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Local Discontinuous Galerkin Scheme for Space Fractional Allen–Cahn Equation

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Abstract

This paper is concerned with the efficient 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 efficiency 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.

Introduction

In this paper, we are concerned with the efficient 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]:

$$\begin{aligned}&u_{t}=\varepsilon \mathscr {L}_\alpha u +f(u),\quad x\in \Omega ,\, t\in (0,T], \end{aligned}$$
(1)
$$\begin{aligned}&u(x,t)=0 ,\quad \,\quad \quad x\in \partial \Omega , \end{aligned}$$
(2)
$$\begin{aligned}&u(x,0) = u_{0}(x), \,\,\quad x\in \Omega , \end{aligned}$$
(3)

where \(\Omega =(a,b)\), f(u) is a nonlinear function, \(\varepsilon \) is a positive constant, and \(\mathscr {L}_\alpha \) denotes the nonlocal operator

$$\begin{aligned} \mathscr {L}_\alpha u(x) = -\frac{1}{2\cos \left( \frac{\alpha \pi }{2}\right) } \left( {_{a}\text{D}_x^\alpha }u(x) + {_x\text{D}_{b}^\alpha }u(x) \right) \end{aligned}$$
(4)

with left and right Riemann–Liouville fractional derivatives of order \(\alpha \), defined by

$$\begin{aligned} _{a}\text{D}_{x}^{\alpha } u(x)= \frac{1}{\Gamma (2-\alpha )}\frac{{\rm {d}}^2}{{\rm{d}} x^2}\int ^{x}_{a}(x-\xi )^{1-\alpha }u(\xi ){{\rm {d}}}\xi , \end{aligned}$$

and

$$\begin{aligned} {_x}\text{D}_{b}^{\alpha } u(x)= \frac{1}{\Gamma (2-\alpha )}\frac{{\rm {d}}^2}{{\rm{d}}x^2}\int ^{b}_{x}(\xi -x)^{1-\alpha }u(\xi ){{\rm{d}}}\xi . \end{aligned}$$

Model (1) is the generalization of the classic AC equation which was originally introduced in 1979 by Allen and Cahn [2]. Usually, \(f(u)=F^{\prime }(u)\) for a given energy F(u), such as the Ginzburg–Landau double well potential \(F(u)=\frac{1}{4}(u^2-1)^2\), is widely applied. To well understand the physical mechanisms of the phase transition in materials science, many efforts have been made to develop efficient numerical methods for the classic AC equation [14,15,16, 23,24,25].

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 difficulty arises in the nonlocality of the fractional derivative operator. Recently, Li et al. [20] developed a fast finite 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 finite difference and spectral methods have been used to solve the nonlinear fractional reaction–diffusion 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 semi-discrete LDG scheme for model (1) in this work. Several attempts were made to develop the LDG methods for solving space fractional diffusion 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 first derivative and a weakly singular integral. For the potential function f(u), we shall suppose that there exists a positive constant L, such that

$$\begin{aligned} \max _{u\in \mathbb{R}}|f^\prime (u)|\le L. \end{aligned}$$
(5)

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 influence of the fractional derivative term is investigated using the proposed numerical scheme. For very small diffusion coefficient, the numerical results show that the influence of the nonlocal dispersive is out of action. For the same diffusion coefficient, 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.

The Semi-discrete LDG Scheme

In the following section, we will denote the standard \(L^2(\Omega )\)-inner product as \((u,v)_{\Omega }=\int _{\Omega }u(x)v(x)\text{d}x\), and the \(L^2\)-norm \( ||v||=\sqrt{(v,v)_{\Omega }}\). For \(\gamma >0\), we denote \(|\cdot|_{H^{\gamma}(\Omega)}\) and \(\Vert \cdot \Vert _{H^{-\gamma }(\Omega )}\) 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 1

The solutionu(xt) of the problem (1)–(3) satisfies the following energyinequality:

$$\begin{aligned} \Vert u(\cdot ,t)\Vert ^2 \le C_{T,\Omega }\Vert u_0\Vert ^2, \end{aligned}$$
(6)

where\(C_{T,\Omega }\)is a constant which depends onTand domain\(\Omega \).

Proof

Taking the \(L^2\)-inner product for both sides of (1), we obtain

$$\begin{aligned} \frac{1}{2}\Vert u(\cdot ,t)\Vert ^2=\varepsilon \left( \mathscr {L}_\alpha u,u \right) +\left( f(u),u\right) . \end{aligned}$$

Since [11, 13]

$$\begin{aligned} \left( \mathscr {L}_\alpha u,u \right) =-|u(\cdot,t)|^2_{H_0^{\frac{\alpha}{2}}} \end{aligned}$$

and

$$\begin{aligned} \left( f(u),u\right) =\left( u-u^2,u\right) \le \frac{b-a}{4}, \end{aligned}$$

we have

$$\begin{aligned} \frac{1}{2}\frac {\rm {d}}{{\rm {d}}t}\Vert u(\cdot ,t)\Vert ^2 +\varepsilon |u(\cdot,t)|^2_{H_0^{\frac{\alpha}{2}}}\le \frac{b-a}{4}. \end{aligned}$$

Integrating from 0 to t, we get

$$\begin{aligned} \Vert u(\cdot ,t)\Vert ^2 +\varepsilon \int_0^t|u(\cdot,s)|^2_{H_0^{\frac{\alpha}{2}}}\text{d}s\le \frac{(b-a)t}{4}+\Vert u_0\Vert ^2. \end{aligned}$$

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:

$$\begin{aligned} \mathscr {L}_\alpha u(x)=-(-\varDelta )^{\alpha /2}u(x) = \frac{\rm {d}}{{\rm {d}}x}\left( \varDelta _{(\alpha -2)/2}\frac{{\rm{d}}u}{{\rm{d}}x}\right) , \end{aligned}$$
(7)

where \(\varDelta _{(\alpha -2)/2}\) expresses the fractional Riesz potential [13, 29]

$$\begin{aligned} \varDelta _{(2-\alpha )/2}u(x) = \frac{-1}{2\cos \left( \frac{\alpha \pi }{2}\right) }\left( {_{a}\text{D}_x^{-(2-\alpha )}}u(x) + {_x\text{D}_{b}^{-(2-\alpha )}}u(x)\right) \end{aligned}$$

with

$$\begin{aligned} _{a}\text{D}_{x}^{-(2-\alpha )} u(x,t)= \frac{1}{\Gamma (2-\alpha )}\int ^{x}_{a}(x-\xi )^{1-\alpha }u(\xi ,t)\mathrm{\mathrm{{d}}}\xi, \end{aligned}$$

and

$$\begin{aligned} {_x}\text{D}_{b}^{-(2-\alpha )} u(x,t)= \frac{1}{\Gamma (2-\alpha )}\int ^{b}_{x}(\xi -x)^{1-\alpha }u(\xi ,t)\mathrm{\mathrm{{d}}}\xi . \end{aligned}$$

By introducing the variables \( w(x,t) = -\varDelta _{(\alpha -2)/2}v(x,t) \) and \( v(x,t) = u_{x}(x,t), \) the nonlocal model (1) is transformed into the system:

$$\begin{aligned}&u_{ t} =\sqrt{\varepsilon } w_{x}+f(u), \end{aligned}$$
(8)
$$\begin{aligned}&w =\varDelta _{(\alpha -2)/2}v, \end{aligned}$$
(9)
$$\begin{aligned}&v = \sqrt{\varepsilon }u_{x}. \end{aligned}$$
(10)

Given a partition of domain \(\Omega \), such that \(a=x_\frac{1}{2}<x_\frac{3}{2}<\cdots <x_{N+\frac{1}{2}}=b\). Denote \(I_j = [x_{j-\frac{1}{2}},x_{j+\frac{1}{2}}]\), \(h_j = x_{j+\frac{1}{2}}-x_{j-\frac{1}{2}}\) with \(\,h=\max \limits_{1\le j\le N} h_j\) being the maximum mesh size. We consider the LDG solutions in a polynomial space \(V^k_h\), which is defined as \(V_h^k = \{v: v\in P^k(I_j), x \in I_j,j=1,2,\cdots ,N\}\). Let \(u_h\) be the finite-element solution of the numerical scheme. The semi-LDG formulation of (8)–(10) gives: find \(u_h,v_h,w_h\in V_h^k\), such that, for all \(\phi (x),\varphi (x),\psi (x) \in V_h^k\), satisfy

$$\begin{aligned}&\left( (u_h)_t, \phi \right) _{I_j} +\left( f(u_h),\phi \right) _{I_j}+\sqrt{\varepsilon }\left( w_h,\phi _x\right) _{I_j} -\sqrt{\varepsilon }\hat{w}_{h}\phi \bigg |^{x^-_{j+\frac{1}{2}}}_{x^+_{j-\frac{1}{2}}} = 0, \end{aligned}$$
(11)
$$\begin{aligned}&\left( w_h, \varphi \right) _{I_j}-\left( \varDelta _{(\alpha -2)/2}v_h, \varphi \right) _{I_j}= 0, \end{aligned}$$
(12)
$$\begin{aligned}&\left( v_h, \psi \right) _{I_j}+\sqrt{\varepsilon }\left( u_h, \psi _x \right) _{I_j}- \sqrt{\varepsilon } \hat{u}_{h}\psi \bigg |^{x^-_{j+\frac{1}{2}}}_{x^+_{j-\frac{1}{2}}}=0, \end{aligned}$$
(13)
$$\begin{aligned}&\left( u_h(x,0),\phi (x)\right) _{I_j}-\left( u_0(x),\phi (x)\right) _{I_j}=0. \end{aligned}$$
(14)

The hat terms in (11)–(14) denote the numerical flux. For the cell \(I_j\subset \Omega \), we use the alternating direction flux [10]:

$$\begin{aligned} (\hat{u}_h)_{j+\frac{1}{2}} = (u_h)^-_{j+\frac{1}{2}}, \, (\hat{w}_h)_{j+\frac{1}{2}}=(w_h)^+_{j+\frac{1}{2}},\quad 0\le j \le N-1, \end{aligned}$$
(15)

or

$$\begin{aligned} (\hat{u}_h)_{j+\frac{1}{2}} = (u_h)^+_{j+\frac{1}{2}}, \, (\hat{w}_h)_{j+\frac{1}{2}}=(w_h)^-_{j+\frac{1}{2}}, \quad 0\le j \le N-1 \end{aligned}$$
(16)

with

$$\begin{aligned} (u_h)^{\pm }_j=u_h^{\pm }(x_j) = \lim _{x\rightarrow x^{\pm }_j}u_h(x), \;\quad [ u_h] = u_h^+-u_h^-. \end{aligned}$$

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

$$\begin{aligned} (\hat{u}_h)_{N+\frac{1}{2}}=u(b,t),\, (\hat{w}_h)_{N+\frac{1}{2}}=(w_h)^-_{N+\frac{1}{2}}+\frac{\beta }{h}\big[ (u_h)_{N+\frac{1}{2}}\big] \end{aligned}$$
(17)

or

$$\begin{aligned} \hat{u}_{\frac{1}{2}}=u(a,t),\, (\hat{w}_h)_{\frac{1}{2}}=(w_h)^-_{\frac{1}{2}}+\frac{\beta }{h}\big[ (u_h)_{\frac{1}{2}}\big] , \end{aligned}$$
(18)

where \(\beta \) is a positive constant, which plays the role of stabilization. Let \(\mathbf {u}_{h}\) be the unknown coefficients of the local discontinuous Galerkin approximation, and then, the semi-discretized ODE system of the scheme (11)–(14) gives the following:

$$\begin{aligned} \frac{\partial \mathbf {u}_{h}}{\partial t}=\mathscr {N}(\mathbf {u}_{h},t), \end{aligned}$$
(19)

where \(\mathscr {N}(\mathbf {u}_{h},t)\) is produced by (11)–(14). Let \(\{t_n=n\varDelta t\}_{n=0}^{N_t}\) be the nodes of the partition of [0, T], where the temporal mesh size \(\Delta t=\frac{T}{N_t}\). For the temporal discretization, we employ the explicit third-order Runge–Kutta method [17]

$$\begin{aligned} {\left\{ \begin{array}{ll} &{}\mathbf {u}_{h}^{(1)}=\mathbf u _h^n+\varDelta t\mathscr {N}( \mathbf {u}_{h}^{n},t_{n}) ,\\ &{}\mathbf {u}_{h}^{(2)}=\frac{4}{3}{} \mathbf u _h^n+\frac{1}{4}\mathbf {u}_{h}^{(1)}+\frac{1}{4}\varDelta t\mathscr {N}( \mathbf {u}_{h}^{(1)},t^{n}+\varDelta t) ,\\ &{}\mathbf {u}_{h}^{n+1}=\frac{1}{3}\mathbf {u}_{h}^{n}+\frac{2}{3}\mathbf {u}_{h}^{(2)} +\frac{2}{3}\varDelta t\mathscr {N}( \mathbf {u}_{h}^{(2)},t_n+\frac{1}{2}\varDelta t) . \end{array}\right. } \end{aligned}$$
(20)

Stability and Convergence

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 [9,10,11, 26, 28].

Stability

Suppose that \(\tilde{u}_h,\tilde{v}_h,\tilde{w}_h\) are the approximations of the solutions \(u_h,v_h,w_h\), respectively. If we denote \(e_u=u_h-\tilde{u}_h,e_v=v_h-\tilde{v}_h,\) and \(e_w=w_h-\tilde{w}_h\), then we have the following theorem.

Theorem 1

The semi-discrete LDG scheme (11)–(14) with the flux (15) is stable, and holds the following:

$$\begin{aligned} \Vert e_u(T)\Vert \le C\Vert e_u(0)\Vert ,\,T>0. \end{aligned}$$
(21)

Proof

Summing all the cells \(I_j\) in the LDG scheme (11)–(14), and taking \(\phi =e_u,\varphi =-e_v,\) and \(\psi =e_w\) yield the following:

$$\begin{aligned}&\sum _{j=1}^N\left( (e_u)_t, e_u\right) _{I_j} +\sum _{j=1}^N\left( f(u_h)-f(\tilde{u}_h),e_h\right) _{I_j} \\&- \sum _{j=1}^N\left( \varDelta _{(\alpha -2)/2}e_v, e_v \right) _{I_j}+\sqrt{\varepsilon }\sum _{j=1}^N\left( e_w,e_{ux}\right) _{I_j} \\& +\sqrt{\varepsilon }\sum _{j=1}^N\left( e_u, e_{wx} \right) _{I_j}+ \sqrt{\varepsilon }\sum _{j=1}^{N-1}(\hat{e}_w)_{j+\frac{1}{2}}[e_u]_{j+\frac{1}{2}} \\& + \sqrt{\varepsilon }(\hat{e}_w)_{\frac{1}{2}}(e_u)^+_{\frac{1}{2}} -\sqrt{\varepsilon }(\hat{e}_w)_{N+\frac{1}{2}}(e_u)^-_{N+\frac{1}{2}}\\& +\sqrt{\varepsilon }\sum _{j=1}^{N-1}(\hat{e}_u)_{j+\frac{1}{2}} [e_w]_{j+\frac{1}{2}}+ \sqrt{\varepsilon }(\hat{e}_u)_{\frac{1}{2}}(e_w)^+_{\frac{1}{2}}\\& -\sqrt{\varepsilon }(\hat{e}_u)_{N+\frac{1}{2}}(e_w)^-_{N+\frac{1}{2}}=0. \end{aligned}$$

In view of

$$\begin{aligned}&\sum _{j=1}^N\left( e_w,e_{ux}\right) _{I_j}+\sum _{j=1}^N \left( e_u, e_{wx} \right) _{I_j} \\& = \sum _{j=1}^{N-1} [e_we_u]_{j+\frac{1}{2}}- (e_w)^+_{\frac{1}{2}}(e_u)^+_\frac{1}{2}+ (e_w)^-_{N+\frac{1}{2}}(e_u)^-_{N+\frac{1}{2}}, \end{aligned}$$

and

$$\begin{aligned} \sum _{j=1}^{N-1}({e_w})^+_{j+\frac{1}{2}}[e_u]_{j+\frac{1}{2}} +\sum _{j=1}^{N-1}(e_u)^-_{j+\frac{1}{2}} [e_w]_{j+\frac{1}{2}} = \sum _{j=1}^{N-1} [e_we_u]_{j+\frac{1}{2}} \end{aligned}$$

for numerical fluxes (15)–(17), we get the following:

$$\begin{aligned}&\left( e_{ut}, e_u\right) _{\Omega }+(\varDelta _{(\alpha -2)/2}e_v,e_v)_{\Omega }+ \frac{\sqrt{\varepsilon }\beta }{h}\left( (e_u)^-_{N+\frac{1}{2}}\right) ^2\\& = ( f(u)-f(\tilde{u}_h),e_u) _{\Omega } . \end{aligned}$$

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

$$\begin{aligned} \left( f(u)-f(\tilde{u}_h),e_u\right) _{\Omega }=\int _{\Omega }(\,f(u)-f(\tilde{u}_h)e_u)\mathrm{{d}}x \le L\Vert e_u\Vert ^2. \end{aligned}$$

Thus, we conclude that

$$\begin{aligned} \frac{1}{2}\frac {\rm {d }}{{\rm {d}}t}\Vert e_u\Vert ^2 + \Vert e_v\Vert ^2_{H^{-(1-\frac{\alpha }{2})}}\le L\Vert e_u\Vert ^2. \end{aligned}$$

Moreover, using the Gronwall’s inequality, we finally get the desired estimate (21).

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 \(\pi ^{\pm }\) and \(\mathscr {P}\), which satisfy the Gronwall’s inequality [9]:

$$\left\{\begin{aligned} & \left( (\pi ^{+}u(x)-u(x)),v(x)\right) _{I_j} = 0,\, \forall v\in P^{k-1}(I_j),\, \nonumber \\& \pi ^{+}u_{j+\frac{1}{2}} = u(x^+_{j+\frac{1}{2}}), \end{aligned}\right.$$
(22)
$$\left\{\begin{aligned} & \left( (\pi ^{-}u(x)-u(x)),v(x)\right) _{I_j} = 0,\, \forall v\in P^{k-1}(I_j), \nonumber \\& \pi ^{-}u_{j+\frac{1}{2}} = u(x^-_{j+\frac{1}{2}}), \end{aligned}\right.$$
(23)
$$\begin{aligned} \left( (\mathscr {P}u(x)-u(x)),v(x)\right) _{I_j}& = 0,\, \forall v\in P^{k}(I_j). \end{aligned}$$
(24)

If we split the errors into the projection error and the interpolation error, i.e., \(e_u=u-u_h=(\pi ^-u-u_h)-(\pi ^-u-u)=:\eta _u-\rho _u,\)\(e_v=v-v_h=(\pi ^+v-v_h)-(\pi ^+v-v)=:\eta _v-\rho _v,\)\(e_w=w-w_h=(\mathscr {P}w-w_h)-(\mathscr {P}w-w)=:\eta _w-\rho _w,\) then the following estimate holds [9]:

$$\begin{aligned} \Vert \rho _u\Vert \le Ch^{k + 1}, \end{aligned}$$
(25)

where \(\rho _u=\pi ^\pm u-u\) or \(\rho _u=\mathscr {P} u-u\), and the positive constant C is dependent of u but independent of h.

Theorem 2

Letube the solution of the problem (1)–(3). If\(u_h\)is the solution of thesemi-LDG scheme (11)–(14), then thereexists a positive constantCindependent ofh, such that

$$\begin{aligned} \Vert u-u_h\Vert \leqslant Ch^{k+1}. \end{aligned}$$
(26)

Proof

From (1)–(3) and (11)–(14) with \(\phi =\eta _u,\varphi =-\eta _v\) and \(\psi =\eta _w\), we have the error equation

$$\begin{aligned}&\left( (\eta _u)_t, \eta _u\right) _{I_j} +\sqrt{\varepsilon } \left( \eta _w,(\eta _u)_{x}\right) _{I_j}-\left( \varDelta _{(\alpha -2)/2}\eta _v, \eta _v \right) _{I_j}\nonumber \\& +\sqrt{\varepsilon }\left( \eta _u, (\eta _w)_{x} \right) _{I_j} - \sqrt{\varepsilon }(\eta _w)^{+}_{j+\frac{1}{2}}(\eta _u)^-_{j+\frac{1}{2}}\nonumber \\& +\sqrt{\varepsilon }(\eta _w)^{+}_{j-\frac{1}{2}}(\eta _u)^+_{j-\frac{1}{2}} \nonumber \\& -\sqrt{\varepsilon }(\eta _u)^{-}_{j+\frac{1}{2}}(\eta _w)^-_{j+\frac{1}{2}} +\sqrt{\varepsilon }(\eta _u)^{-}_{j-\frac{1}{2}}(\eta _w)^+_{j-\frac{1}{2}}\nonumber \\& =\left( (\rho _u)_t, \eta _u\right) _{I_j}+\left( \rho _v, \eta _w \right) _{I_j}-\left( \rho _w, \eta _v \right) _{I_j}\nonumber \\& +\sqrt{\varepsilon } \left( \rho _w,(\eta _u)_{x}\right) _{I_j}-\left( \varDelta _{(\alpha -2)/2}\rho _v, \eta _v \right) _{I_j}\nonumber \\& +\sqrt{\varepsilon }\left( \rho _u, (\eta _w)_{x} \right) _{I_j}- \sqrt{\varepsilon }(\rho _w)^{+}_{j+\frac{1}{2}}(\eta _u)^-_{j+\frac{1}{2}} \nonumber \\& +\sqrt{\varepsilon }(\rho _w)^{+}_{j-\frac{1}{2}}(\eta _u)^+_{j-\frac{1}{2}} -\sqrt{\varepsilon }(\rho _u)^{-}_{j+\frac{1}{2}}(\eta _w)^-_{j+\frac{1}{2}}\nonumber \\& +\sqrt{\varepsilon }(\rho _u)^{-}_{j-\frac{1}{2}}(\eta _w)^+_{j-\frac{1}{2}} - \left( f(u)-f(u_h),\eta _u\right) _{I_j}. \end{aligned}$$
(27)

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

$$\begin{aligned} \text{LHS}~\text{of}~{(3.7)}=\frac{1}{2}\Vert \eta _u\Vert ^2+\Vert \eta _v\Vert ^2_{H^{1-\frac{\alpha }{2}}}+ \frac{\sqrt{\varepsilon } \beta }{h} |(\eta _u)_{N+\frac{1}{2}}|^2. \end{aligned}$$
(28)

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

$$\begin{aligned}&\sum _{j=1}^N\left( (\rho _u)_t, \eta _u\right) _{I_j}+\sum _{j=1}^N\left( \rho _v, \eta _w \right) _{I_j}-\sum _{j=1}^N\left( \rho _w, \eta _v \right) _{I_j}\\& +\sum _{j=1}^N\sqrt{\varepsilon } \left( \rho _w,(\eta _u)_{x}\right) _{I_j}-\sum _{j=1}^N\left( \varDelta _{(\alpha -2)/2}\rho _v, \eta _v \right) _{I_j}\\& +\sqrt{\varepsilon }\sum _{j=1}^N\left( \rho _u, (\eta _w)_{x} \right) _{I_j}+\sqrt{\varepsilon }\sum _{j=1}^N(\rho _w)^{+}_{j+\frac{1}{2}}[\eta _u]_{j+\frac{1}{2}} \\& +\sqrt{\varepsilon }(\rho _w)^{+}_{\frac{1}{2}}((\eta _u)^{+})_{\frac{1}{2}} -\sqrt{\varepsilon }\sum _{j=1}^N(\rho _w)^{-}_{N+\frac{1}{2}}((\eta _u)^{-})_{N+\frac{1}{2}} \\& +\frac{\sqrt{\varepsilon }\beta }{h} (\rho _u)^{-}_{j+\frac{1}{2}}[(\eta _u)^{-}]_{N+\frac{1}{2}} -\sum _{j=1}^N \left( f(u)-f(u_h),\eta _u\right) _{I_j}. \end{aligned}$$

Because of \((\eta _u)_x\in P^{k-1}(I_j),(\eta _w)_x\in P^{k-1}(I_j),(\eta _v)_x\in P^{k-1}(I_j),\eta _v\in P^{k}(I_j)\), we have \(\left( \rho _w, (\eta _u)_x\right) _{I_j} =0,\)\( \,\left( \rho _u,(\eta _w)_x\right) _{I_j}=0,\)\(\left( \rho _v,\eta _w\right) _{I_j}=0,\)\(\left( \rho _v,(\eta _w)_x\right) _{I_j}=0,\)\((\rho _u)_{j+\frac{1}{2}}=0,\,\) and \((\rho _w)_{j+\frac{1}{2}} =0.\) Then, we obtain the following:

$$\begin{aligned}&\sum _{j=1}^N\left( (\rho _u)_t, \eta _u\right) _{I_j}-\sum _{j=1}^N\left( \rho _w, \eta _v \right) _{I_j}\\& -\sum _{j=1}^N\left( \varDelta _{(\alpha -2)/2}\rho _v, \eta _v \right) _{I_j} -\sqrt{\varepsilon }\sum _{j=1}^N(\rho _w)^{-}_{N+\frac{1}{2}}((\eta _u)^{-})_{N+\frac{1}{2}} \\& -\sum _{j=1}^N \left( f(u)-f(u_h),\eta _u\right) _{I_j}\\& \le \left( (\rho _u)_t, \eta _u\right) +\frac{\varepsilon _1}{2}\Vert \eta _v\Vert ^2+\frac{1}{2\varepsilon _1}\Vert \rho _w\Vert ^2 \\& +\frac{\varepsilon _2}{2}\Vert \eta _v\Vert ^2+\frac{1}{2\varepsilon _2} \Vert \varDelta _{(\alpha -2)/2}(\rho _v)\Vert ^2\\& +\frac{\varepsilon _3\sqrt{\varepsilon }}{2}|(\rho _w)^{-}_{N+\frac{1}{2}}|^2 +\frac{\sqrt{\varepsilon }}{2\epsilon _3}|((\eta _u)^{-})_{N+\frac{1}{2}}|^2 \\& +L\Vert \eta _u\Vert ^2+\frac{\varepsilon _4L}{2}\Vert \eta _v\Vert ^2 +\frac{L}{2\varepsilon _4}\Vert \rho _w\Vert ^2. \end{aligned}$$

Recalling the approximation properties [11, 28]

$$\begin{aligned} \Vert \varDelta _{(\alpha -2)/2}(\rho _v)\Vert \le Ch^{k+1}, \end{aligned}$$

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

$$\begin{aligned} \text{RHS}~\text{of}~{(3.7)}& \le \left( (\rho _u)_t, \eta _u\right) +\left( \frac{\varepsilon _1}{2}+ \frac{\varepsilon _2}{2}+\frac{\varepsilon _4L}{2}\right) \Vert \eta _v\Vert ^2 \\&\quad +\,Ch^{2k+2}\frac{\sqrt{\varepsilon }}{2\varepsilon _4}|((\eta _u)^{-})_{N+\frac{1}{2}}|^2 +L\Vert \eta _u\Vert ^2, \end{aligned}$$

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

$$\begin{aligned}& \frac{1}{2}\frac{\rm{d}}{{\rm{d}}t}\Vert \eta _u(t)\Vert ^2+\left( 1-\frac{\varepsilon _1 \bar{C} }{2} -\frac{\varepsilon _2\bar{C}}{2} -\frac{\varepsilon _4\bar{C}L}{2}\right) \Vert \eta _v\Vert ^2_{H^{-(1-\frac{\alpha }{2})}} \\& \le \left( (\rho _u)_t, \eta _u\right) +Ch^{2k+2} +L\Vert \eta _u\Vert ^2. \end{aligned}$$

Furthermore, using (25), the Cauchy–Schwartz inequality and selecting \(\varepsilon _1,\varepsilon _2,\varepsilon _3\) and \(\epsilon _4\) properly, we obtain the following:

$$\begin{aligned} \frac{1}{2}\frac{\rm {d}}{{\rm{d}}t}\Vert \eta _u(t)\Vert ^2 \le C h^{2k+2}+L \Vert \eta _u \Vert ^2. \end{aligned}$$

If we chose \(\Vert \eta _u(0)\Vert \le Ch^{k+1}\), with the help of the Gronwall’s inequality, we deduce that \( \Vert \eta _u\Vert \le C h^{k+1}. \) Finally, using the triangle inequality \(\Vert e_u\Vert \le \Vert \eta _u\Vert +\Vert \rho _u\Vert \), we arrive at the result (26).

Numerical Results

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 \(\Delta t\leqslant C_\mathrm{CFL} h\) is satisfied. 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 different initial values.

Example 1

Consider the fractional AC equation [18] \( u_{t} = -(-\varDelta )^{\alpha /2} u+u-u^3+g(x,t)\) on \( (x,t)\in [0, 1]\times (0, 1]\) with the initial value \(u_0(x) = x^3(1-x)^3\) and the source term \(g(x,t) = \mathrm{{e}}^{-t}\left( -u_0(x)+ (-\varDelta )^{\frac{\alpha }{2}}u_0(x)\right) -u+u^3. \) A simple calculation yields \(u(x,t) = \mathrm{{e}}^{-t} x^3\cdot(1-x)^3\).

Table 1 The \(L^{2}\) errors and \(L^{\infty }\) errors of Example 1 for \(P^1\) element
Table 2 The \(L^{2}\) errors and \(L^{\infty }\) errors of Example 1 for \(P^2\) element

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 \(\Delta t=0.01h\).

Example 2

Consider the nonlocal AC equation \( u_{t} = -(-\varDelta )^{\alpha /2} u+u-u^3+g(x,t)\) on \( (x,t)\in [0, 1]\times (0, 1]\) subject to the initial condition \(u_0(x) = x^7(1-x)^7\), and the source term \(g(x,t) = \mathrm{{e}}^{-t}\left( -u_0(x)+ (-\varDelta )^{\frac{\alpha }{2}}u_0(x)\right) -u+u^3. \) The exact solution gives \(u(x,t) = \mathrm{{e}}^{-t} x^7\cdot(1-x)^7\).

In the implementation of this example, we fixed the time and space steps as \(\Delta t=0.001h\). 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 \(L^{2}\) errors and \(L^{\infty }\) errors of Example 2 for \(P^1\) element
Table 4 The \(L^{2}\) errors and \(L^{\infty }\) errors of Example 2 for \(P^2\) element
Table 5 The \(L^{2}\) errors and \(L^{\infty }\) errors of Example 2 for \(P^3\) element

Example 3

To test the dissipative influence of the fractional derivative term, we consider the nonlocal AC equation (1) on finite domain \(x\in (-1,1), t\in (0,T]\) with the initial data:

$$\begin{aligned} u_0(x)= \left\{ \begin{array}{ll} 0.1\cos (2\pi x),&{} \quad -1<x<1,\\ 0,&{} \quad \mathrm{otherwise}. \end{array}\right. \end{aligned}$$
(29)

Figures 1, 2, and 3 show the evolution of the numerical solutions with different \(\alpha \). In the numerical experiment, we fixed the time step \(\Delta t=1/10\) and the space step \(h=1/100\) for \(\varepsilon =0.005\). From Figs. 1, 2, and 3 we can see that the dissipation is different for different \(\alpha \). The dissipative mechanism of nonlocal term seems to enhance the smoothness of numerical solutions when \(\alpha \) tends to 2. Figures 1, 2, and 3 suggest that the numerical solutions with small \(\alpha \) tend to steady-state solutions faster than big \(\alpha \).

Fig. 1
figure1

The evolution of the initial condition (29) with \(\alpha =1.2\)

Fig. 2
figure2

The evolution of the initial condition (29) with \(\alpha =1.5\)

Fig. 3
figure3

The evolution of the initial condition (29) with \(\alpha =1.9\)

Fig. 4
figure4

The evolution of initial value (29) with \(\alpha =1.2,1.5,1.8,1.99\) and different diffusion coefficients \(\varepsilon \)

Fig. 5
figure5

The numerical solutions with \(\alpha =1.2, 1.5, 1.8, 1.99\) and fixed \(\varepsilon =0.005\)

Furthermore, we test the effect of the diffusion coefficient \(\varepsilon \) in Figs. 4 and 5. Here, we take \(h=1/40,\)\(\Delta t=0.001\) and \(T=5\). The numerical results indicate that the effects of \(\alpha \) and \(\varepsilon \) are different. For the same \(\alpha \), the effect of dissipation becomes weaker when \(\varepsilon \) is decreasing. And the effect of \(\alpha \) will be lost when \(\varepsilon \) is small enough. In other words, the dissipative influence of the fractional nonlocal term is not strong for a small parameter \(\varepsilon \).

Example 4

In this example, we consider traveling wave solutions of the nonlocal AC equation (1) on finite domain \(x\in (-1,1), t\in (0,T]\) with the initial value [16]:

$$\begin{aligned} u_0(x)=\frac{1}{2}\left( 1-\tanh \left( \frac{x}{2\sqrt{2}\epsilon }\right) \right) . \end{aligned}$$
(30)

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 different anomalous diffusion coefficients \(\alpha \) at \(T=5\). From Fig. 6, we observe that as \(\alpha \) increases, the waveform keeps with the initial value. Figure 7 shows the numerical traveling wave solutions with different \(\epsilon \) and \(\alpha \) at \(T=3\). For small diffusion coefficient \(\varepsilon \), the influence 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 different chosen parameters \(\alpha \) and \(\varepsilon \). Figure 8 displays a rapid smoothing effect of the solution for different diffusion coefficients. This behavior is consistent with the existing works’ findings [3, 6, 27].

Fig. 6
figure6

Traveling wave solutions of the fractional AC equation with \(\epsilon =0.002 \,5\) for different \(\alpha \) at \(T=5\)

Fig. 7
figure7

Numerical traveling wave solutions with different \(\epsilon \) and \(\alpha \) at \(T=3\)

Fig. 8
figure8

Evolutions from the initial condition (30) for traveling wave solutions with \(\alpha =1.5\) for \(\varepsilon = 0.005\), \(\varepsilon =0.025\), and \(\varepsilon =0.002\) at \(T=3\), where the first row gives the numerical solutions and the second row gives the contours

Conclusions

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 influence of the nonlocal term works for some \(\varepsilon \). It is observed that the nonlocal term plays an important role in the solutions of the nonlocal AC equation. The effect will be lost for small \(\varepsilon \). In addition, the nonlocal model (1) can be represented as follows [18, 27]:

$$\begin{aligned} u_t=\frac{\delta \mathscr {L}(t)}{\delta u}, \end{aligned}$$

where the Lyapunov energy functional \(\mathscr {L}(t)\) gives the following:

$$\begin{aligned} \mathscr {L}(t)=\int _{\Omega }\left(F(u)-\frac{\varepsilon }{2}u\mathscr {L}_\alpha u\right)\mathrm{{d}}x, \end{aligned}$$

which satisfies \(\frac{{\rm{d}}\mathscr {L}(t)}{{\rm{d}}t}\le 0\). To investigate the long time behavior of model (1), it is important to design the numerical scheme that preserves energy stability. In this paper, our goal is to develop high-order LDG methods for the space fractional AC equation, but, in the future, we will design energy preserving LDG schemes for the considered model in two and three dimensions.

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Acknowledgements

The 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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Correspondence to Can Li.

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Li, C., Liu, S. Local Discontinuous Galerkin Scheme for Space Fractional Allen–Cahn Equation. Commun. Appl. Math. Comput. 2, 73–91 (2020). https://doi.org/10.1007/s42967-019-00034-9

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Keywords

  • Fractional Allen–Cahn equation
  • Local discontinuous Galerkin scheme
  • Error estimates

Mathematics Subject Classification

  • 26A33
  • 35R11
  • 65M60
  • 65M12