Zhao et al. Advances in Difference Equations 2013, 2013:156 http://www.advancesindifferenceequations.com/content/2013/1/156
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Fourier spectral method for the modified Swift-Hohenberg equation Xiaopeng Zhao1* , Bo Liu1 , Peng Zhang2 , Wenyu Zhang1 and Fengnan Liu1 *
Correspondence:
[email protected] College of Mathematics, Jilin University, Changchun, 130012, China Full list of author information is available at the end of the article 1
Abstract In this paper, we consider the Fourier spectral method for numerically solving the modified Swift-Hohenberg equation. The semi-discrete and fully discrete schemes are established. Moreover, the existence, uniqueness and the optimal error bound are also considered.
1 Introduction In [], Doelman et al. studied the modified Swift-Hohenberg equation ut = –k( + ) u + μu – b|∇u| – u .
()
Setting a = k – μ, considering () in D case, we find that ut + kuxxxx + kuxx + au + b|ux | + u = ,
(x, t) ∈ (, ) × (, T).
()
On the basis of physical considerations, as usual, Eq. () is supplemented with the following boundary value conditions: u(x, t) = uxx (x, t) = ,
x = ,
()
and the initial condition u(x, ) = u (x),
x ∈ (, ),
()
where k > and a, b are constants. u (x) is a given function from a suitable phase space. The Swift-Hohenberg equation is one of the universal equations used in the description of pattern formation in spatially extended dissipative systems (see []), which arise in the study of convective hydrodynamics [], plasma confinement in toroidal devices [], viscous film flow and bifurcating solutions of the Navier-Stokes []. Note that the usual Swift-Hohenberg equation [] is recovered for b = . The additional term b|ux | , reminiscent of the Kuramoto-Sivashinsky equation, which arises in the study of various pattern formation phenomena involving some kind of phase turbulence or phase transition (see [–]), breaks the symmetry u → –u. © 2013 Zhao et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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During the past years, many authors have paid much attention to the Swift-Hohenberg equation (see, e.g., [, , ]). However, only a few people have been devoted to the modified Swift-Hohenberg equation. It was Doelman et al. [] who first studied the modified Swift-Hohenberg equation for a pattern formation system with two unbounded spatial directions that are near the onset to instability. Polat [] also considered the modified Swift-Hohenberg equation. In his paper, the existence of a global attractor is proved for the modified Swift-Hohenberg equation as ()-(). Recently, Song et al. [] studied the long time behavior for the modified Swift-Hohenberg equation in an H k (k ≥ ) space. By using an iteration procedure, regularity estimates for the linear semigroups and a classical existence theorem of a global attractor, they proved that problem ()-() possesses a global attractor in the Sobolev space H k for all k ≥ , which attracts any bounded subset of H k () in the H k -norm. The spectral methods are essentially discretization methods for the approximate solution of partial differential equations. They have the natural advantage in keeping the physical properties of primitive problems. During the past years, many papers have already been published to study the spectral methods, for example, [–]. However, for the other boundary condition, can we also use the Fourier spectral method? The answer is ‘Yes’. Choose a good finite dimensional subspace SN (here, we set SN = span{sin kπx; k = , , . . . , N}), we can also have the basic results for the orthogonal projecting operator PN . In this paper, we consider the Fourier spectral method for the modified Swift-Hohenberg equation. The existence of a solution locally in time is proved by the standard Picard iteration, global existence results are obtained by proving a priori estimate for the appropriate norms of u(x, t). Adjusted to our needs, the results are given in the following form. Theorem . Assume that u ∈ HE (, ) = {v; v ∈ H (, ), v(, t) = v(, t) = } and b ≤ k, then there exists a unique global solution u(x, t) of the problem ()-() for all T ≥ such that u(x, t) ∈ L∞ , T; HE (, ) ∩ L , T; H (, ) . Furthermore, it satisfies
∂u , v + k(uxx , vxx ) – k(ux , vx ) + γ (u, v) ∂t + b |ux | , v + u , v = , ∀v ∈ HE (, ), u(·, ), v = (u , v), ∀v ∈ HE (, ).
()
This paper is organized as follows. In the next section, we consider a semi-discrete Fourier spectral approximation, prove its existence and uniqueness of the numerical solution and derive the error bound. In Section , we consider the full-discrete approximation for problem ()-(). Furthermore, we prove convergence to the solution of the associated continuous problem. In the last section, some numerical experiments which confirm our results are performed. Throughout this paper, we denote L , Lp , L∞ , H k norm in simply by · , · p , · ∞ and · H k .
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2 Semi-discrete approximation In this section, we consider the semi-discrete approximation for problem ()-(). First of all, we recall some basic results on the Fourier spectral method which will be used throughout this paper. For any integer N > , introduce the finite dimensional subspace of HE (, ) SN = span{sin kπx; k = , . . . , N}. Let PN : L (, ) → SN be an orthogonal projecting operator which satisfies (u – PN u, v) = ,
∀v ∈ SN .
()
For the operator PN , we have the following result (see [, ]): (B) PN commutes with derivation on HE (, ), i.e., PN uxx = (PN u)xx ,
∀u ∈ HE (, ).
Using the same method as [, ], we can obtain the following result (B) for problem ()-(): (B) For any real ≤ μ ≤ , there is a constant c such that
u – PN u μ ≤ cN μ– uxx ,
∀u ∈ HE (, ).
Define the Fourier spectral approximation: Find uN (t) =
N
∂uN , vN + k(uNxx , vNxx ) + k(uN , vNxx ) + a(uN , vN ) ∂t + b |uNx | , vN + uN , vN = , ∀vN ∈ SN
j= aj (t) cos jπx ∈ SN
such that
()
for all T ≥ with uN () = PN u . Now, we are going to establish the existence, uniqueness etc. of the Fourier spectral approximation solution uN (t) for all T ≥ . Lemma . Let u ∈ L (, ) and b ≤ k, then problem () has a unique solution uN (t) satisfying the following inequalities: uN (t) ≤ c u ,
where c = e(|a|+)T and c =
T
uNxx (τ ) dτ ≤ c u ,
(|a|+)Tc + k–b
()
for all T ≥ .
Proof Set vN = cos jπx in () for each j ( ≤ j ≤ N ) to obtain d aj (t) = fj a (t), a (t), . . . , aN (t) , dt
j = , , . . . , N,
()
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where all fj : RN → R ( ≤ j ≤ N ) are smooth and locally Lipschitz continuous. Noticing that uN () = PN u , then aj () = (u , cos jπx),
j = , , . . . , N.
()
Using the theory of initial-value problems of the ordinary differential equations, there is a time TN > such that the initial-value problem ()-() has a unique smooth solution (a (t), . . . , aN (t)) for t ∈ [, TN ]. Setting vN = uN in (), we have d
uN + k uNxx + uN ≤ |a| uN + k uNx + |b| |uNx | , uN . dt
()
Noticing that |b| b uN , uNxx ≤ uN + uNxx |b| |uNx | , uN = – and k k uNx = –k(uN , uNxx ) ≤ uNxx + k uN . Summing up, we get d b
uN + k –
uNxx ≤ |a| + k uN . dt
()
Using Gronwall’s inequality, we deduce that
uN ≤ e(|a|+k)t uN () ≤ e(|a|+k)T u = c u . Integrating () from to t, we derive that T T b
uN – uN () + k –
uNxx dt ≤ |a| + k
uN dt.
Hence T b
uNxx dt ≤ |a| + k c T u + u . k– From the above inequality, we obtain the second inequality of () immediately. Therefore, Lemma . is proved. Lemma . Let u ∈ H (, ) and b ≤ k, then the solution uN (t) of problem () satisfying uNx (t) ≤ c ,
T
unxxx (τ ) dτ ≤ c
()
for all T ≥ , where c and c are positive constants depending only on k, a, b, T and u H .
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Proof Setting vN = uNxx in (), we obtain d
uNx + k uNxxx + a uNx = k uNxx + b |uNx | , uNxx + uN , uNxx . dt
()
Notice that
uN , uNxx = – uN uNx , uNx = – uN uNx ≤
and k uNxx = –k(uNx , uNxxx ) ≤
k
uNxxx + uNx .
On the other hand, by Nirenberg’s inequality, we have
uNx ≤ C uNxxx uN , where C is a positive constant independent of N . Hence |b| |b| |b| |uNx | , uNxx ≤ uNxx +
uNx k |b|
uNx + C uNxxx uN ≤ uNxxx + k k |b|
uNx + c uN
≤ uNxxx + k k |b| ≤ uNxxx +
uNx + c a, k, T, u . k
()
Summing up, we get b d
uNx + k uNxxx ≤ |a| + +
uNx + c a, k, T, u . dt k
()
Using Gronwall’s inequality, we immediately obtain
uNx ≤ e(|a|++ ≤ e(|a|++
b )t k
uNx () + kc(a, k, T, u ) |a|k + b + k
b )T k
ux +
kc(a, k, T, u ) ≤ c k, a, b, T, u H . |a|k + b + k
Integrating () from to t, we obtain
T
b |a| + + c T + cT + ux
k = c k, a, b, T, u H .
uNxxx (τ ) dτ ≤ k
Then Lemma . is proved.
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Lemma . Let u ∈ HE (, ) and b ≤ k, then the solution uN (t) of problem (), satisfying
uNxx (t) ≤ c ,
T
uNxxxx (τ ) dτ ≤ c
()
for all T ≥ , where c and c are positive constants, depending only on k, a, b, T and
u H . Proof Setting vN = uNxxxx in (), we obtain d
uNxx + k uNxxxx dt
= uNxxx – a uNxx – b |uNx | , uNxxxx – uN , uNxxxx .
()
Using Nirenberg’s inequality, we obtain
uN ≤ C uNxxxx uN ,
uNx ≤ C uNxxxx uNx ,
where C > is a constant depending only on the domain. Therefore k |b| |uNx | , uNxxxx ≤ uNxxxx + k ≤ uNxxxx +
b
uNx k k
uNxxxx + c a, k, b, T, u H
and
k uN , uNxxxx ≤ uNxxxx + k ≤ uNxxxx +
uN k k
uNxxxx + c a, k, T, u .
On the other hand, we have k uNxxx = –k(uNxx , uNxxxx ) ≤
k
uNxxxx + uNxx .
Summing up, we get d
uNxx + k uNxxxx ≤ + |a| uNxx + c a, k, b, T, u H . dt
()
Using Gronwall’s inequality, we have c(a, k, b, T, u H )
uNxx ≤ e(+|a|)t uNxx () + + |a| ≤ e(+|a|)T uxx +
c(a, k, b, T, u H ) = c a, k, b, T, uxx . + |a|
()
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Integrating () from to t, we obtain
uNxxxx (τ ) dτ ≤ + |a| c T + Tc a, k, b, T, u H + uNxx () k = c k, a, b, T, u H .
T
Therefore, Lemma . is proved.
Remark . Basing on the above Lemmas .-., we can get the H -norm estimate for problem (). Then, by Sobolev’s embedding theorem, we immediately conclude that sup uN (x, t) = uN (x, t)∞ ≤ c k, a, b, T, u H ,
()
sup uNx (x, t) = uNx (x, t)∞ ≤ c k, a, b, T, u H .
()
x∈[,]
x∈[,]
Now, we give the following theorem. Theorem . Suppose that u ∈ HE (, ) and b ≤ k. Suppose further that u(x, t) is the solution of problem ()-() and uN (x, t) is the solution of semi-discrete approximation (). Then there exist a constant c depending on k, a, b, T and u H such that u(x, t) – uN (x, t) ≤ c N – + u – uN () . Proof Denote ηN = u(t) – PN u(t) and eN = PN u(t) – uN (t). From () and (), we get (eNt , vN ) + k(eNxx , vNxx ) – k(eNx , vNx ) + a(eN , vN ) + b |ux | – |uNx | , vN + u – uN , vN = , ∀vN ∈ SN .
()
Set vN = eN in (), we derive that d
eN + k eNxx = k eNx – a eN – b |ux | – |uNx | , eN – u – uN , eN . dt By Theorem ., we have sup u(x, t) ≤ c k, a, b, u H . x∈[,]
Then – u – uN , eN = – (eN + ηN ) u + uuN + uN , eN ≤ sup |u| + |uuN | + |uN | · eN + ηN
eN
x∈[,]
≤ c + c c + c eN + eN + ηN .
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By Theorem ., we have sup ux (x, t) ≤ c k, a, b, u H ,
uxx (x, t) ≤ c k, a, b, u H .
()
x∈[,]
Using Sobolev’s embedding theorem, we have sup |eNx | ≤ C eN H ≤ C eNxx ,
()
x∈[,]
where C and C are positive constants depending only on the domain. Then, using the method of integration by parts, we have –b |ux | – |uNx | , eN = –b (eNx + ηNx )(ux + uNx ), eN = b (eN + ηN )(ux + uNx ), eNx + b (eN + ηN )(uxx + uNxx ), eN .
()
Hence, by ()-() and Hölder’s inequality, we get –b |ux | – |uNx | , eN
≤ |b| sup |ux + uNx | · eN
eNx + ηN
eNx
x∈[,]
+ |b| sup |eN | · eN + ηN
uxx + uNxx
x∈[,]
≤ |b| sup |ux + uNx | · eN
eNx + ηN
eNx
x∈[,]
+ C|b| eNx · eN + ηN
uxx + uNxx
≤ (c + c )|b| eNx + eN + ηN + C(c + c )|b| ε eNx + eN + ηN
ε ε ≤ (c + c )|b| ε eNxx + +
eN + ηN ε ε + C(c + c )|b| ε eNxx + +
eN + ηN , ε ε where ε ∈ R+ is a constant. Summing up, we get
d
eN + k – (c + c )|b| + C(c + c )|b| ε eNxx dt ≤ k eNx + c eN + c ηN = –k(eN , eNxx ) + c eN + c ηN k + c eN + c ηN , ≤ ε eNxx + ε
()
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where ε c = |γ | + c + c c + c + + (c + c )|b| + + C(c + c )|b|, ε ε c = c + c c + c + (c + c )|b| + C(c + c )|b|. ε From Theorem . and (B), we have
ηN ≤ cN – uxx ≤ c k, a, b, T, u H N – .
()
Then a simple calculation shows that
d
eN + k – (c + c )|b| + C(c + c )|b| ε – ε eNxx dt k k + c eN + c ηN ≤ + c eN + c c N – , ≤ ε ε
()
where ε is small enough, it satisfies (k – [(c + c )|b| + C(c + c )|b|]ε) – ε > . Therefore, by Gronwall’s inequality, we deduce that c c N – .
eN ≤ e( ε +c )T eN () + ( + c )ε
()
Hence, the proof is completed.
3 Fully discrete scheme In this section, we set up a full-discretization scheme for problem ()-() and consider the fully discrete scheme which implies the pointwise boundedness of the solution. Let t be the time-step. The full-discretization spectral method for problem ()-() is j read as: find uN ∈ SN (j = , , , . . . , N ) such that for any vN ∈ SN , the following holds:
j+ j j+ j+ uN – uN , vN + k u¯ Nxx , vNxx – k u¯ Nx , vNx t j+ j+ j+ + a u¯ N , vN + b u¯ Nx , vN + u¯ N , vN = , j+
j
()
j+
with uN () = PN u , where u¯ N = (uN + uN ). j The solution uN has the following property. j
Lemma . Assume that u ∈ HE (, ) and b ≤ k. Suppose that uN is a solution of problem (), then there exist positive constants c , c , c , c , c depending only on k, a, b, T and u H such that j u ≤ c , N
j u ≤ c , Nx
j u ≤ c . Nxx
Furthermore, we have j sup uN ≤ c ,
x∈[,]
j sup uNx ≤ c .
x∈[,]
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Proof It can be proved the same as Lemmas .-.. Since the proof is so easy, we omit it. j
In the following, we analyze the error estimates between numerical solution uN and exact solution u(tj ). According to the properties of the projection operator PN , we only j j need to analyze the error between PN u(tj ) and uN . Denoted by uj = u(tj ), ej = PN uj – uN and ηj = uj – PN uj . Therefore j
uj – uN = ηj + ej .
If no confusion occurs, we denote the average of the two instant errors en and en+ by e¯ n+ , j j+ n n+ where e¯ n+ = e +e . On the other hand, we let η¯ j+ = η +η . Firstly, we give the following error estimates for the full discretization scheme. Lemma . For the instant errors ej+ and ej , we have j+ j j+ j e ≤ e + t ut (t ) – uN – uN , e¯ j+ j+ t tj+ (t) +
uttt dt + t e¯ j+ . tj
()
Proof Applying Taylor’s expansion about tj+ , using Hölder’s inequality, we can prove the lemma immediately. Since the proof is the same as [], we omit it.
Taking the inner product of () with e¯ j+ , and letting t = tj+ , we obtain
j+ j+ j+ j+ j+ ut , e¯ j+ + kuxx , e¯ xx – k ux , e¯ x + a uj+ , e¯ j+ j+ + b ux , e¯ j+ + uj+ , e¯ j+ = .
Taking vN = e¯ n+ in (), we obtain
j+ j j+ j+ j+ j+ j+ uN – uN j+ , e¯ + k u¯ Nxx , e¯ xx – k u¯ Nx , e¯ x + a u¯ N , e¯ j+ t j+ j+ + b u¯ Nx , e¯ j+ + u¯ N , e¯ j+ = .
Comparing the above two equations, we get j+ j uN – uN j+ j+ ¯ ut – ,e t j+ j+ j+ j+ j+ j+ j+ = –k uxx – u¯ Nxx , e¯ xx + k ux – u¯ Nx , e¯ x – a uj+ – u¯ N , e¯ j+ j+ j+ j+ – b ux – u¯ Nx , e¯ j+ – uj+ – u¯ N , e¯ j+ . So, we investigate the error estimates of the five items on the right-hand side of the previous equation.
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Lemma . Suppose that u ∈ HE (, ) and b ≤ k, u is the solution for problem ()-() j and uN is the solution for problem (), then j+ k j+ k(t) j+ j+ –k uxx – u¯ Nxx , e¯ xx ≤ – e¯ xx +
tj+
uxxtt dt.
tj
Proof Using Taylor’s expansion, we obtain j+
j
u =u
t j+ u + – t
uj+ = uj+ +
t j+
(t – tj )utt dt,
tj
t j+ u + t
tj+
(tj – t)utt dt. t j+
Hence j u + uj+ – uj+ =
t j+
(t – tj )utt dt +
tj
tj+
(tj – t)utt dt . t j+
By Hölder’s inequality, we have t tj+ j+ j j+ uxx – u + uj+ = (t – tj )utt dt + (tj – t)utt dt xx xx tj t xx ≤ j+
(t)
j+
tj+
uxxtt dt.
tj
j+
Noticing that (η¯ xx , e¯ xx ) = . Therefore j+ j+ j j j+ j j+ uxx + uxx j+ uxx + uxx uNxx + uNxx j+ j+ j+ j+ – uxx – u¯ Nxx , e¯ xx = – uxx – , e¯ xx – – , e¯ xx j j+ j+ uxx + uxx j+ j+ j+ e¯ xx – η¯ xx + e¯ j+ ¯ xx ≤ xx , e uxx – j+ j+ j+ j+ (t) tj+ ≤
uxxtt dt e¯ xx – η¯ xx , e¯ xx – e¯ xx tj (t) tj+ j+ ≤
uxxtt dt – e¯ xx . tj
Then Lemma . is proved.
Lemma . Suppose that u ∈ HE (, ) and b ≤ k, u is the solution for problem ()-() j and uN is the solution for problem (), then j+ j+ j+ k ux – u¯ Nx , e¯ x ≤
k(t) k e¯ j+ + k e¯ j+ + kc N – + xx
where c is the same constant as ().
tj+ tj
utt dt,
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Proof Noticing that η¯ j+ ≤ cN – . Hence j+ j+ j+ ux – u¯ Nx , e¯ x j+ j+ = – uj+ – u¯ N , e¯ xx j+ j+ j u + uj uN + uN j+ j+ = – uj+ – u¯ j+ , e¯ xx – – , e¯ xx j j+ uj+ + uj j+ uj+ + uj uj+ N + uN j+ e¯ xx + – – ≤ u e¯ xx t tj+ j+ j+ j+ e¯ xx + e¯ + η¯ j+ e¯ j+ (t – t )u dt + (t – t)u dt ≤ xx j tt j+ tt tj
≤
t j+
(t)
tj+
utt dt
j+ e¯ xx + e¯ j+ e¯ j+ + η¯ j+ e¯ j+ xx xx
tj
j+ (t) ≤ εe¯ xx + e¯ j+ + c N – + ε ε In the above inequality, setting ε =
,
tj+
utt dt.
tj
we get the conclusion.
Lemma . Suppose that u ∈ HE (, ) and b ≤ k, u is the solution for problem ()-() j and uN is the solution for problem (), then j+ –a uj+ – u¯ N , e¯ j+ |a|(t) ≤ |a|e¯ j+ + |a|c N – +
tj+
utt dt,
tj
where c is the same constant as (). Proof We have j+ –a uj+ – u¯ N , e¯ j+ j+ uj+ + uj j+ u – uj j+ , e¯ + a – u¯ N , e¯ j+ = –a uj+ – j+ j+ j j+ u + u j+ u – uj j+ j+ ≤ |a|u – + |a| – u¯ N e¯ e¯ t tj+ j+ j+ + |a|e¯ j+ + η¯ j+ e¯ j+ ≤ |a| (t – tj )utt dt + (tj+ – t)utt dt e¯ tj
t j+
(t) tj+ ≤ |a|
utt dt e¯ j+ + |a| e¯ j+ + η¯ j+ e¯ j+ tj |a|(t) tj+
utt dt. ≤ |a|e¯ j+ + |a|c N – + tj Then Lemma . is proved.
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Lemma . Suppose that u ∈ HE (, ) and b ≤ k, u is the solution for problem ()-() j and uN is the solution for problem (), then j+ j+ –b ux – u¯ Nx , e¯ j+ ≤
c |b|(t) k e¯ j+ + xx
tj+
uxtt dt + c c N – + c e¯ j+ ,
tj
where C ∈ R+ , c = c |b| + c |b| + C|b|(c + c ) and c = c |b| + c +
c . k
Proof Notice that sup ux (x, t) ≤ c ,
uxx (x, t) ≤ c ,
x∈[,]
j u ≤ c , Nxx
j sup uNx ≤ c .
x∈[,]
Hence j+ j+ –b ux – u¯ Nx , e¯ j+ j+ j j+ j ux + ux ux + ux j+ j+ ux – , e¯ j+ = –b ux + j+ j j+ j ux + ux ux + ux j+ j+ –b + u¯ Nx – u¯ Nx , e¯ j+ j+ j j+ j ux + ux ux + ux j+ j+ = –b ux + ux – , e¯ j+ j+ j j+ j ux + ux uj+ + uj uN + uN j+ j+ +b + u¯ Nx – , e¯ x j+ j+ j+ j j j+ j u + uj uN + uN uxx + uxx uNxx + uNxx j+ + – , e¯ . +b We have used the method of integration by parts in (). Then j+ j+ –b ux – u¯ Nx , e¯ j+ j j+ j j+ uj+ ux + ux x + ux uj+ e¯ j+ · – ≤ |b| sup ux + x x∈[,] j+ j+ j j+ ux + ujx + uj uN + uN j+ u e¯ j+ + u¯ Nx · – + |b| sup x x∈[,] j+ j+ j j j+ j j uj+ uNxx + uNxx xx + uxx u + u – uN + uN + + |b| sup e¯ j+ · x∈[,] t tj+ j+ j+ (t – tj )uxtt dt + (tj+ – t)uxtt dt ≤ c |b| e¯ tj
t j+
j+ + c |b| + c |b| + C|b|(c + c ) e¯ j+ + η¯ j+ e¯ x j+ (t) tj+
uxtt dt + c |b|e¯ j+ + εe¯ xx ≤ c |b| tj
()
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c j+ e¯ + c e¯ j+ + η¯ j+ + ε j+ c |b|(t) tj+ c e¯ j+ + c c N – . ≤ ε e¯ xx +
uxtt dt + c |b| + c + ε tj Setting ε =
k
in the above inequality, we obtain the conclusion.
Lemma . Suppose that u ∈ HE (, ) and b ≤ k, u is the solution for problem ()-() j and uN is the solution for problem (), then j+ (t) – uj+ – u¯ N , e¯ j+ ≤ where c =
c
tj+
utt dt + c e¯ j+ + c c N – ,
tj
+ (c + c c + c ), c = c + c c + c .
Proof Notice that sup uj ≤ c , x∈[,]
j sup uN ≤ c .
x∈[,]
Hence j+ – uj+ – u¯ N , e¯ j+ j+ j+ u + uj j+ j+ u + uj j+ , e¯ – – u¯ N , e¯ = – uj+ – j+ j+ + uj uj+ + uj uj+ + uj j+ u j+ e¯ j+ u – + +u ≤ sup u x∈[,] j+ j+ u + uj uj+ + uj + uj j+ j+ u j+ j+ + u¯ N – u¯ N + u¯ N + sup e¯ x∈[,] t tj+ j+ j+ e¯ (t – t )u dt + (t – t)u dt ≤ c j tt j+ tt tj
t j+
j+ e¯ + η¯ j+ e¯ j+ + j+ c (t) tj+ + c + c c + c e¯ ≤
utt dt + tj + c + c c + c η¯ j+ (t) tj+ ≤
utt dt + c e¯ j+ + c c N – . tj
c
+ c c + c
Then Lemma . is proved. Now, we obtain the following theorem.
Theorem . Suppose that u ∈ HE (, ) and b ≤ k, u(x, t) is the solution for problem ()-() satisfying u ∈ L∞ , T; H (, ) ,
utt ∈ L , T; H (, ) ,
uttt ∈ L , T; L (, ) .
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j
Suppose further that uN is the solution for problem (). Then if t is sufficiently small, there exist positive constants c depending on k, a, b, T, u H and c depending on k, T T a, b, T, u H , utt H dt and uttt dt such that, for j = , , , . . . , N , j+ e ≤ c N – + e + c (t) . Proof By Lemmas .-., we obtain j+ j e ≤ e + tc ej+ + ej + N – tj+
utt + uxtt + uxxtt + uttt dt, + (t) c tj
where c and c are positive constants depending only on k, a, b, T and u H . For t being sufficiently small such that c t ≤ , setting c = (c + c ), we get j+ e ≤ ( + c t)ej + c tN – + (t) Bj , where
tj+
Bj =
utt + uxtt + uxxtt + uttt dt.
tj
Using Gronwall’s inequality for the discrete form, we have
j j+ i e ≤ ec (j+)t e + c jtN – + (t) . B i=
Direct computation shows that j i=
tj+
B ≤ i
utt + uxtt + uxxtt + uttt dt.
Thus, Theorem . is proved. Furthermore, we have the following theorem.
Theorem . Suppose that u ∈ HE (, ) and b ≤ k, u(x, t) is the solution for problem ()-() satisfying u ∈ L∞ , T; H (, ) ,
utt ∈ L , T; H (, ) ,
j
uttt ∈ L , T; L (, ) .
Suppose further that uN ∈ SN (j = , , , . . .) is the solution for problem () and the initial value uN satisfies e = PN u – uN ≤ cN – uxx . Then there exist positive constants T c depending on k, a, b, T, u H and c depending on k, a, b, T, u H , utt H dt, T uttt dt such that u(x, tj ) – uj ≤ c N – + c (t) , N
j = , , , . . . , N.
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4 Numerical results In this section, using the spectral method described in (), we carry out some numerical computations to illustrate out results in previous section. The full-discretization spectral method is read as: For vl = sin lπx, l = , . . . , N , find unN =
N
αin sin iπx,
i = , . . . , N,
i=
such that () holds. Noticing that under the inner product (·, ·), {sin iπx, i = , , . . .} is the system of orthogonal functions, then
sin i πx · sin i πx dx =
, i = i , , i = i .
()
Therefore, the terms of () are
j+ j j+ j α – αl uN – uN , vl = l , t t
j+ k j+ k j+ j j k u¯ Nxx , vlxx = uNxx + uNxx , vlxx = (lπ) αl + αl , j+ j+ j+ k j j –k u¯ Nx , vlx = k uN + uN , vlxx = – (lπ) αl + αl , a j+ j+ a j+ j j a u¯ N , vl = uN + uN , vl = αl + αl , j+ b j+ j b u¯ Nx , vl = uNx + uNx , vl b p p π αpj+ αpj+ + αpj αpj + αpj+ αpj ξp p l p ,p = N
=
and
j+ u¯ N , vl =
N j+ j+ j+ αp αp αp + αpj+ αpj+ α j + αpj+ αpj αpj + αpj αpj αpj ηp p p l , p p ,p ,p =
where ξp p l =
cos p πx · cos p πx · sin lπx dx,
ηp p p l =
sin p πx · sin p πx · sin p πx · sin lπx dx.
Thus, () can be transformed as j+ j αl – αl b k k a j+ j + (lπ) – (lπ) + αl + αl + ρl + σl = , t
()
Zhao et al. Advances in Difference Equations 2013, 2013:156 http://www.advancesindifferenceequations.com/content/2013/1/156
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where l = , . . . , N , and ρl =
N
p p π αpj+ αpj+ + αpj αpj + αpj+ αpj ξp p l ,
p ,p =
σl =
N
αpj+ αpj+ α j+ + αpj+ αpj+ α j + αpj+ αpj αpj + αpj αpj αpj ηp p p l . p p
p ,p ,p =
If αkn (k = , , . . . , N ) is known, there exists an N variable nonlinear system of equations j+ for αl (l = , , . . . , N ) which can be seen as ⎞ f (α j+ ) ⎜ j+ ⎟ j+ ⎜ f (α ) ⎟ ⎜ =⎜ . ⎟ F α ⎟ = . ⎝ .. ⎠ fN (α j+ ) ⎛
We use the simple Newton method to seek the solutions. Initialization yields j j j+ j T α() = α , α , . . . , αN .
()
The iterative formulation is as follows: j+
j+
j+
α(k+) = α(k) + α(k) , j+ j+ j+ F (α() ) · α(k) + F(α(k) ) = ,
()
k = , , , . . . ,
j+
j+
where F (α() ) is the N × N order Jacobi matrix for F(α j+ ) when α j+ = α() , ⎛
j+
∂ f (α() ), j+ ⎜ .. ⎜ F α() = ⎝ . j+ ∂ fN (α() ),
..., .. . ..., j+
⎞ j+ ∂N f (α() ) ⎟ .. ⎟. . ⎠ j+ ∂N fN (α() ) j+
()
j+
Give accuracy > , when α(k+) – α(k) < , stop the iteration, α j+ ≈ α(k+) . As an example, we choose k = , a = , b = , u = ( – x) x , t = ., N = , and get the solution which evolves from t = to t = . (cf. Figure ).
Figure 1 Expanded property of solution when N = 32, t = 0.0005.
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Table 1 Errors of different time steps at t = 0.1 t 0.001 0.001 × 0.001 × 0.001 × 0.001 ×
1 2 1 4 1 8 1 16
err(0.1, t)
err(0.1,t) (t)2
1.6808 × 10–6 2.2439 × 10–7 2.6068 × 10–8 2.8494 × 10–9 2.8754 × 10–10
1.6808 0.8976 0.4171 0.1824 0.0736
Table 2 Errors of different basic function numbers at t = 0.1 N
err(0.1, t0 )
err(0.1,t0 ) N–2
24 28 32 36 40
2.28708 × 10–10 1.3588 × 10–10 7.32182 × 10–11 3.87087 × 10–11 2.03256 × 10–11
1.32 × 10–7 1.07 × 10–7 7.50 × 10–8 5.02 × 10–8 3.25 × 10–8
Now, we consider the variation of error. Since there is no exact solution for ()-() known to us, we make a comparison between the solution of () on coarse meshes and a fine mesh. Choose t = ., . × , . × , . × , . × , . × , respectively, to solve (). Set umin N (x, .) as the solution for tmin = . × . Denote
err(., t) =
ukN (x, .) – umin N (x, .) dx
,
k = , , . . . , .
()
Then the error is showed in Table at t = .. On the other hand, choose N = , , , , , t = . × , respectively, to solve (). Then the error is showed in Table at t = .. It is easy to see that the third column err(.,t) of Table is monotone decreasing along (t) ) with the time step’s waning, the third column err(.,t of Table is monotone decreasing N – along with N ’s magnifying. Hence, we can find positive constants C = ., C = . × – such that
err(., t) ≤ C, (t)
k = , , . . . ,
and err(., t ) ≤ C, N –
N = , , , , .
Thus, the order of error estimates is O((t) + N – ) proved in Theorem ..
Competing interests The authors declare that they have no competing interests. Authors’ contributions XZ wrote the first draft, PZ made the figure of numerical solution and results on errors of different time steps, WZ made the results on errors of different basic function numbers, BL and FL corrected and improved the final version. All authors read and approved the final draft.
Zhao et al. Advances in Difference Equations 2013, 2013:156 http://www.advancesindifferenceequations.com/content/2013/1/156
Author details 1 College of Mathematics, Jilin University, Changchun, 130012, China. 2 Institute of Software, Chinese Academy of Sciences, Beijing, 100190, China. Acknowledgements This work is supported by the Graduate Innovation Fund of Jilin University (Project 20121059). The authors would like to express their deep thanks for the referee’s valuable suggestions about the revision and improvement of the manuscript. Received: 28 December 2012 Accepted: 16 May 2013 Published: 4 June 2013 References 1. Doelman, A, Standstede, B, Scheel, A, Schneider, G: Propagation of hexagonal patterns near onset. Eur. J. Appl. Math. 14, 85-110 (2003) 2. Song, L, Zhang, Y, Ma, T: Global attractor of a modified Swift-Hohenberg equation in Hk space. Nonlinear Anal. 72, 183-191 (2010) 3. Swift, J, Hohenberg, PC: Hydrodynamics fluctuations at the convective instability. Phys. Rev. A 15, 319-328 (1977) 4. La Quey, RE, Mahajan, PH, Rutherford, PH, Tang, WM: Nonlinear saturation of the trapped-ion mode. Phys. Rev. Lett. 34, 391-394 (1975) 5. Shlang, T, Sivashinsky, GL: Irregular flow of a liquid film down a vertical column. J. Phys. France 43, 459-466 (1982) 6. Kuramoto, Y: Diffusion-induced chaos in reaction systems. Prog. Theor. Phys. Suppl. 64, 346-347 (1978) 7. Polat, M: Global attractor for a modified Swift-Hohenberg equation. Comput. Math. Appl. 57, 62-66 (2009) 8. Sivashinsky, GL: Nonlinear analysis of hydrodynamic instability in laminar flames. Acta Astron. 4, 1177-1206 (1977) 9. Lega, J, Moloney, JV, Newell, AC: Swift-Hohenberg equation for lasers. Phys. Rev. Lett. 73, 2978-2981 (1994) 10. Peletier, LA, Rottschäfer, V: Large time behavior of solution of the Swift-Hohenberg equation. C. R. Math. Acad. Sci. Paris, Sér. I 336, 225-230 (2003) 11. Chai, S, Zou, Y, Gong, C: Spectral method for a class of Cahn-Hilliard equation with nonconstant mobility. Commun. Math. Res. 25, 9-18 (2009) 12. He, Y, Liu, Y: Stability and convergence of the spectral Galerkin method for the Cahn-Hilliard equation. Numer. Methods Partial Differ. Equ. 24, 1485-1500 (2008) 13. Ye, X, Cheng, X: The Fourier spectral method for the Cahn-Hilliard equation. Appl. Math. Comput. 171, 345-357 (2005) 14. Yin, L, Xu, Y, Huang, M: Convergence and optimal error estimation of a pseudo-spectral method for a nonlinear Boussinesq equation. J. Jilin Univ. Sci. 42, 35-42 (2004) 15. Canuto, C, Hussaini, MY, Quarteroni, A, Zang, TA: Spectral Methods in Fluid Dynamics. Springer, New York (1988) 16. Xiang, X: The Numerical Analysis for Spectral Methods. Science Press, Beijing (2000) (in Chinese)
doi:10.1186/1687-1847-2013-156 Cite this article as: Zhao et al.: Fourier spectral method for the modified Swift-Hohenberg equation. Advances in Difference Equations 2013 2013:156.
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