- Open Access
Fourier spectral method for the modified Swift-Hohenberg equation
Advances in Difference Equations volume 2013, Article number: 156 (2013)
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.
In , Doelman et al. studied the modified Swift-Hohenberg equation
Setting , considering (1) in 1D case, we find that
On the basis of physical considerations, as usual, Eq. (2) is supplemented with the following boundary value conditions:
and the initial condition
where and a, b are constants. 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 . The additional term , 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 [6–8]), breaks the symmetry .
During the past years, many authors have paid much attention to the Swift-Hohenberg equation (see, e.g., [3, 9, 10]). 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 (2)-(4). Recently, Song et al.  studied the long time behavior for the modified Swift-Hohenberg equation in an () 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 (2)-(4) possesses a global attractor in the Sobolev space for all , which attracts any bounded subset of in the -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, [11–14]. However, for the other boundary condition, can we also use the Fourier spectral method? The answer is ‘Yes’. Choose a good finite dimensional subspace (here, we set ), we can also have the basic results for the orthogonal projecting operator .
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 . Adjusted to our needs, the results are given in the following form.
Theorem 1.1 Assume that and , then there exists a unique global solution of the problem (2)-(4) for all such that
Furthermore, it satisfies
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 3, we consider the full-discrete approximation for problem (2)-(4). 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 , , , norm in Ω simply by , , and .
2 Semi-discrete approximation
In this section, we consider the semi-discrete approximation for problem (2)-(4). First of all, we recall some basic results on the Fourier spectral method which will be used throughout this paper. For any integer , introduce the finite dimensional subspace of
Let be an orthogonal projecting operator which satisfies
(B1) commutes with derivation on , i.e.,
(B2) For any real , there is a constant c such that
Define the Fourier spectral approximation: Find such that
for all with .
Now, we are going to establish the existence, uniqueness etc. of the Fourier spectral approximation solution for all .
Lemma 2.1 Let and , then problem (7) has a unique solution satisfying the following inequalities:
where and for all .
Proof Set in (7) for each j () to obtain
where all () are smooth and locally Lipschitz continuous. Noticing that , then
Using the theory of initial-value problems of the ordinary differential equations, there is a time such that the initial-value problem (9)-(10) has a unique smooth solution for .
Setting in (7), we have
Summing up, we get
Using Gronwall’s inequality, we deduce that
Integrating (12) from 0 to t, we derive that
From the above inequality, we obtain the second inequality of (8) immediately. Therefore, Lemma 2.1 is proved. □
Lemma 2.2 Let and , then the solution of problem (7) satisfying
for all , where and are positive constants depending only on k, a, b, T and .
Proof Setting in (7), we obtain
On the other hand, by Nirenberg’s inequality, we have
where C is a positive constant independent of N. Hence
Summing up, we get
Using Gronwall’s inequality, we immediately obtain
Integrating (16) from 0 to t, we obtain
Then Lemma 2.2 is proved. □
Lemma 2.3 Let and , then the solution of problem (7), satisfying
for all , where and are positive constants, depending only on k, a, b, T and .
Proof Setting in (7), we obtain
Using Nirenberg’s inequality, we obtain
where is a constant depending only on the domain. Therefore
On the other hand, we have
Summing up, we get
Using Gronwall’s inequality, we have
Integrating (19) from 0 to t, we obtain
Therefore, Lemma 2.3 is proved. □
Remark 2.1 Basing on the above Lemmas 2.1-2.3, we can get the -norm estimate for problem (7). Then, by Sobolev’s embedding theorem, we immediately conclude that
Now, we give the following theorem.
Theorem 2.1 Suppose that and . Suppose further that is the solution of problem (2)-(4) and is the solution of semi-discrete approximation (7). Then there exist a constant c depending on k, a, b, T and such that
Proof Denote and . From (2) and (7), we get
Set in (23), we derive that
By Theorem 1.1, we have
By Theorem 1.1, we have
Using Sobolev’s embedding theorem, we have
where and C are positive constants depending only on the domain. Then, using the method of integration by parts, we have
Hence, by (24)-(26) and Hölder’s inequality, we get
where is a constant. Summing up, we get
From Theorem 1.1 and (B2), we have
Then a simple calculation shows that
where ε is small enough, it satisfies . Therefore, by Gronwall’s inequality, we deduce that
Hence, the proof is completed. □
3 Fully discrete scheme
In this section, we set up a full-discretization scheme for problem (2)-(4) and consider the fully discrete scheme which implies the pointwise boundedness of the solution.
Let Δt be the time-step. The full-discretization spectral method for problem (2)-(4) is read as: find () such that for any , the following holds:
with , where .
The solution has the following property.
Lemma 3.1 Assume that and . Suppose that is a solution of problem (31), then there exist positive constants , , , , depending only on k, a, b, T and such that
Furthermore, we have
Proof It can be proved the same as Lemmas 2.1-2.3. Since the proof is so easy, we omit it. □
In the following, we analyze the error estimates between numerical solution and exact solution . According to the properties of the projection operator , we only need to analyze the error between and . Denoted by , and . Therefore
If no confusion occurs, we denote the average of the two instant errors and by , where . On the other hand, we let .
Firstly, we give the following error estimates for the full discretization scheme.
Lemma 3.2 For the instant errors and , we have
Proof Applying Taylor’s expansion about , 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 (2) with , and letting , we obtain
Taking in (31), we obtain
Comparing the above two equations, we get
So, we investigate the error estimates of the five items on the right-hand side of the previous equation.
Lemma 3.3 Suppose that and , u is the solution for problem (2)-(4) and is the solution for problem (31), then
Proof Using Taylor’s expansion, we obtain
By Hölder’s inequality, we have
Noticing that . Therefore
Then Lemma 3.3 is proved. □
Lemma 3.4 Suppose that and , u is the solution for problem (2)-(4) and is the solution for problem (31), then
where is the same constant as (28).
Proof Noticing that . Hence
In the above inequality, setting , we get the conclusion. □
Lemma 3.5 Suppose that and , u is the solution for problem (2)-(4) and is the solution for problem (31), then
where is the same constant as (28).
Proof We have
Then Lemma 3.5 is proved. □
Lemma 3.6 Suppose that and , u is the solution for problem (2)-(4) and is the solution for problem (31), then
where , and .
Proof Notice that
We have used the method of integration by parts in (33). Then
Setting in the above inequality, we obtain the conclusion. □
Lemma 3.7 Suppose that and , u is the solution for problem (2)-(4) and is the solution for problem (31), then
where , .
Proof Notice that
Then Lemma 3.7 is proved. □
Now, we obtain the following theorem.
Theorem 3.1 Suppose that and , is the solution for problem (2)-(4) satisfying
Suppose further that is the solution for problem (31). Then if Δt is sufficiently small, there exist positive constants depending on k, a, b, T, and depending on k, a, b, T, , and such that, for ,
Proof By Lemmas 3.2-3.7, we obtain
where and are positive constants depending only on k, a, b, T and . For Δt being sufficiently small such that , setting , we get
Using Gronwall’s inequality for the discrete form, we have
Direct computation shows that
Thus, Theorem 3.1 is proved. □
Furthermore, we have the following theorem.
Theorem 3.2 Suppose that and , is the solution for problem (2)-(4) satisfying
Suppose further that () is the solution for problem (31) and the initial value satisfies . Then there exist positive constants depending on k, a, b, T, and depending on k, a, b, T, , , such that
4 Numerical results
In this section, using the spectral method described in (31), we carry out some numerical computations to illustrate out results in previous section. The full-discretization spectral method is read as: For , , find
such that (31) holds.
Noticing that under the inner product , is the system of orthogonal functions, then
Therefore, the terms of (31) are
Thus, (31) can be transformed as
where , and
If () is known, there exists an N variable nonlinear system of equations for () which can be seen as
We use the simple Newton method to seek the solutions. Initialization yields
The iterative formulation is as follows:
where is the order Jacobi matrix for when ,
Give accuracy , when , stop the iteration, .
As an example, we choose , , , , , , and get the solution which evolves from to (cf. Figure 1).
Now, we consider the variation of error. Since there is no exact solution for (2)-(4) known to us, we make a comparison between the solution of (31) on coarse meshes and a fine mesh.
Choose , respectively, to solve (31). Set as the solution for . Denote
Then the error is showed in Table 1 at .
On the other hand, choose , , respectively, to solve (31). Then the error is showed in Table 2 at .
It is easy to see that the third column of Table 1 is monotone decreasing along with the time step’s waning, the third column of Table 2 is monotone decreasing along with N’s magnifying. Hence, we can find positive constants , such that
Thus, the order of error estimates is proved in Theorem 3.2.
Doelman A, Standstede B, Scheel A, Schneider G: Propagation of hexagonal patterns near onset. Eur. J. Appl. Math. 2003, 14: 85-110.
Song L, Zhang Y, Ma T:Global attractor of a modified Swift-Hohenberg equation in space. Nonlinear Anal. 2010, 72: 183-191. 10.1016/j.na.2009.06.103
Swift J, Hohenberg PC: Hydrodynamics fluctuations at the convective instability. Phys. Rev. A 1977, 15: 319-328. 10.1103/PhysRevA.15.319
La Quey RE, Mahajan PH, Rutherford PH, Tang WM: Nonlinear saturation of the trapped-ion mode. Phys. Rev. Lett. 1975, 34: 391-394. 10.1103/PhysRevLett.34.391
Shlang T, Sivashinsky GL: Irregular flow of a liquid film down a vertical column. J. Phys. France 1982, 43: 459-466. 10.1051/jphys:01982004303045900
Kuramoto Y: Diffusion-induced chaos in reaction systems. Prog. Theor. Phys. Suppl. 1978, 64: 346-347.
Polat M: Global attractor for a modified Swift-Hohenberg equation. Comput. Math. Appl. 2009, 57: 62-66. 10.1016/j.camwa.2008.09.028
Sivashinsky GL: Nonlinear analysis of hydrodynamic instability in laminar flames. Acta Astron. 1977, 4: 1177-1206. 10.1016/0094-5765(77)90096-0
Lega J, Moloney JV, Newell AC: Swift-Hohenberg equation for lasers. Phys. Rev. Lett. 1994, 73: 2978-2981. 10.1103/PhysRevLett.73.2978
Peletier LA, Rottschäfer V: Large time behavior of solution of the Swift-Hohenberg equation. C. R. Math. Acad. Sci. Paris, Sér. I 2003, 336: 225-230. 10.1016/S1631-073X(03)00021-9
Chai S, Zou Y, Gong C: Spectral method for a class of Cahn-Hilliard equation with nonconstant mobility. Commun. Math. Res. 2009, 25: 9-18.
He Y, Liu Y: Stability and convergence of the spectral Galerkin method for the Cahn-Hilliard equation. Numer. Methods Partial Differ. Equ. 2008, 24: 1485-1500. 10.1002/num.20328
Ye X, Cheng X: The Fourier spectral method for the Cahn-Hilliard equation. Appl. Math. Comput. 2005, 171: 345-357. 10.1016/j.amc.2005.01.050
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. 2004, 42: 35-42.
Canuto C, Hussaini MY, Quarteroni A, Zang TA: Spectral Methods in Fluid Dynamics. Springer, New York; 1988.
Xiang X: The Numerical Analysis for Spectral Methods. Science Press, Beijing; 2000. (in Chinese)
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.
The authors declare that they have no competing interests.
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.
Authors’ original submitted files for images
Below are the links to the authors’ original submitted files for images.
About this article
Cite this article
Zhao, X., Liu, B., Zhang, P. et al. Fourier spectral method for the modified Swift-Hohenberg equation. Adv Differ Equ 2013, 156 (2013). https://doi.org/10.1186/1687-1847-2013-156
- Fourier Spectral Methods
- Swift-Hohenberg Equation
- Fully-discrete Scheme
- Variable Nonlinear Systems
- Viscous Film Flow