Journal of Computational Physics 358 (2018) 256–282
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Journal of Computational Physics www.elsevier.com/locate/jcp
A fast linearized conservative finite element method for the strongly coupled nonlinear fractional Schrödinger equations Meng Li a , Xian-Ming Gu b,c,∗ , Chengming Huang d,e , Mingfa Fei d , Guoyu Zhang d a
School of Mathematics and Statistics, Zhengzhou University, Zhengzhou, Henan 450001, PR China School of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu, Sichuan 611130, PR China Johann Bernoulli Institute of Mathematics and Computer Science, University of Groningen, Nijenborgh 9, P.O. Box 407, 9700 AK Groningen, The Netherlands d School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, Hubei 430074, PR China e Hubei Key Laboratory of Engineering Modeling and Scientific Computing, Huazhong University of Science and Technology, Wuhan, Hubei 430074, PR China b c
a r t i c l e
i n f o
Article history: Received 20 July 2017 Received in revised form 31 October 2017 Accepted 29 December 2017 Available online 5 January 2018 Keywords: Coupled nonlinear fractional Schrödinger equations Finite element method Mass and energy conservation Unconditional convergence Circulant preconditioner Krylov subspace methods
a b s t r a c t In this paper, a fast linearized conservative finite element method is studied for solving the strongly coupled nonlinear fractional Schrödinger equations. We prove that the scheme preserves both the mass and energy, which are defined by virtue of some recursion relationships. Using the Sobolev inequalities and then employing the mathematical induction, the discrete scheme is proved to be unconditionally convergent in the sense of L 2 -norm and H α /2 -norm, which means that there are no any constraints on the grid ratios. Then, the prior bound of the discrete solution in L 2 -norm and L ∞ -norm are also obtained. Moreover, we propose an iterative algorithm, by which the coefficient matrix is independent of the time level, and thus it leads to Toeplitz-like linear systems that can be efficiently solved by Krylov subspace solvers with circulant preconditioners. This method can reduce the memory requirement of the proposed linearized finite element scheme from O ( M 2 ) to O ( M ) and the computational complexity from O ( M 3 ) to O ( M log M ) in each iterative step, where M is the number of grid nodes. Finally, numerical results are carried out to verify the correction of the theoretical analysis, simulate the collision of two solitary waves, and show the utility of the fast numerical solution techniques. © 2018 Elsevier Inc. All rights reserved.
1. Introduction Recently, a noticeable growth of the attention has been drawn to the fractional Schrödinger equation (FSE). Originally, in papers [1,2], Laskin developed the FSE with quantum Riesz space-fractional derivative by extending the Feynman path integral to Lévy one. Naber [3] derived the time fractional Schrödinger equation by replacing the first order time derivative to a Caputo fractional one. In [4], Guo and Xu discussed the physical applications of the FSEs. Secchi [5] discussed the ground state solution of the FSEs in R N . Along the mathematical front, there are extensive theoretical researches which have been carried out for the FSEs and coupled ones. Guo [6–9] studied the existence and well-posedness criterion of the global
*
Corresponding author at: School of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu, Sichuan 611130, PR China. E-mail addresses:
[email protected] (M. Li),
[email protected],
[email protected] (X.-M. Gu),
[email protected] (C. Huang).
https://doi.org/10.1016/j.jcp.2017.12.044 0021-9991/© 2018 Elsevier Inc. All rights reserved.
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smooth solution of the FSEs, and obtained the conservation properties including the mass and energy. The authors in [10] considered the existence and uniqueness of the global solution to the coupled FSEs by using the Faedo–Galerkin method. For more theoretical analysis about the FSEs and coupled ones, such as the soliton dynamics, ground states, attractor and consistency problem, readers can refer to [11–15] and references therein. Unlike the classical Schrödinger equations, there is more difficult to find the analytical solutions of the FSEs explicitly. Therefore, it is necessary to use numerical methods. Up to now, the FSEs have drawn a growing number of researchers’ attention along the numerical front. Initially, Amore et al. [16] studied a collocation approach based on little sinc functions for the linear FSE. In [17,18], Wang et al. established two difference schemes for the coupled FSE. Subsequently, the authors in [19] discussed the convergence property in the maximum norm for the modified Crank–Nicolson scheme of the coupled FSE. Wang and Huang constructed and analyzed a series of difference schemes for the single cubic nonlinear FSEs [20–23]. Zhao et al. [24] presented a fourth-order compact ADI scheme for two-dimensional FSE. Wei et al. [25,26] investigated the implicit fully discrete local discontinuous Galerkin methods for the FSE and coupled one which involve the Caputo-type time fractional derivative. In [27], Liu et al. proposed a difference scheme for numerically solving the time-space FSEs. Ran and Zhang [28] proposed an implicit conservative difference scheme for numerically solving the strongly coupled nonlinear FSEs. Li et al. [29,30] discussed the Galerkin finite element schemes for the FSE, and analyzed their well-posedness, conservation and convergence properties. In this paper, we present a Galerkin finite element method for solving the strongly coupled nonlinear fractional Schrödinger equations α
iut − β(−) 2 u + [κ1 |u |2 + (κ1 + 2κ2 )| v |2 ]u + γ u + ζ v = 0, x ∈ R, 0 < t ≤ T , α
2
2
(1.1)
i v t − β(−) v + [κ1 | v | + (κ1 + 2κ2 )|u | ] v + γ v + ζ u = 0, x ∈ R, 0 < t ≤ T ,
(1.2)
u (x, 0) = u 0 , v (x, 0) = v 0 , x ∈ R,
(1.3)
2
where i = −1, u and v denote the complex amplitudes or envelopes of two wave packets, u 0 and v 0 are two known initial functions, the parameter β > 0 describes the group velocity dispersion, κ1 is the self-focusing of a signal for pulses in birefringent media, κ1 + 2κ2 is the cross-phase modulation defining the integrability of the nonlinear system (1.1)–(1.2), γ is the normalized birefringence constant, and the linear coupling parameter ζ explains the effects that arise from twisting α of the fibre and elliptic deformation of the fibre [31]. In one dimensional case, the fractional Laplacian operator (−) 2 is equivalent to the following Riesz-type fractional derivative 2
1
α
(−) 2 u (x, t ) =
2cos(απ /2)
[−∞ D αx u (x, t ) + x D α+∞ u (x, t )],
(1.4)
α where −∞ D α x u (x, t ) and x D +∞ u (x, t ) denote the left and right Riemann–Liouville fractional derivatives, respectively [32]. In this paper, we restrict that when |x| → ∞, the analytical solutions and initial values of the system (1.1)–(1.3) decay to zero. In this case, the system is referred to as the volume constraint problem [33,34], and the corresponding Riesz-type fractional derivative is defined as
1
α
(−) 2 u (x, t ) =
2cos(απ /2)
[a D αx u (x, t ) +x D bα u (x, t )], x ∈ Ω,
and the boundary conditions are given by
u (x, t ) = v (x, t ) = 0, x ∈ R\Ω, 0 ≤ t ≤ T ,
(1.5)
where Ω = (a, b) with a 0 and b 0. Proposition 1. If the wave functions u (x, t ) and v (x, t ) are the solutions of (1.1)–(1.3) with the boundary conditions (1.5), then we have the following conservation results: (1) mass conservation:
M (t ) ≡ M (0), t > 0,
(1.6)
(2) energy conservation:
E (t ) ≡ E (0), t > 0,
(1.7)
where M (t ) := u 2 + v 2 and
u (·, t ) 4L 4 + v (·, t ) 4L 4 2 − (κ1 + 2κ2 ) |u |2 | v |2dx − γ u (·, t ) 2 + v (·, t ) 2 − 2ζ Re u v¯ dx .
α
α
E (t ) := β (−) 4 u (·, t ) 2 + (−) 4 u (·, t ) 2
Ω
−
κ1
Ω
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Hereinafter, · and · L 4 denote the L 2 -norm and L 4 -norm, “Re( f )” and “Im( f )” represent the real part and imaginary part of f , and ¯f is the complex conjugate of f . Proof. Making the inner product of (1.1) with u, and taking the imaginary part of the resulting equation, we obtain
d dt
u 2 + ζ Im( v , u ) = 0.
(1.8)
Similarly, if we make the inner product of (1.2) with v, and take the imaginary part of the resulting equation, one can deduce that
d dt
v 2 + ζ Im(u , v ) = 0.
(1.9)
Then, taking the sum of (1.8) and (1.9), and using the relation Im( v , u ) + Im(u , v ) = 0, we have
d dt
( u 2 + v 2 ) = 0,
which implies that the mass conservation (1.6) holds for t > 0. Next, Making the inner product of (1.1)–(1.2) with ut and v t , respectively, and then taking the real part of each resulting equation, we obtain that
−
β d
−
β d
2 dt
α
(−) 4 u 2 +
κ1 d 4 dt
u 4L 4 + (κ1 + 2κ2 ) Re (| v |2 u , ut ) +
γ d 2 dt
u 2 + ζ Re ( v , ut ) = 0,
(1.10)
v 2 + ζ Re (u , v t ) = 0.
(1.11)
and
2 dt
α
(−) 4 v 2 +
κ1 d 4 dt
v 4L 4 + (κ1 + 2κ2 ) Re (|u |2 v , v t ) +
γ d 2 dt
Observe that
1 d Re (| v | u , ut ) + (|u | v , v t ) = |u |2 | v |2 dx, 2
2
2 dt
(1.12)
Ω
and
Re ( v , ut ) + (u , v t ) = Re
d
u v¯ dx .
dt
(1.13)
Ω
Taking the sum of (1.10) and (1.11), and by using (1.12) and (1.13), it follows that
d dt
E (t ) = 0,
which shows that the energy E (t ) is invariant for t > 0.
(1.14)
2
Since the conservative scheme performs better than the nonconservative ones, the researchers are of interest to construct some schemes which can preserve some invariant properties of the original differential equations. Among the existing numerical methods, the schemes in [17,20,30] are conservative in the mass sense, and the schemes in [18,19,21,23,29,28] preserve both the mass and energy conservation. What is more, in order to avoid heavy iterative costs at each temporal step, we want to present a linearly implicit conservative scheme for the FSEs. However, up to now, there exist few works focusing on the above two aspects, just the Ref. [18]. As far as we know, there are few studies on the Galerkin finite element methods for the FSEs and it should be pointed out that the finite element method is a great approach to solve FPDEs since that it is always with a higher convergence order and needs few domain limitations [35–41]. Recently, Li and Huang [29] derived a Crank–Nicolson finite element scheme for the nonlinear FSE, which conserves both the mass and energy. However, it must ask for the grid ratio τ = o(h1/4 ) with the time step τ and mesh size h. To avoid this requirement, Li et al. [30] proposed a linearized scheme for the nonlinear FSE and adopted a splitting technique to get the optimal L 2 -norm error estimate without any restrictions on the grid ratio. Nevertheless, the linearized scheme only preserves the mass conservation. Meanwhile, for above proposed finite element schemes for the FSEs, we also observed that the authors only obtained the L 2 -norm error estimates, and there are no any error estimates in the sense of the fractional norm. Moreover, to the best of our knowledge, in terms of the Galerkin finite element method, no work is proposed to numerically solve the strongly coupled FSEs. Motivated by the difference scheme [42], we in this paper propose a new linearized finite element scheme for the strongly coupled FSEs, which treats the time derivative by a modified Crank–Nicolson difference scheme, and the space derivative by the Galerkin finite element method. One can prove that the proposed linearized scheme can preserve both the discrete mass and energy which are defined by
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recursion relationships. We also observe that at each time step of the new discrete scheme, only one linear system needs to be solved and therefore, the computational cost will be significantly saved. In addition, the unconditionally convergent properties containing both the L 2 -norm and fractional norm are also obtained, in the sense that there are no any restrictions on the size of the time step τ in terms the space mesh size h. On the other hand, after giving an iterative algorithm, the coefficient matrix of the linearized discretized system is unchanged along the time marching, which leads to Toeplitz-like linear systems that can be solved by Krylov subspace solvers with circulant preconditioners. This method can efficiently reduce the memory requirement of the proposed iterative scheme from O ( M 2 ) to O ( M ) and the computational complexity from O ( M 3 ) to O ( M log M ) in each iterative step, in which M is the number of grid nodes. Moreover, both theoretical and numerical results are reported to show the effectiveness of the fast linearized conservative FEM for the strongly coupled nonlinear fractional Schrödinger equations. The current paper is arranged as follows. In Section 2, the required notations, the fractional function spaces and corresponding properties are introduced. In Section 3, we propose a linearized conservative Galerkin finite element scheme for the system (1.1)–(1.3). Section 4 is devoted to a rigorous theoretical analysis, including the mass and energy conservation, unconditional convergence and boundedness. In Section 5, in order to efficiently reduce the memory requirement and the computational complexity form of the proposed linearized scheme, we develop an iterative algorithm to solve the nonlinear systems, and then introduce a fast solution method. Some numerical examples are presented in Section 6 to illustrate the effectiveness of the scheme. Finally, we conclude the paper with some remarks. 2. Preliminaries In this section, we introduce some useful notations, and recall definitions and lemmas of the fractional derivative spaces which we will use thereafter. 2.1. Notations Define the L p -norm (1 ≤ p < ∞) and L ∞ -norm as
p
u L p :=
1p
|u (x)| dx
, u L ∞ := ess sup |u (x)|. x∈Ω
Ω
Specially, when p = 2, we can replace u L 2 by u for short. Let Th be a regular partition of Ω with Ω = ∪e i = ∪[xi , xi +1 ], 0 ≤ i ≤ M − 1 and h be the maximum diameter of the elements in Ω . For a given Th , we introduce the finite element space α
¯ ∩ H 2 (Ω) : ω| K ∈ P r ( K ), ∀ K ∈ Th }, S hr := {ω ∈ V := C (Ω) 0 α /2
where H 0 (Ω) is defined as the Sobolev space given in the following subsection, and P r ( K ) denotes the set of polynomials of which degrees are no more than r. For a given positive T and any positive integer N, let τ be the time step size such that τ = T / N. Denote tn = nτ , n = 0, 1, · · · N be a uniform partition of the time interval [0, T ]. Given any sequence of function {ωn } defined on Ω , we denote 1
δt ωn+ 2 =
ωn+1 − ωn n+1/2 ωn+1 + ωn n+1/2 3ωn − ωn−1 , ωˆ = , ω˜ = , 0 ≤ n ≤ N − 1. τ 2 2
2.2. Fractional derivative spaces In this subsection, we introduce some basic definitions and lemmas which are extremely useful for subsequent theoretical analysis. Firstly, one assumes that the following functional spaces consist of the complex-valued functions. For convenience, μ μ μ μ we will denote x D L u := a D x u and x D R u := x D b u. Definition 1. ([43,44]). For spaces
μ > 0, define the semi-norms and norms of the left, right and symmetric fractional derivative
12 μ |u | J μ (Ω) := x D L u , u J μ (Ω) := u 2 + |u |2J μ (Ω) , L
L
μ
2
|u | J μ (Ω) := x D R u , u J μ (Ω) := u
R
(2.1)
L
R
+ |u |2J μ (Ω) R
12 (2.2)
,
12 2 2 |u | J (Ω) := |(x D L u ,x D R u )| , u J (Ω) := u + |u | J μ (Ω) , μ S
μ
μ
1 2
μ S
S
(2.3)
260
M. Li et al. / Journal of Computational Physics 358 (2018) 256–282
μ
μ
μ
and J L ,0 (Ω), J R ,0 (Ω), J S ,0 (Ω) denote the closure of C 0∞ (Ω) with respect to the above semi-norms and norms.
μ > 0, define the semi-norm μ |u | H μ (Ω) := |ξ | u˜ (ξ ) L 2 (R) ,
Definition 2. ([43,44]). For
(2.4)
and norm
12
u H μ (Ω) := u 2 + |u |2H μ (Ω) ,
(2.5)
μ
and let H 0 (Ω) denote the closure of C 0∞ (Ω) with respect to · H μ (Ω) , where ξ is the Fourier transform parameter, and u˜ ˆ where uˆ is the zero extension of u outside of Ω . is the Fourier transform of u, We have the following lemmas about the equivalent relations between above proposed spaces and norms. μ
μ
μ
Lemma 1. ([43,44]). For μ > 0 and μ = n − 12 , n ∈ N, then J L (Ω), J R (Ω), J S (Ω) and H μ (Ω) are equal with equivalent norms and μ
μ
μ
μ
semi-norms, and J L ,0 (Ω), J R ,0 (Ω), J S ,0 (Ω) and H 0 (Ω) are equal with equivalent norms and semi-norms. μ
μ
Lemma 2. ([45]). For 1 < μ ≤ 2, if u , v ∈ J L (Ω) (or J R (Ω)), u |∂Ω = 0, v |∂Ω = 0, then μ
μ/2
(x D L u , v ) = (x D L
μ/2
u, x D R
μ
μ/2
v ), (x D R u , v ) = (x D R
μ/2
u, x D L
v ).
(2.6)
μ
Lemma 3 (Fractional Poincaré–Friedrichs inequality, [43,44]). For u ∈ J L ,0 (Ω), 0 < γ < μ, then
u ≤ C |u | J μ (Ω) , |u | J γ (Ω) ≤ C |u | J μ (Ω) . L
(2.7)
L
L
μ
For u ∈ J R ,0 (Ω), 0 < γ < μ, then
u ≤ C |u | J μ (Ω) , |u | J γ (Ω) ≤ C |u | J μ (Ω) . R
(2.8)
R
R
μ
There are similar results for u ∈ H 0 (Ω). With these preparations, we turn to construct an effective linearized conservative finite element method for numerically solving the strongly coupled FSE (1.1)–(1.3) in the forthcoming section. 3. The finite element discretized scheme In this section, we construct a linearized conservative fully discrete finite element scheme for the strongly coupled FSEs (1.1)–(1.3) with the boundary conditions (1.5). To this end, we begin with the variational formulation of the system (1.1)–(1.3), which is to find u , v ∈ V , such that for all χ ∈ V ,
κ1 |u |2 + (κ1 + 2κ2 )| v |2 u , χ + γ (u , χ ) + ζ ( v , χ ) = 0,
i (u t , χ ) − β B (u , χ ) +
i(v t , χ ) − β B (v , χ ) +
(3.1)
κ1 | v |2 + (κ1 + 2κ2 )|u |2 u , χ + γ ( v , χ ) + ζ (u , χ ) = 0,
(3.2)
u (x, 0) = u 0 , v (x, 0) = v 0 , x ∈ Ω, where the bilinear function
B (u , χ ) :=
1 2cos( α2 π )
α /2
(3.3)
α /2
α /2
α /2
(x D L u , x D R χ ) + (x D R u , x D L
χ) .
For convenience of theoretical analysis, we define the following semi-norm and norm 1
1
|u | α2 := B (u , u ) 2 , u α2 := ( u 2 + |u |2α ) 2 .
(3.4)
2
α
α
α
We observe from Lemma 1 that |u | α and u α are equivalent with the semi-norms and norms of J L2 (Ω), J R2 (Ω), H 2 (Ω) α
2
and J S2 (Ω). From [45], one can obtain
2
M. Li et al. / Journal of Computational Physics 358 (2018) 256–282
261
B (u , v ) ≤ C 1 u α v α , B (u , u ) ≥ C 2 u 2α , 2
2
(3.5)
2
where C 1 and C 2 are positive constants. Let t = t n+1/2 in (3.1)–(3.3). Then, using the finite difference method in time and the Galerkin finite element method in space, we derive the fully discrete finite element scheme, which is to find U n , V n ∈ S hr , n = 1, 2, · · · , N − 1, such that for all χh ∈ S hr ,
i (δt U n+ 2 , χh ) − β B (Uˆ n+ 2 , χh ) + ( L n1+1 , χh ) + γ (Uˆ n+ 2 , χh ) + ζ ( V˜ n+ 2 , χh ) = 0, i (δt V
1
1
n+ 12
1 ˆ n+ 2
, χh ) − β B ( V
where
L n1+1 :=
1
L n2+1 :=
1
2
n +1 , h ) + (L2
,χ
1
1
1 ˆ n+ 2
1 ˜ n+ 2
χh ) + γ ( V
, χh ) + ζ (U
κ1 3|U n |2 − |U n−1 |2 + (κ1 + 2κ2 ) 3| V n |2 − | V n−1 |2 κ1 3| V n |2 − | V n−1 |2 + (κ1 + 2κ2 ) 3|U n |2 − |U n−1 |2
2
(3.6)
, χh ) = 0,
(3.7)
Uˆ n+ 2 , 1
Vˆ n+ 2 . 1
Noting that above scheme is not selfstarting, thus the first step values U 1 and V 1 need to be provided by other scheme. We adopt the following implicit scheme, which is to find U 1 , V 1 ∈ S hr , such that
i (δt U 2 , χh ) − β B (Uˆ 2 , χh ) + ( L 11 , χh ) + γ (Uˆ 2 , χh ) + ζ ( Vˆ 2 , χh ) = 0,
(3.8)
i (δt V , χh ) − β B ( Vˆ , χ
(3.9)
1
1
1 2
1
1 2
0
1 h ) + (L2 ,
1
χh ) + γ ( Vˆ , χh ) + ζ (Uˆ , χh ) = 0, 1 2
1 2
0
U = Ih u0 , V = Ih v 0 , where
L 11 := L 12 :=
1 2 1
2
α /2
and I h : H 0
(3.10)
κ1 (|U 0 |2 + |U 1 |2 ) + (κ1 + 2κ2 )(| V 0 |2 + | V 1 |2 ) Uˆ 2 , 1
κ1 (| V 0 |2 + | V 1 |2 ) + (κ1 + 2κ2 )(|U 0 |2 + |U 1 |2 ) Vˆ 2 , 1
(Ω) ∩ H m+1 (Ω) → S hr is the commonly used Lagrange interpolation satisfying
u − I h u l ≤ Chm−l u m , 0 ≤ l ≤ m ≤ r + 1.
(3.11)
Here, C > 0 is a constant independent of h (see [46]). 4. Theoretical analysis In this section, we study theoretical properties of the discrete schemes (3.6)–(3.10), including the mass and energy conservation, unconditional convergence and boundedness. 4.1. Conservation Lemma 4. For the fully discrete solution ωn ∈ X h , 0 ≤ n ≤ N, we have
1
1
ˆ n+ 2 , δt ωn+ 2 ) = Re B (ω
1 2τ
|ωn+1 |2α − |ωn |2α . 2
(4.1)
2
Proof. It follows from the definition of B (·, ·) that 1
1
ˆ n+ 2 , δt ωn+ 2 ) B (ω = =
1 2cos( α2 π ) 1
4τ cos( α2 π )
−
1 α /2 n+ 12 α /2 ˆ , x D R δt ωn+ 2 xDL ω
α /2 n+1 α /2 , x D R ω n +1 xDL ω
α /2 n ω , x D αR /2 ωn
xDL
+
+
−
1 α /2 n+ 12 α /2 ˆ , x D L δt ωn+ 2 xDR ω
α /2 n+1 α /2 , x D R ωn xDL ω
α /2 n+1 ω , x D αL /2 ωn+1
xDR
−
+
α /2 n α /2 n+1 xDL ω , xDR ω
α /2 n+1 ω , x D αL /2 ωn
xDR
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M. Li et al. / Journal of Computational Physics 358 (2018) 256–282
=
1
+
2τ
α /2 n α /2 n+1 xDR ω , xDL ω
|ω
n +1 2 |α 2
− |ω
n 2 |α 2
+
1 2τ
−
α /2 n α /2 n xDR ω , xDL ω n
B (ω , ω
n +1
) − B (ω
n +1
,ω ) . n
Since B (ωn , ωn+1 ) is conjugate with B (ωn+1 , ωn ), one obtains
1
1
ˆ n+ 2 , δt ωn+ 2 ) = Re B (ω
1 2τ
|ωn+1 |2α − |ωn |2α . 2
2
2
This completes the proof.
Theorem 1. The discrete scheme (3.6)–(3.10) is conservative in the sense
M n := U n 2 + V n 2 + 2τ ζ G n ≡ M 0 , 0 ≤ n ≤ N , n
E :=
β(|U n |2α
+ | V n |2α
2
2
n 2
(4.2)
n 2
n
0
) − γ ( U + V ) − 2τ F ≡ E , 0 ≤ n ≤ N ,
(4.3)
where G n and F n are defined by the following recursive way
G 0 = G 1 = 0, G n+1 − G n = Im( V˜ n+ 2 , Uˆ n+ 2 ) + Im(U˜ n+ 2 , Vˆ n+ 2 ), 1 ≤ n ≤ N − 1, κ1 κ 1 + 2 κ2 0 0 2 ζ F0 = ( U 0 4L 4 + V 0 4L 4 ) +
U V + Re (U 0 , V 0 ), 4τ 2τ τ κ1 κ 1 + 2 κ2 1 1 2 ζ 1 1 4 1 4 F = ( U L 4 + V L 4 ) +
U V + Re (U 1 , V 1 ), 4τ 2τ τ 1
1
1
1
F n+1 − F n = Re ( L n1+1 , δt U n+ 2 ) + Re ( L n2+1 , δt V n+ 2 ) + ζ Re ( V˜ n+ 2 , δt U n+ 2 ) 1
1
1
1
1 1 + Re (U˜ n+ 2 , δt V n+ 2 ) , 1 ≤ n ≤ N − 1.
χh = Uˆ n+ 2 , n = 1, 2, · · · , N − 1 in (3.6) and taking the imaginary part of the resulting equation, we have 1
Proof. Denoting
Re (δt U n+ 2 , Uˆ n+ 2 ) + Im( L n1+1 , Uˆ n+ 2 ) + ζ Im( V˜ n+ 2 , Uˆ n+ 2 ) = 0. 1
1
1
1
1
(4.4)
As the result that
Re (δt U n+ 2 , Uˆ n+ 2 ) = 1
1
U n+1 2 − U n 2 2τ
, Im( Ln1+1 , Uˆ n+ 2 ) = 0, 1
one obtains from (4.4),
U n+1 2 − U n 2 2τ Similarly, we denote
+ ζ Im( V˜ n+ 2 , Uˆ n+ 2 ) = 0. 1
1
(4.5)
χh = Vˆ n+ 2 in (3.7) and take the imaginary part of the resulting equation to arrive at 1
V n+1 2 − V n 2 2τ
+ ζ Im(U˜ n+ 2 , Vˆ n+ 2 ) = 0. 1
1
(4.6)
Adding (4.5) and (4.6) yields
( U n+1 2 + V n+1 2 ) − ( U n 2 + V n 2 ) 2τ
+ ζ ( G n +1 − G n ) = 0,
which immediately implies (4.2) for n ≥ 1. Denoting we obtain
−β
1
χh = δt U n+ 2 , n = 1, 2, · · · , N − 1 in (3.6), taking the real part of the resulting equation, and by using Lemma 4,
|U n+1 |2α − |U n |2α 2
2τ
2
+ Re ( Ln1+1 , δt U n+ 2 ) + γ
Meanwhile, denoting χh = δt V Lemma 4 again, we have
1
n+ 12
U n+1 2 − U n 2 2τ
+ ζ Re ( V˜ n+ 2 , δt U n+ 2 ) = 0. 1
1
, n = 1, 2, · · · , N − 1 in (3.7), taking the real part of the resulting equation, and by using
M. Li et al. / Journal of Computational Physics 358 (2018) 256–282
−β
| V n+1 |2α − | V n |2α 2
2
2τ
+ Re ( Ln2+1 , δt V n+ 2 ) + γ 1
V n+1 2 − V n 2 2τ
263
+ ζ Re (U˜ n+ 2 , δt V n+ 2 ) = 0. 1
1
Adding above two equations yields (4.3) for n ≥ 1. The conservation results (4.2)–(4.3) for n = 0 can be obtained following similar way as above. For the clarity of the paper, we don’t state here. Therefore, we complete the proof. 2 4.2. Convergence and boundedness In this subsection, we analyze the unconditional convergence and boundedness properties of the discrete schemes. To this end, we first introduce the following two lemmas which show some relations between the L p -norms and the fractional Sobolev norms. Lemma 5. (See [41].) For 0 ≤ μ0 ≤ μ ≤ 1, there exists a constant C > 0, such that μ0
μ 1− μ0
u H μ0 ≤ C u Hμμ u
1 2
Lemma 6. (See [41].) For
−
1 p
.
< μ0 ≤ 1, and 2 ≤ p ≤ +∞, then there exists a constant C > 0, such that
u L p ≤ C u H μ0 . Remark 1. From Lemma 5 and Lemma 6, it is observed that if 0 ≤ μ0 ≤ μ ≤ 1, exists C μ0 > 0, such that μ0
μ 1− μ0
u L p ≤ C μ0 u Hμμ u
1 2
−
1 p
< μ0 ≤ 1 and 2 ≤ p ≤ +∞, there
(4.7)
.
Let P h : V → S hr denote the Ritz projection operator defined by
B (u − P h u , v h ) = 0, ∀u ∈ V , v h ∈ S hr . From [35,36], we have the following approximate properties. Lemma 7. For u ∈ V ∩ H λ (Ω),
α /2 < λ ≤ r + 1, we have
u − P h u ≤ Chλ u λ , α = 3/2;
u − P h u ≤ Chλ−σ u λ , α = 3/2, σ ∈ (0, 1/2), and for α /2-norm of u − P h u, the following estimate holds α
|u − P h u | α2 ≤ Chλ− 2 u λ , where C > 0 is a constant independent of h. Before giving the error estimates, we need to obtain the error equations. For 1 ≤ n ≤ N − 1, we write the system (1.1)–(1.3) at the time t = tn+1/2 into its equivalent form n+ 12
i (δt un+ 2 , χ ) − β B (uˆ n+ 2 , χ ) + ( K 1n+1 , χ ) + γ (uˆ n+ 2 , χ ) + ζ ( v˜ n+ 2 , χ ) = ( R 1 1
1
1
1
n+ 12
i (δt v n+ 2 , χ ) − β B ( vˆ n+ 2 , χ ) + ( K 2n+1 , χ ) + γ ( vˆ n+ 2 , χ ) + ζ (u˜ n+ 2 , χ ) = ( R 2 1
where
1
1
1
, χ ),
(4.8)
, χ ),
(4.9)
χ ∈ V , and 1 1 K 1n+1 := κ1 (3|un |2 − |un−1 |2 ) + (κ1 + 2κ2 )(3| v n |2 − | v n−1 |2 ) uˆ n+ 2 , 2 1 1 K 2n+1 := κ1 (3| v n |2 − | v n−1 |2 ) + (κ1 + 2κ2 )(3|un |2 − |un−1 |2 ) vˆ n+ 2 , 2
n+ 1 R1 2
1
n+ 1
α
1
1
1
1
1
1
:= i (δt un+ 2 − ut 2 ) − β(−) 2 (uˆ n+ 2 − un+ 2 ) + γ (uˆ n+ 2 − un+ 2 ) + ζ ( v˜ n+ 2 − v n+ 2 ) 1 1 1 + K 1n+1 − κ1 |un+ 2 |2 + (κ1 + 2κ2 )| v n+ 2 |2 un+ 2 ,
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M. Li et al. / Journal of Computational Physics 358 (2018) 256–282
n+ 12
n+ 1
1
α
1
1
1
1
1
1
:= i (δt v n+ 2 − v t 2 ) − β(−) 2 ( vˆ n+ 2 − v n+ 2 ) + γ ( vˆ n+ 2 − v n+ 2 ) + ζ (u˜ n+ 2 − un+ 2 ) n +1 n+ 12 2 n+ 12 2 n+ 12 . | + (κ1 + 2κ2 )|u | v + K 2 − κ1 | v
R2
For n = 1, one gets 1
1
1 2
1 2
1
1
1
1 2
1 2
1 2
i (δt u 2 , χ ) − β B (uˆ 2 , χ ) + ( K 11 , χ ) + γ (uˆ 2 , χ ) + ζ ( vˆ 2 , χ ) = ( R 12 , χ ), i (δt v , χ ) − β B ( vˆ , χ where
) + ( K 21 ,
(4.10)
χ ) + γ ( vˆ , χ ) + ζ (uˆ , χ ) = ( R 2 , χ ),
(4.11)
χ ∈ V , and 1 1 1 0 2 1 2 0 2 1 2 K 1 := κ1 (|u | + |u | ) + (κ1 + 2κ2 )(| v | + | v | ) uˆ 2 , 2 1 1 K 21 := κ1 (| v 0 |2 + | v 1 |2 ) + (κ1 + 2κ2 )(|u 0 |2 + |u 1 |2 ) vˆ 2 , 2
1 2
1
1
α
1
1
1
1
1
1
1
1
1
R 1 := i (δt u 2 − ut2 ) − β(−) 2 (uˆ 2 − u 2 ) + γ (uˆ 2 − u 2 ) + ζ ( vˆ 2 − v 2 )
1 1 1 + K 11 − κ1 |u 2 |2 + (κ1 + 2κ2 )| v 2 |2 u 2 ,
1
1
1
α
1
1
1
R 22 := i (δt v 2 − v t2 ) − β(−) 2 ( vˆ 2 − v 2 ) + γ ( vˆ 2 − v 2 ) + ζ (uˆ 2 − u 2 )
1
1
+ K 21 − κ1 | v 2 |2 + (κ1 + 2κ2 )|u 2 |2
1 2 . v
Suppose that the exact solutions of (1.1)–(1.3) are smooth enough, then by using Taylor’s Theorem, it follows that n+ 12
R1
n+ 12
≤ C R τ 2,
R2
≤ C R τ 2 , 0 ≤ n ≤ N − 1,
(4.12)
where C R > 0 is a constant independent of τ but dependent of u and v. The deduction of (4.12) is similar as Refs. [42,29]. Therefore, for brevity of the paper, we don’t state here. The next things we need do is to construct the error equations. For attaining this purpose, we firstly define the error functions as
enu = un − U n = (un − P h un ) + ( P h un − U n ) := ρun + ηnu , env := v n − V n = ( v n − P h v n ) + ( P h v n − V n ) := ρ vn + ηnv . For 1 ≤ n ≤ N − 1, taking χ = χh in (4.8)–(4.9), subtracting the discrete systems (3.6)–(3.7) from the resulting equations, and by using the definition of the Ritz projection P h , we obtain n+ 12
i (δt ηu
n+ 12
, χh ) − β B (ηˆ u
n+ 1 i (δt v 2 ,
η
n+ 12
, χh ) + ( K 1n+1 − Ln1+1 , χh ) + γ (ηˆ u
n+ 12 , h) − β B( ˆ v
χ
η
n+ 12
, χh ) + ζ (η˜ v
n+ 1 ( ˆv 2 ,
χh ) + ( K 2n+1 − Ln2+1 , χh ) + γ η
n+ 12 , h) + ζ ( ˜u
χ
η
n+ 12
, χh ) = ( R u
χh ) =
, χh ),
n+ 1 (R v 2 ,
χh ),
(4.13) (4.14)
where n+ 12
Ru
n+ 12
:= R 1
n+ 12
− i δt ρu
n+ 12
− γ ρˆu
n+ 12
− ζ ρ˜ v
n+ 12
, Rv
n+ 12
:= R 2
n+ 12
− i δt ρ v
n+ 12
− γ ρˆ v
n+ 12
− ζ ρ˜u
.
Meanwhile, for n = 1, we have 1
1
1
1
1
1 2
1 2
1 2
1 2
1 2
i (δt ηu2 , χh ) − β B (ηˆ u2 , χh ) + ( K 11 − L 11 , χh ) + γ (ηˆ u2 , χh ) + ζ (ηˆ v2 , χh ) = ( R u2 , χh ), i (δt η v , χh ) − β B (ηˆ v , χh ) + ( K 21 − L 12 , χh ) + γ (ηˆ v , χh ) + ζ (ηˆ u , χh ) = ( R v , χh ),
(4.15) (4.16)
where 1
1
1
1
1
1
1
1
1
1
R u2 := R 12 − i δt ρu2 − γ ρˆu2 − ζ ρˆ v2 , R v2 := R 22 − i δt ρ v2 − γ ρˆ v2 − ζ ρˆu2 . Applying Lemma 7 and (4.12), one easily gets that for n+ 12
Ru and for
n+ 12
≤ C (τ 2 + hr +1 ), R v
α = 3/2,
α = 3/2,
≤ C (τ 2 + hr +1 ), 0 ≤ n ≤ N − 1,
(4.17)
M. Li et al. / Journal of Computational Physics 358 (2018) 256–282
n+ 12
Ru
n+ 12
≤ C (τ 2 + hr +1−σ ), R v
≤ C (τ 2 + hr +1−σ ), 0 ≤ n ≤ N − 1, σ ∈ (0, 1/2),
where C > 0 is a constant independent of h and
265
(4.18)
τ.
Theorem 2. Suppose that the exact solutions u and v of the system (1.1)–(1.3) with the boundary conditions (1.5) are smooth enough. Then, there exist sufficient small h0 > 0 and τ0 > 0, such that when 0 < h ≤ h0 and 0 < τ ≤ τ0 , we have the following optimal error estimates of the discrete scheme (3.6)–(3.7) and the initial-step scheme (3.8)–(3.10),
3
max ( enu + env ) ≤ C (τ 2 + hr +1 ),
α = ,
1≤n≤ N
(4.19)
2
3
max ( enu + env ) ≤ C (τ 2 + hr +1−σ ),
α = , σ ∈ (0, 1/2),
1≤n≤ N
(4.20)
2
and α
max (|enu | α + |env | α ) ≤ C (τ 2 + hr +1− 2 ),
1≤n≤ N
2
(4.21)
2
where C > 0 is a positive constant which is independent of h and τ . Proof. We firstly consider the case
α = 32 . For n = 1, by the mass conservation (4.2), we have
M 1 = U 1 2 + V 1 2 = M 0 ≤ C M 1 .
(4.22)
Then, the energy conservation (4.3) yields
E 1 = −β |U 1 |2α + | V 1 |2α 2
+ γ ( U 1 2 + V 1 2 ) + 2τ F 1 = E 0 ,
2
(4.23)
which means one by using (4.22),
|β| |U 1 |2α + | V 1 |2α ≤ |γ |C M 1 + | E 0 | + 2τ | F 1 |. 2
(4.24)
2
It follows from the definition of F 1 and (4.22) that
| F 1| = ≤
|κ1 | 4τ
( U 1 4L 4 + V 1 4L 4 ) +
| κ 1 | + |κ 1 + 2 κ 2 | 4τ
| κ 1 + 2 κ2 | 2τ
U 1 V 1 2 +
( U 1 4L 4 + V 1 4L 4 ) +
|ζ |C M 1 2τ
|ζ |
τ
Re (U 1 , V 1 ) (4.25)
.
Taking p = 4 and
μ = α /2 in (4.7), and by using Young’s inequality and (4.22), one concludes that 2α −4μ0 α −4μ
U 1 4L 4 + V 1 4L 4 ≤ C μ0 ε U 1 2 α + V 1 2 α + 2C μ0 C (ε )C M 1 0 . H
H
2
2
From Lemma 1 and Lemma 3, we can assume that |U 1 |
U 1 2
H
α 2
= U 1 2 + |U 1 |2
H
α 2
H
α 2
(4.26)
≤ C α |U 1 | α2 , where C α > 0 is a constant. Thus, we have
≤ U 1 2 + C α2 |U 1 |2α . 2
Therefore, it follows from (4.26) that
U 1 4L 4
+ V 1 4L 4
≤ C μ0 ε C M 1 +
2
C α |U 1 |2α 2
+
2
C α | V 1 |2α 2
2α −4μ0 α −4μ0
+ 2C μ0 C (ε )C M 1
.
(4.27)
Substituting (4.27) into (4.25), and then by (4.24), we obtain that
|U 1 |2α + | V 1 |2α ≤ C 1 ε (|U 1 |2α + | V 1 |2α ) + C 2 , 2
2
2
2
where
(|κ1 | + |κ1 + 2κ2 |)C μ0 C α2 , 2|β| 2α −4μ0 1 C M1 ε α −4μ (|γ | + |ζ |)C M 1 + | E 0 | + (|κ1 | + |κ1 + 2κ2 |)C μ0 . C 2 := + C (ε )C M 1 0 |β| 2 C 1 :=
(4.28)
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M. Li et al. / Journal of Computational Physics 358 (2018) 256–282
Taking
ε = C 1−1 /2 in (4.28), we conclude that
|U 1 |2α + | V 1 |2α ≤ 2C 2 . 2
(4.29)
2
Then, from Lemma 6 and above analysis, if we take p = +∞ and
μ0 = α /2, one leads to
U 1 2L ∞ + V 1 2L ∞ ≤ C (|U 1 |2α + | V 1 |2α ), 2
(4.30)
2
which immediately implies one by using (4.29)–(4.30),
U 1 L ∞ + V 1 L ∞ ≤ C (|U 1 | α2 + | V 1 | α2 ) ≤ C M 2 . 1
Next, substituting arrive at
(4.31)
1
χh = ηˆ u2 in (4.15), and χh = ηˆ v2 in (4.16), and taking the imaginary part of the resulting equations, we
ηu1 2 − ηu0 2 2τ
η1v 2 − η0v 2 2τ
1
1
1
1
1
+ Im( K 11 − L 11 , ηˆ u2 ) + ζ Im(ηˆ v2 , ηˆ u2 ) = Im( R u2 , ηˆ u2 ), 1
1
1
1
(4.32)
1
+ Im( K 21 − L 12 , ηˆ v2 ) + ζ Im(ηˆ u2 , ηˆ v2 ) = Im( R v2 , ηˆ v2 ).
(4.33)
Adding (4.32) and (4.33), and then using the Cauchy–Schwarz inequality and Young’s inequality, it holds that
( ηu1 2 + η1v 2 ) − ( ηu0 2 + η0v 2 ) 2τ
≤ C ( ηu1 2 + η1v 2 ) + C ( ηu0 2 + η0v 2 ) + C K 11 − L 11 2 + C K 21 − L 12 2 1
1
+ C R u2 2 + C R v2 2 . Notice that
+ (| v 0 |2 + | v 1 |2 )e 1v , 4 4 κ1 κ 1 + 2 κ2 1 1 1 1 1 1 ˆ 12 0 2 1 2 1 1 1 1 1 ˆ 12 0 2 1 2 1 ¯ ¯ ¯ ¯ K 2 − L2 = 2(e v v + V e v ) V + (| v | + | v | )e v + 2(e u u + U e u )U + (|u | + |u | )e u .
K 11 − L 11 =
(4.34)
κ1
2(e 1u u¯ 1 + U 1 e¯ 1u )Uˆ 2 + (|u 0 |2 + |u 1 |2 )e 1u + 1
κ 1 + 2 κ2
4
2(e 1v v¯ 1 + V 1 e¯ 1v ) Vˆ
1 2
4
Since the exact solutions of (1.1)–(1.3) are smooth enough, by (4.31) and Lemma 7, we obtain
K 11 − L 11 2 + K 21 − L 12 2 ≤ C ( e 1u 2 + e 1v 2 ) ≤ C ( ηu1 2 + η1v 2 ) + Ch2r +2 .
(4.35)
Substituting (4.17) and (4.35) into (4.34) yields
( ηu1 2 + η1v 2 ) − ( ηu0 2 + η0v 2 ) 2τ Then for a sufficiently small
ηu1 2 + η1v 2 ≤
≤ C ( ηu1 2 + η1v 2 ) + C ( ηu0 2 + η0v 2 ) + C (τ 4 + h2r +2 )
(4.36)
τ , by Lemma 7, we obtain
1 + 2C τ 1 − 2C τ
( ηu0 2 + η0v 2 ) +
2C τ 1 − 2C τ
(τ 4 + h2r +2 ) ≤ C (τ 4 + h2r +2 ),
(4.37)
which immediately implies
e 1u + e 1v ≤ ( ρu1 + ρ v1 ) + ( ηu1 + η1v ) ≤ C (τ 2 + hr +1 ). Next, we estimate |e 1u | α + |e 1v | α . Denote 2
1
2
1
(4.38)
1
χh = δt ηu2 in (4.15), and take the real part of the resulting equation to arrive at
1
1
1
1
1
1
1
−β B (ηˆ u2 , δt ηu2 ) + Re ( K 11 − L 11 , δt ηu2 ) + γ Re (ηˆ u2 , δt ηu2 ) + ζ Re (ηˆ v2 , δt ηu2 ) = Re ( R u2 , δt ηu2 ), which means
|η
1 2 u| α 2
− |η
0 2 u| α 2
=
2
β
Re
K 11
−
L 11
1 2
1 2
1 2
+ γ ηˆ u + ζ ηˆ v − R u , η − η 1 u
0 u
.
(4.39)
We observe from (4.17), (4.35) and (4.37) that 1 1 1 1 K − L 1 + γ ηˆ u2 + ζ ηˆ v2 − R u2 ≤ C (τ 2 + hr +1 ). 1 1
(4.40)
M. Li et al. / Journal of Computational Physics 358 (2018) 256–282
267
From (4.39), by Cauchy–Schwarz inequality and Lemma 3, we have
|ηu1 |2α − |ηu0 |2α ≤ 2
2
2
|β|
1 1 1 K 1 − L 1 + γ ηˆ u2 + ζ ηˆ v2 − R u2 (|η1 | α + |η0 | α ). u 2 u 2 1 1
(4.41)
It follows from (4.40)–(4.41) and Lemma 7 that
|ηu1 | α2 ≤ |ηu0 | α2 +
2
|β|
1 1 1 α
K 11 − L 11 + γ ηˆ u2 + ζ ηˆ v2 − R u2 ≤ C (τ 2 + hr +1− 2 ).
(4.42)
Similarly, we can also get α
|η1v | α2 ≤ C (τ 2 + hr +1− 2 ).
(4.43)
Therefore, by Lemma 7, we have α
|e 1u | α2 + |e 1v | α2 ≤ C (τ 2 + hr +1− 2 ).
(4.44)
It follows from (4.38) and (4.44) that we have obtained the error estimates (4.19) and (4.21) for n = 1. Subsequently, we use mathematical induction to prove (4.19). Suppose that (4.19) holds for 2 ≤ j ≤ n, 2 ≤ n ≤ N − 1, and we prove that it also holds for j = n + 1. According to the assumptions, we have α
e u L ∞ + e v L ∞ ≤ C (|e u | α2 + |e v | α2 ) ≤ C (τ 2 + hr +1− 2 ), j
j
j
j
2 ≤ j ≤ n.
(4.45)
Hence, we have
U j L ∞ + V j L ∞ ≤ C (|U j | α2 + | V j | α2 ) ≤ C M , 2 ≤ j ≤ n.
(4.46)
We observe that
K 1n+1 − L n1+1
=
κ1 2
n 2
(3|u | − |u
n −1 2
| )uˆ
n+ 12
n 2
− (3|U | − |U
1 κ 1 + 2 κ2 n+ 12 ˆ + (3| v n |2 − | v n−1 |2 )uˆ n+ 2 | )U
n −1 2
2
− (3| V n |2 − | V n−1 |2 )Uˆ n+ 2 1 κ1 n+ 1 = 3(|un |2 − |U n |2 ) − (|un−1 |2 − |U n−1 |2 ) uˆ n+ 2 + (3|U n |2 − |U n−1 |2 )ˆe u 2 2 1 κ 1 + 2 κ2 n+ 1 3(| v n |2 − | V n |2 ) − (| v n−1 |2 − | V n−1 |2 ) uˆ n+ 2 + (3| V n |2 − | V n−1 |2 )ˆe u 2 . . + 1
2
Therefore, from (4.46), it follows that
K 1n+1 − Ln1+1 ≤ C ( enu+1 + enu + enu−1 + env + env−1 ).
(4.47)
According to (4.47) and Lemma 7, one arrives at
K 1n+1 − Ln1+1 ≤ C ηnu+1 + ηnu + ηnu−1 + ηnv + ηnv−1 + C (τ 2 + hr +1 ).
(4.48)
Similarly, we have
K 2n+1 − Ln2+1 ≤ C ηnv+1 + ηnv + ηnv−1 + ηnu + ηnu−1 + C (τ 2 + hr +1 ). n+ 12
Substituting χh = ηˆ u equations, we obtain
ηnu+1 2 − ηnu 2 2τ
ηnv+1 2 − ηnv 2 2τ
and
n+ 12
χh = ηˆ v
(4.49)
into (4.13) and (4.14) respectively, and taking the imaginary parts of the resulting
n+ 12
+ Im( K 1n+1 − Ln1+1 , ηˆ u
n+ 12
+ Im( K 2n+1 − Ln2+1 , ηˆ v
n+ 12
) + ζ Im(η˜ v
n+ 12
) + ζ Im(η˜ u
n+ 12
, ηˆ u
n+ 12
, ηˆ v
n+ 12
) = Im( R u
n+ 12
) = Im( R v
n+ 12
, ηˆ u
n+ 12
, ηˆ v
),
(4.50)
).
(4.51)
Adding (4.50) and (4.51), by using Cauchy–Schwarz inequality and Young’s inequality to the resulting equations, then according to (4.17) and (4.48)–(4.49), we have
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M. Li et al. / Journal of Computational Physics 358 (2018) 256–282
ηnu+1 2 + ηnv+1 2 ≤ ηnu 2 + ηnv 2 + C τ ( ηnu+1 2 + ηnu 2 + ηnu−1 2 + ηnv+1 2 + ηnv 2 + ηnv−1 2 ) + C τ (τ 2 + hr +1 )2 .
(4.52)
Hence, it follows from discrete Gronwall inequality that
ηnu+1 2 + ηnv+1 2 ≤ C (τ 2 + hr +1 )2 .
(4.53)
Therefore, from Lemma 7, one obtains
enu+1 + env+1 ≤ C (τ 2 + hr +1 ). Next, we turn to prove the the resulting equation, we get
−
β 2τ
n+ 12
2
n+ 12
Similarly, we denote
β 2τ
α /2-norm of the error functions. Denoting χh = δt ηu
(|ηnu+1 |2α − |ηnu |2α ) + Re ( K 1n+1 − Ln1+1 , δt ηu
= Re ( R u
−
(4.54) n+ 12
2
n+ 12
, δt ηu
(4.55)
χh = δt η v
in (4.14), and take the real part of the resulting equation to arrive at n+ 12
(|ηnv+1 |2α − |ηnv |2α ) + Re ( K 2n+1 − Ln2+1 , δt η v n+ 12
= Re ( R v
2
n+ 12
, δt η v
γ n+ 1 n+ 1 ( ηnu+1 2 − ηnu 2 ) + ζ Re (η˜ v 2 , δt ηu 2 ) 2τ
). n+ 12
2
)+
in (4.13), and taking the real part of
)+
γ n+ 1 n+ 1 ( ηnv+1 2 − ηnv 2 ) + ζ Re (η˜ u 2 , δt η v 2 ) 2τ (4.56)
).
Then, it follows from (4.55) and (4.56) that
|ηnu+1 |2α + |ηnv+1 |2α −
γ
( ηnu+1 2 + ηnv+1 2 ) γ 2τ n+ 1 = |ηnu |2α + |ηnv |2α − ( ηnu 2 + ηnv 2 ) + Re ( K 1n+1 − L n1+1 , δt ηu 2 ) 2 2 β β 2
2
β
n+ 12
+ Re ( K 2n+1 − Ln2+1 , δt η v
n+ 12
) + ζ Re (η˜ v
n+ 12
, δt ηu n+ 12 n+ 12 n+ 12 n+ 12 + Re ( R u , δt ηu ) + Re ( R v , δt η v ) .
n+ 12
) + ζ Re (η˜ u
n+ 12
, δt η v
) (4.57)
We now start to estimate the items at the right hand of (4.57). By (4.13), one obtains n+ 12
δt ηu
α
n+ 12
+ i ( K 1n+1 − Ln1+1 ) + i γ ηˆ u
α
n+ 12
+ i ( K 2n+1 − Ln2+1 ) + i γ ηˆ v
= −i β(−) 2 ηˆ u
n+ 12
n+ 12
+ i ζ η˜ v
n+ 12
− i Ru
,
(4.58)
.
(4.59)
and n+ 12
δt η v
= −i β(−) 2 ηˆ v
n+ 12
n+ 12
+ i ζ η˜ u
n+ 12
− iRv
Therefore, by Lemma 3, (4.17) and (4.48), we have
1
Re ( K n+1 − Ln+1 , δt ηnu+ 2 )
1 1
α n+ 1 n+ 12 n+ 12 n+ 12
n +1 n +1 n +1 n +1 2
2 = Re ( K 1 − L 1 , −i β(−) ηˆ u + i ( K 1 − L 1 ) + i γ ηˆ u + i ζ η˜ v − i R u )
n +1 n +1 2 ≤ C | K 1 − L 1 | α + C |ηnu+1 |2α + |ηnu |2α + K 1n+1 − Ln1+1 2 + ηnu+1 2 + ηnu 2 2 2 2 1 n+ + ηnv 2 + ηnv−1 2 + R u 2 2 .
(4.60)
Moreover, similar as (4.48), it obtains from Lemma 7 that
α | K 1n+1 − Ln1+1 | α2 ≤ C |ηnu+1 | α2 + |ηnu | α2 + |ηnu−1 | α2 + |ηnv | α2 + |ηnv−1 | α2 + C (τ 2 + hr +1− 2 ). Then, by (4.17), substituting (4.61) into (4.60) yields
(4.61)
M. Li et al. / Journal of Computational Physics 358 (2018) 256–282
1
Re ( K n+1 − Ln+1 , δt ηnu+ 2 ) ≤ C |ηn+1 |2α + |ηn |2α + |ηn−1 |2α + |ηn |2α + |ηn−1 |2α + C (τ 2 + hr +1− α2 )2 . u u u v v 1 1
2 2 2 2 2
269
(4.62)
Similarly, we have
1
Re ( K n+1 − Ln+1 , δt ηnv+ 2 ) ≤ C |ηn+1 |2α + |ηn |2α + |ηn−1 |2α + |ηn |2α + |ηn−1 |2α + C (τ 2 + hr +1− α2 )2 . v v v u u 2 2
2 2 2 2 2
(4.63)
From (4.58), it follows that
1 1
1 1 1 1 1
Re (η˜ nv+ 2 , δt ηnu+ 2 ) = Re (η˜ nv+ 2 , −i β(−) α2 ηˆ nu+ 2 + i ( K n+1 − Ln+1 ) + i γ ηˆ nu+ 2 + i ζ η˜ nv+ 2 − i R nu+ 2 )
1 1
n+ 1 n+ 1 n+ 1 ≤ C |η˜ v 2 |2α + |ηˆ u 2 |2α + C K 1n+1 − Ln1+1 2 + R u 2 2 . 2
(4.64)
2
Hence, by (4.17), (4.48), (4.53) and Lemma 3, we have
1 1
Re (η˜ nv+ 2 , δt ηnu+ 2 ) ≤ C |ηn+1 |2α + |ηn |2α + |ηn |2α + |ηn−1 |2α + C (τ 2 + hr +1 )2 . u u v v
2 2 2 2
(4.65)
Similarly, one obtains that
1 1
Re (η˜ nu+ 2 , δt ηnv+ 2 ) ≤ C |ηn+1 |2α + |ηn |2α + |ηn |2α + |ηn−1 |2α + C (τ 2 + hr +1 )2 . v v u u
2 2 2 2
(4.66)
In addition, we have
1 1
1 1 1 1 1
Re ( R nu+ 2 , δt ηnu+ 2 ) = Re ( R nu+ 2 , −i β(−) α2 ηˆ nu+ 2 + i ( K n+1 − Ln+1 ) + i γ ηˆ nu+ 2 + i ζ η˜ nv+ 2 − i R nu+ 2 )
1 1
n+ 1 n+ 1 n+ 1 ≤ C | R u 2 |2α + |ηˆ u 2 |2α + C K 1n+1 − Ln1+1 2 + η˜ v 2 2 . 2
(4.67)
2
It would clearly be easy to obtain one by analogy with (4.17), n+ 12
|Ru
α
n+ 12
| α ≤ C (τ 2 + hr +1− 2 ), | R v 2
α
| α ≤ C (τ 2 + hr +1− 2 ), 0 ≤ n ≤ N − 1. 2
(4.68)
Therefore, it follows from (4.53), (4.67) and (4.68) that
1 1
Re ( R nu+ 2 , δt ηnu+ 2 ) ≤ C |ηn+1 |2α + |ηn |2α + C (τ 2 + hr +1− α2 )2 . u u
2 2
(4.69)
Similarly, we have
1 1
Re ( R nv+ 2 , δt ηnv+ 2 ) ≤ C |ηn+1 |2α + |ηn |2α + C (τ 2 + hr +1− α2 )2 . v v
2 2
(4.70)
Substituting (4.58)–(4.70) into (4.57) concludes
|ηnu+1 |2α + |ηnv+1 |2α −
γ
( ηnu+1 2 + ηnv+1 2 ) γ ≤ |ηnu |2α + |ηnv |2α − ( ηnu 2 + ηnv 2 ) + C τ |ηnu+1 |2α + |ηnu |2α + |ηnu−1 |2α + |ηnv+1 |2α + |ηnv |2α + |ηnv−1 |2α 2 2 2 2 2 2 2 2 β 2
2
β
α
+ C (τ 2 + hr +1− 2 )2 . Accordingly, for a sufficiently small
τ , it follows from the discrete Gronwall inequality and (4.53) that α
|ηnu+1 |2α + |ηnv+1 |2α ≤ C (τ 2 + hr +1− 2 )2 . 2
2
Then, by the triangular inequality and Lemma 7, we obtain α
|enu+1 | α2 + |env+1 | α2 ≤ C (τ 2 + hr +1− 2 ).
(4.71)
Combining (4.54) with (4.71), and by virtue of the mathematical induction method, we have completed the proof of (4.19). According to Lemma 7, the stated convergence results for α = 3/2 hold analogously. Thus, the proof is complete. 2
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We also observe from Theorem 2 that the discrete solutions are bounded in the L 2 -norm and L ∞ -norm. Theorem 3. The solutions of the Galerkin finite element scheme (3.6)–(3.7) and the initial step scheme (3.8)–(3.10) are bounded, i.e., there exist positive constants C M 1 and C M 2 , such that
U n + V n ≤ C M 1 , and
U n L ∞ + V n L ∞ ≤ C M 2 . Remark 2. It is founded from the proof process of Theorem 2 that there are no any requirements for the grid ratios, that is to say, the discrete schemes (3.6)–(3.7) and (3.8)–(3.10) are unconditionally convergent and bounded. Therefore, in terms of the Galerkin finite element methods, the schemes in this paper are more benefit than ones of Refs. [29,30]. Remark 3. In order to obtain the unconditional error estimates in the L 2 - and H α -norms, it is observed along the proof 2 process of Theorem 2 that we need not utilize the conservation properties for n > 1. In this point, we can say that the proof method is better than one in Ref. [42]. 5. Fast implementation of the iterative numerical scheme In this section, we discuss the practical implementations of our proposed numerical scheme. First of all, we introduce the linearized iterative scheme for implementing the proposed conservative finite element method. 5.1. The linearized iterative scheme We observe that the coefficient matrix of the linearized discrete scheme (3.6)–(3.7) should be generated at each time step, which is time consuming. Therefore, one can propose a new iterative algorithm as
i
U n +1 ( s +1 ) − U n
τ +
i
γ 2
2
(U n+1(s+1) + U n , χh ) + ζ ( V˜ n+ 2 , χh ) = 0, 1
V n +1 ( s +1 ) − V n
τ +
γ 2
β n +1 ( s ) , χh − B (U n+1(s+1) + U n , χh ) + ( L 1 , χh )
, χh −
β 2
(5.1) n +1 ( s )
B ( V n+1(s+1) + V n , χh ) + ( L 2
, χh )
( V n+1(s+1) + V n , χh ) + ζ (U˜ n+ 2 , χh ) = 0, 1
(5.2)
where n +1 ( s )
L1
n +1 ( s ) L2
:= :=
1 4 1 4
κ1 3|U n |2 − |U n−1 |2 + (κ1 + 2κ2 ) 3| V n |2 − | V n−1 |2
n 2
κ1 3 | V | − | V
n −1 2
|
(U n+1(s) + U n ),
n 2 n −1 2 + (κ1 + 2κ2 ) 3|U | − |U ( V n+1(s) + V n ), |
with the boundary conditions
U n +1 ( s +1 ) = 0,
V n +1 ( s +1 ) = 0,
x ∈ R\Ω,
(5.3)
and the initial iterative conditions
U n +1 ( 0) =
U n, n = 0, 2U n − U n−1 , n ≥ 1,
(5.4)
M. Li et al. / Journal of Computational Physics 358 (2018) 256–282
271
and
V
n +1 ( 0)
=
V n, n = 0, 2V n − V n−1 , n ≥ 1.
(5.5)
5.2. Fast solution method based on the structure of coefficient matrices In order to establish a fast algorithm in numerically solving the system (1.1)–(1.3), we rewrite the iterative scheme (5.1)–(5.5) in matrix form. To this end, we adopt the finite element space based on the Lagrange piecewise-linear shape functions. Let us suppose that the partition in the spatial direction is uniform, i.e., h = xi +1 − xi , 0 ≤ i ≤ M − 1. Denote
U n |e i = f in L 1i + g ni L 2i ,
V n |e i = pni L 1i + qni L 2i ,
0 ≤ n ≤ N,
and n( s ) i L1
U n(s) |e i = f i
n( s ) i L2,
n( s ) i L1
V n(s) |e i = p i
+ gi
n( s ) i L2,
+ qi
0 ≤ n ≤ N,
s = 0, 1 , 2 , . . . ,
where
L 1i
=
x i +1 − x , h
x ∈ ei ,
0,
else,
and
L 2i
=
x−xi , h
x ∈ ei ,
0,
else.
Rewrite (5.1)–(5.2) as
i+
γτ 2
= i− i+
γτ
2
2
βτ
(U n , χh ) +
2
γτ
= i−
βτ (U n+1(s+1) , χh ) − B (U n+1(s+1) , χh ) n +1 ( s )
B (U n , χh ) − τ ( L 1
2
, χh ) − ζ τ ( V˜ n+ 2 , χh ),
(5.6)
, χh ) − ζ τ (U˜ n+ 2 , χh ).
(5.7)
1
βτ ( V n+1(s+1) , χh ) − B ( V n+1(s+1) , χh )
γτ
2
βτ
( V n , χh ) +
2
n +1 ( s )
B ( V n , χh ) − τ ( L 2
2
1
Therefore, the iterative scheme (5.6)–(5.7) can be written as the following matrix forms
i+
γτ
2
M 1 U n +1 ( s +1 ) −
βτ 2
M 2 U n +1 ( s +1 )
1 βτ γτ τ n +1 ( s ) n +1 ( s ) = i− M1 U n + (U + U n ) + ζ τ M 1 V˜ n+ 2 , M2 U n − Mu 2
i+
γτ
2
= i−
2
M 1 V n +1 ( s +1 ) −
γτ 2
4
M1 V n + n+1(s)
βτ 2
βτ 2
M 2 V n +1 ( s +1 )
M2 V n −
n+1(s)
where the M 1 , M u and M v Toeplitz-like structure expressed as
(5.8)
τ 4
n +1 ( s )
Mv
( V n+1(s) + V n ) + ζ τ M 1 U˜ n+ 2 , 1
(5.9)
are ( M + 1) × ( M + 1) tridiagonal matrices [46,29], and M 2 has ( M + 1) × ( M + 1)
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M. Li et al. / Journal of Computational Physics 358 (2018) 256–282
⎛
c1
ξ
c2
⎜ ⎜ c1 d0 d1 ⎜ ⎜ ⎜ c d1 d0 ⎜ 2 ⎜ . . . ⎜ . .. .. ⎜ . M2 = ⎜ . .. ⎜ .. . ⎜ c M −3 ⎜ ⎜ .. ⎜ c M −2 d M −3 . ⎜ ⎜ ⎝ c M −1 d M −2 d M −3 g δ1 δ2
· · · c M −3 c M −2 c M −1 .. . d M −3 d M −2 d2 .. . d M −3 d1 d2 .. .. .. .. . . . . d1
d0
..
..
.
..
d1
d2
.
d0
d1
. ···
···
..
. δ3
d1
d0
δ M −1
g
⎞
⎟ δ1 ⎟ ⎟ ⎟ δ2 ⎟ ⎟ .. ⎟ ⎟ . ⎟ ⎟. ⎟ δ M −3 ⎟ ⎟ ⎟ δ M −2 ⎟ ⎟ ⎟ δ M −1 ⎠
ς
For convenience, let us denote
⎧ γτ βτ ⎪ ⎪ ⎨ Aˆ = i + 2 M 1 − 2 M 2 , ⎪ ⎪ ⎩ Bˆ = i − γ2τ M 1 + β2τ M 2 . ˆ and Bˆ are also Toeplitz-like matrices, and thus we need only O ( M + 1) memory to store Aˆ or B. ˆ It implies Obviously, both A ˆ (or B) ˆ can be evaluated by FFTs in O (( M + 1) log( M + 1)) operations. In fact, we that the matrix–vector product involving A can remove some elements in the coefficient matrices and right-hand sides corresponding to the discretization of boundary value conditions, then the FEM discretized scheme can be simplified as follows,
=
=
i+
M1 −
2
γτ
i−
i+
γτ ˜ 2
γτ 2
i−
β τ ˜ ˜ n +1 ( s +1 ) M2 U 2
˜1− M
2
2
M1 +
1 β τ ˜ ˜ n τ ˜ n +1 ( s ) ˜ n +1 ( s ) ˜ n ˜ 1 U˜ n+ 2 , M2 V − M v (V + V ) + ζτ M
2
n+1(s)
V˜ n , U˜
and V˜
(5.11)
4
˜ i = M i (2 : M , 2 : M ), i = 1, 2 and M ˜ where M n+ 12
(5.10)
4
β τ ˜ ˜ n +1 ( s +1 ) M2 V
γτ ˜ 2
1 β τ ˜ ˜ n τ ˜ n +1 ( s ) ˜ n +1 ( s ) ˜ n ˜ 1 V˜ n+ 2 , M2 U − Mu (U + U ) + ζτ M
˜1+ M
n+ 12
= M n+1(s) (2 : M , 2 : M ), ∈ {u , v }. Moreover, U˜ n+1(s+1) , V˜ n+1(s+1) , U˜ n ,
are obtained by deleting the elements corresponding to U n+1(s+1) , V n+1(s+1) , U n , V n , U˜ n+2 and V˜ n+ 2 γτ ˜ βτ ˜ from the discretization of boundary value conditions. In addition, it is meaningful to remark that A = i + 2 M 1 − 2 M2
and B = i −
γτ 2
1
˜1+ M
βτ 2
1
˜ 2 are both complex symmetric Toeplitz matrices. In other words, we need to solve several M
complex symmetric Toeplitz systems in each time-level, whereas these coefficient matrices A are time independent [47–49]. Meanwhile, the Toeplitz matrix–vector product B v (v ∈ C M −1 is an arbitrary vector) can be quickly evaluated by FFTs in O (( M − 1) log( M − 1)) operations. We have the following theorem [50] to compute the inverse of complex symmetric Toeplitz matrix [51,52] via solving one fundamental Toeplitz linear system, i.e., 1 Theorem 1. Let T = {a p −q }kp− , p =0 be a complex symmetric Toeplitz matrix. If the Toeplitz system T x = e 1 (where e 1 is the first column vector of the identity matrix of order k) is solvable and x0 = 0, x = (x0 , x2 , . . . , xk−1 ), then T is invertible and
⎡⎛
T −1 =
x0 ⎢ ⎜ 1 ⎢⎜ x1
⎢⎜
. x0 ⎣⎝ ..
xk−1
=
1 x0
0 x0
.. . xk−2
( Lˆ Lˆ T − Rˆ Rˆ T ),
⎞⎛ ⎞ ⎛ ⎞⎛ ⎞⎤ ··· 0 x0 x1 · · · xk−1 0 ··· 0 0 0 xk−1 · · · x1 ⎟ ⎜ ⎜ ⎜. .. .. ⎟⎥ ··· 0 ⎟ 0 0⎟ .. ⎟ ⎜ 0 x0 · · · xk−2 ⎟ ⎜ xk−1 · · · ⎟ ⎜ .. ⎥ . . . ⎟ ⎟ ⎟⎥ ⎜ .. ⎟ ⎜ .. .. . . .. ⎟ − ⎜ .. . . .. . . . . ⎠ ⎠ ⎝ ⎠ ⎠ ⎝ ⎝ . . . . . . . . . . 0 0 · · · xk−1 ⎦ · · · x0 0 0 ··· x0 x1 · · · xk−1 0 0 0 ··· 0
Lˆ , Rˆ ∈ Ck×k .
According to Theorem 1, it notes that the inverse matrix A −1 can be easily computed without additional storage for dense matrices. We only need to solve one Toeplitz linear system Ax = e 1 rather than several Toeplitz systems in each
M. Li et al. / Journal of Computational Physics 358 (2018) 256–282
273
Algorithm 1 Compute y = A −1 v, where v ∈ C M −1 is the arbitrarily given vector. 1: Solve the fundamental system of linear equations,
Ax = e 1 , 2: 3: 4: 5: 6:
e 1 = [1, 0 . . . , 0] T ∈ C M −1
(5.12)
by Krylov subspace methods with suitable preconditioners; Compute the Toeplitz matrix–vector product y 2 = Rˆ T v via FFTs; Compute the Toeplitz matrix–vector product y 1 = Rˆ y 2 via FFTs; Compute the Toeplitz matrix–vector product z2 = Lˆ T v by FFTs; ˆ 2 by FFTs; Compute the Toeplitz matrix–vector product z1 = Lz Calculate the vector y = x1 ( z1 − y 1 ). 0
Fig. 1. The exact solutions and the numerical solutions of Example 1.
time-level, it should significantly reduce the computational cost and memory [47]. For the sake of clarity, we give the following algorithm to compute the matrix–vector product A −1 v. In Algorithm 1, since the coefficient matrix A is a complex symmetric Toeplitz matrix, the conjugate orthogonal conjugate gradient (COCG) method [51,52] with Strang preconditioner is very suitable for solving such systems, i.e., Eq. (5.12). Moreover, high efficiency of Strang circulant preconditioner for handling space (and time) fractional differential equations has been investigated in [53,52,48]. In the next section, we also show the effectiveness of Strang preconditioner in terms of the elapsed CPU time and the clustered eigenvalues of preconditioned matrices. 6. Numerical experiments The numerical experiments presented in this section have a two-fold objective. They illustrate that the proposed scheme α α can indeed converge with the accuracy of O (hr +1 + τ 2 ) in L 2 -norm and O (hr +1− 2 + τ 2 ) in H 2 -norm. At the same time, they assess the computational efficiency of the fast solution techniques (i.e., Algorithm 1) described in Sect. 5. For the direct solver, we choose built-in function for LU factorization of MATLAB. For the COCG method with circulant preconditioners, the stopping criterion is set as r (k) 2 / r (0) 2 < 10−12 , where r (k) is the residual vector of the linear system after k iterations, and the initial guess is chosen as the zero vector. All experiments were performed on a Windows 7 (32 bit) PC-Intel® Core™ i5-3470 CPU 3.20 GHz, 4 GB of RAM using MATLAB 2014a with machine epsilon 10−16 in double precision floating point arithmetic. By the way, all timings are averages over 20 runs of our algorithms. Example 1. First, to test the accuracy of the discrete scheme, we first consider the following system with source terms α
iut − (−) 2 u + 2(|u |2 + | v |2 )u + u + v = f (x, t ), 0 < x < 1, 0 < t ≤ 1, α
i v t − (−) 2 v + 2(| v |2 + |u |2 ) v + v + u = g (x, t ), 0 < x < 1, 0 < t ≤ 1. The initial and boundary conditions, and the source terms f (x, t ) and g (x, t ) are chosen correspondingly to the exact solutions
u (x, t ) = (t + 1)3 x2 (1 − x)2 ,
v (x, t ) = (t + 1)α x2 (1 − x)2 .
Fig. 1 displays the exact solutions and numerical solutions for α = 1.6. Fig. 2 shows the L 2 -norm error estimates and α corresponding convergence orders with different α . Meanwhile, Fig. 3 gives the H 2 -norm error estimates and corresponding
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M. Li et al. / Journal of Computational Physics 358 (2018) 256–282
Fig. 2. The L 2 -norm error estimates of Example 1 with
Fig. 3. The H α /2 -norm error estimates of Example 1 with
τ = 0.1h.
τ = 0.0005 for upper two figures and h = 0.005 for below two figures.
convergence orders in both the temporal and spatial directions. All these results show that the scheme have the accuracies α α of O (hr +1 + τ 2 ) in L 2 -norm and O (hr +1− 2 + τ 2 ) in H 2 -norm. Because of the existences of the non-zero terms f (x, t ) and g (x, t ), one can notice that the aforementioned discrete conservation laws are no longer valid for this example.
M. Li et al. / Journal of Computational Physics 358 (2018) 256–282
Fig. 4. The L 2 -norm error estimates of Example 2 with
275
τ = 0.1h.
Table 1 The error |un − U n | H α/2 and the order of convergence in space using piecewise P 1 elements for different α (τ = 0.005).
α
h = 0.2
h = 0.1
Order
1.2 1.4 1.6 1.8
4.0782e−02 5.2174e−02 5.1330e−02 2.5153e−02
1.4321e−02 1.9802e−02 2.1026e−02 1.2172e−02
1.5098 1.3977 1.2876 1.0472
Table 2 The error |un − U n | H α/2 and the order of convergence in time using piecewise P 1 elements for different α (h = 0.05).
α
τ = 0.02
τ = 0.01
Order
1.2 1.4 1.6 1.8
1.9036e−02 1.0180e−01 2.4335e−01 3.7071e−01
3.8086e−03 2.0402e−02 4.8096e−02 7.2569e−02
2.3214 2.3190 2.3390 2.3529
Example 2. Second, we consider the following strongly coupled system α
iut − (−) 2 u + 2(|u |2 + | v |2 )u + u + v = 0, α
2
2
i v t − (−) 2 v + 2(| v | + |u | ) v + v + u = 0,
x ∈ Ω = (a, b),
0
