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Fourier spectral method for the modified Swift-Hohenberg equation
© Zhao et al.; licensee Springer 2013
Received: 28 December 2012
Accepted: 16 May 2013
Published: 4 June 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.
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.
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
for all with .
Now, we are going to establish the existence, uniqueness etc. of the Fourier spectral approximation solution for all .
where and for all .
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 .
From the above inequality, we obtain the second inequality of (8) immediately. Therefore, Lemma 2.1 is proved. □
for all , where and are positive constants depending only on k, a, b, T and .
Then Lemma 2.2 is proved. □
for all , where and are positive constants, depending only on k, a, b, T and .
Therefore, Lemma 2.3 is proved. □
Now, we give the following theorem.
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.
with , where .
The solution has the following property.
Proof It can be proved the same as Lemmas 2.1-2.3. Since the proof is so easy, we omit it. □
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.
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. □
So, we investigate the error estimates of the five items on the right-hand side of the previous equation.
Then Lemma 3.3 is proved. □
where is the same constant as (28).
In the above inequality, setting , we get the conclusion. □
where is the same constant as (28).
Then Lemma 3.5 is proved. □
where , and .
Setting in the above inequality, we obtain the conclusion. □
where , .
Then Lemma 3.7 is proved. □
Now, we obtain the following theorem.
Thus, Theorem 3.1 is proved. □
Furthermore, we have the following theorem.
4 Numerical results
such that (31) holds.
Give accuracy , when , stop the iteration, .
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.
Errors of different time steps at
1.6808 × 10−6
2.2439 × 10−7
2.6068 × 10−8
2.8494 × 10−9
2.8754 × 10−10
Errors of different basic function numbers at
2.28708 × 10−10
1.32 × 10−7
1.3588 × 10−10
1.07 × 10−7
7.32182 × 10−11
7.50 × 10−8
3.87087 × 10−11
5.02 × 10−8
2.03256 × 10−11
3.25 × 10−8
Thus, the order of error estimates is proved in Theorem 3.2.
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.
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