In the following section, we will denote the standard L2(Ω)-inner product as(u,v)Ω=∫Ωu(x)v(x)dx , and the L2-normFor γ ＞0, we denote | ·|Hγ(Ω)and ‖·‖H-γ(Ω)be the semi-norm and negative norm of the factional Sobolev space [ 1, 11,13], respectively. And we assume that the regularity of solutions is enough for our numerical method. With the similar argument used in [ 1, 11, 13], for the problem ( 1)-( 3), we have the following lemma.

Lemma 1The solution u( x, t) of the problem ( 1)-( 3) satisf ies the following energy inequality:

where CT,Ωis a constant which depends on T and domain Ω.

ProofTaking the L2-inner product for both sides of ( 1), we obtain

Since [ 11, 13]

and

we have

Integrating from 0 to t, we get

Using the Gronwall"s inequality, we get ( 6).

Using the relation of Riemann-Liouville and Caputo fractional derivatives [ 22], we can rewrite the fractional nonlocal term as follows:

where Δ(α-2)∕2expresses the fractional Riesz potential [ 13, 29]

with

and

By introducing the variables w(x,t)=-Δ(α-2)∕2v(x,t) and v(x,t)=ux(x,t), the nonlocal model ( 1) is transformed into the system:

Given a partition of domain Ω , such thatDenote

The hat terms in ( 11)-( 14) denote the numerical f lux. For the cell Ij⊂Ω , we use the alternating direction f lux [ 10]:

or

with

For the right and left boundaries, we use the f luxes suggested in [ 10], which gives the following:

or

where β is a positive constant, which plays the role of stabilization. Let uhbe the unknown coeffi cients of the local discontinuous Galerkin approximation, and then, the semi-discretized ODE system of the scheme ( 11)-( 14) gives the following:

where N(uh,t) is produced by ( 11)-( 14). Let {tn=nΔt}be the nodes of the partition of[0, T], where the temporal mesh size. For the temporal discretization, we employ the explicit third-order Runge-Kutta method [ 17]

In what follows, we shall discuss the stability and convergence of the semi-discrete scheme ( 11)-( 14). The technique used in proving the related results is similar to the traditional LDG scheme in [ 9- 11, 26, 28].

3.1 Stability

Suppose that ~uh,~vh,~whare the approximations of the solutions uh,vh,wh, respectively.If we denote eu=uh-~uh,ev=vh-~vh, and ew=wh-~wh, then we have the following theorem.

Theorem 1The semi-discrete LDG scheme ( 11)-( 14) with the f lux ( 15) is stable, and holds the following:

ProofSumming all the cells Ijin the LDG scheme ( 11)-( 14), and taking φ =eu,φ=-ev,and ψ =ewyield the following:

In view of

and

for numerical f luxes ( 15)-( 17), we get the following:

Furthermore, using the assumption ( 5), we conclude that

Thus, we conclude that

Moreover, using the Gronwall"s inequality, we f inally get the desired estimate ( 21).

3.2 Convergence

Next, we prove the error estimates for the semi-discrete LDG scheme ( 11)-( 14). To carry out our error estimates, we will need the following projections π±and P , which satisfy the Gronwall"s inequality [ 9]:

If we split the errors into the projection error and the interpolation error, i.e.,eu=u-uh=(π-u-uh)-(π-u-u)=∶ηu-ρu, ev=v-vh=(π+v-vh)-(π+v-v)=∶ηv-ρv,ew=w-wh=(Pw-wh)-(Pw-w)=∶ηw-ρw, then the following estimate holds [ 9]:

where ρu=π±u-u or ρu=Pu-u , and the positive constant C is dependent of u but independent of h.

Theorem 2Let u be the solution of the problem ( 1)-( 3). If uhis the solution of the semi-LDG scheme ( 11)-( 14), then there exists a positive constant C independent of h, such that

ProofFrom ( 1)-( 3) and ( 11)-( 14) with φ =ηu,φ=-ηvand ψ =ηw, we have the error equation

Summing the left-hand side of Eq. ( 27), with the similar proving process given in Theorem 1, we get the following:

For the right-hand side of Eq. ( 27), we obtain the following:

Recalling the approximation properties [ 11, 28]

and applying the projection property ( 25), we have the following:

where the assumption ( 5) is also used. Combining ( 28) and the norm-equivalence [ 11, 13,28], we get the following:

Furthermore, using ( 25), the Cauchy-Schwartz inequality and selecting ε1,ε2,ε3and ε4properly, we obtain the following:

If we chose ‖ηu(0)‖≤Chk+1, with the help of the Gronwall"s inequality, we deduce that‖ηu‖≤Chk+1. Finally, using the triangle inequality ‖eu‖≤‖ηu‖+‖ρu‖ , we arrive at the result ( 26).

In this section, we use the LDG scheme ( 11)-( 14) to solve the fractional AC equation.To test numerically the order of convergence of the LDG scheme ( 11)-( 14), we test two numerical examples with exact solutions. As we know that the explicit Runge-Kutta methods are conditionally stable. In our examples, the CFL condition Δt≤CCFLh is satisf ied.

Table 1 The L2 errors and L∞errors of Example 1 for P1 element

Table 2 The L2 errors and L∞errors of Example 1 for P2 element

To understand the dynamics behaviors of the fractional AC equation, we employ the proposed numerical scheme to solve the space fractional AC equation ( 1) with diff erent initial values.

Example 1Consider the fractional AC equation [ 18] ut=-(-Δ)α∕2u+u-u3+g(x,t) on(x,t)∈[0,1]×(0,1] with the initial value u0(x)=x3(1-x)3and the source termA simple calculation yields u(x,t)=e-tx3·(1-x)3.

Tables 1 and 2 show the rates of convergence of the LDG solution. Obviously, our numerical scheme is (k+1)-th order accurate . This is in accordance with our results given in Theorem 2. In this example, we chose the time and space steps which satisfy Δt=0.01h.

Example 2Consider the nonlocal AC equation ut=-(-Δ)α∕2u+u-u3+g(x,t) on(x,t)∈[0,1]×(0,1] subject to the initial condition u0(x)=x7(1-x)7, and the source termThe exact solution gives u(x,t)=e-tx7·(1-x)7.

In the implementation of this example, we f ixed the time and space steps as Δt=0.001h .The numerical results are listed in Tables 3, 4, and 5. It is observed that the convergence rate is in accordance with the theoretical results given in Theorem 2.

Table 3 The L2 errors and L∞errors of Example 2 for P1 element

Table 4 The L2 errors and L∞errors of Example 2 for P2 element

Table 5 The L2 errors and L∞errors of Example 2 for P3 element

Example 3To test the dissipative inf luence of the fractional derivative term, we consider the nonlocal AC equation ( 1) on f inite domain x∈(-1,1),t∈(0,T] with the initial data:

Figures 1, 2, and 3 show the evolution of the numerical solutions with diff erent α . In the numerical experiment, we f ixed the time step Δt=1∕10 and the space step h=1∕100 for ε=0.005 . From Figs. 1, 2, and 3 we can see that the dissipation is diff erent for diff erent α .The dissipative mechanism of nonlocal term seems to enhance the smoothness of numerical solutions when α tends to 2. Figures 1, 2, and 3 suggest that the numerical solutions with small α tend to steady-state solutions faster than big α.

Fig. 1 The evolution of the initial condition ( 29) with α=1.2

Fig. 2 The evolution of the initial condition ( 29) with α=1.5

Fig. 3 The evolution of the initial condition ( 29) with α=1.9

Fig. 4 The evolution of initial value ( 29) with α=1.2,1.5,1.8,1.99 and diff erent diff usion coeffi cients ε

Furthermore, we test the eff ect of the diff usion coeffi cient ε in Figs. 4 and 5. Here, we take h=1∕40, Δt=0.001 and T=5 . The numerical results indicate that the eff ects of α and ε are diff erent. For the same α , the eff ect of dissipation becomes weaker when ε is decreasing. And the eff ect of α will be lost when ε is small enough. In other words, the dissipative inf luence of the fractional nonlocal term is not strong for a small parameter ε.

Example 4In this example, we consider traveling wave solutions of the nonlocal AC equation ( 1) on f inite domain x∈(-1,1),t∈(0,T] with the initial value [ 16]:

Fig. 5 The numerical solutions with α=1.2,1.5,1.8,1.99 and f ixed ε=0.005

Fig. 6 Traveling wave solutions of the fractional AC equation with ε=0.0025 for diff erent α at T=5

Fig. 7 Numerical traveling wave solutions with diff erent ε and α at T=3

In the simulation, we take the space step as h=1∕100 . Figure 6 shows the initial value ( 30) and the numerical results of the fractional AC equation with diff erent anomalous diff usion coeffi cients α at T=5 . From Fig. 6, we observe that as α increases, the waveform keeps with the initial value. Figure 7 shows the numerical traveling wave solutions with diff erent ε and α at T=3 . For small diff usion coeffi cient ε , the inf luence of the fractional derivative term will be lost. To further observe the numerical time simulations of the fractional AC equation with the initial value ( 30), we plot the numerical traveling wave solutions in Fig. 8. The second row shows the contour plot of numerical solutions corresponding to the time evolution for diff erent chosen parameters α and ε .Figure 8 displays a rapid smoothing eff ect of the solution for diff erent diff usion coeffi -cients. This behavior is consistent with the existing works" f indings [ 3, 6, 27].

We have presented an LDG scheme for solving a fractional-in-space AC equation.Detailed stability and convergence analysis are given. The optimal convergence rate is checked by a numerical example. Numerical simulations have shown that the dissipative inf luence of the nonlocal term works for some ε . It is observed that the nonlocal term plays an important role in the solutions of the nonlocal AC equation. The eff ect will be lost for small ε . In addition, the nonlocal model ( 1) can be represented as follows [ 18,27]:

Fig. 8 Evolutions from the initial condition ( 30) for traveling wave solutions with α =1.5 for ε =0.005 ,ε=0.025 , and ε =0.002 at T =3 , where the f irst row gives the numerical solutions and the second row gives the contours

where the Lyapunov energy functional L(t) gives the following:

AcknowledgementsThe authors would like to thank the referees for their valuable comments and suggestions that have vastly improved the original manuscript of this paper. The research is supported by the National Natural Science Foundations of China (Grant number 11426174) and the Natural Science Basic Research Plan in Shaanxi Province of China (Grant number 2018JM1016).

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