Abstract
In the study of pattern formation in bi-stable systems, the extended Fisher-Kolmogorov (EFK) equation plays an important role. In this paper, a Fourier pseudo-spectral method for solving the EFK equation in two space dimensions is presented. Prior bounds are proved using Lyapunov function. Further, optimal error estimates are established for the semi-discrete scheme. Finally, a fully discrete scheme based on Crank-Nicolson method is proposed, and related optimal error estimates are derived and some numerical experiments are presented.
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1 Introduction
In this paper, we consider the following extended Fisher-Kolmogorov (EFK) equation:
where \(f(u)=u^{3}-u\), Ω is a bounded domain in \(\mathbb{R}^{2}\) with boundary ∂Ω, γ is a positive constant. On the basis of physical considerations, equation (1.1) is supplemented with the following boundary conditions:
and the initial condition
When \(\gamma=0\) in (1.1), we obtain the standard Fisher-Kolmogorov equation, which is a kind of classical second-order diffusion equation (see [1, 2] and so on). However, by adding a stabilizing fourth order derivative term to the Fisher-Kolmogorov equation, Coullet et al. [3], Dee and van Saarloos [4–6] proposed (1.1) and called the model the extended Fisher-Kolmogorov equation. Equation (1.1) occurs in a variety of applications like pattern formation in bi-stable systems [7], propagation of domain walls in liquid crystals [8], traveling waves in reaction diffusion systems [9] and a mesoscopic model of a phase transition in a binary system near the Lipschitz point [10].
Regarding computational studies, there are some numerical experiments conducted in [4] without any convergence analysis. In [11], Danumjaya and Pani studied the convergence of a numerical solution of (1.1) using the second-order splitting combined with orthogonal cubic spline collocation method. In [12], a finite element Galerkin method for the two-dimensional EFK equation (1.1) and optimal error estimates are derived. In [13], Khiari and Omrani studied the finite difference scheme for the EFK equation in two dimensions.
In this paper, we consider the Fourier pseudo-spectral method for problem (1.1)-(1.3). In Section 2, some basic results and definitions are recalled. In Section 3, prior bounds and optimal error estimates for semi-discrete scheme are proved. In Section 4, a fully discrete scheme based on the Crank-Nicolson method is proposed, and related theoretical results are proved. Finally, in Section 5, some numerical experiments are presented to confirm our results.
2 Preliminaries
In this section, we recall some basic results as regards the Fourier pseudo-spectral method, which will be used throughout this paper.
Let \(\Omega=[0,\pi]\times[0,\pi]\), \(2N_{1}\), \(2N_{2}\) be any positive integers, in the continuation of this paper. Let \(N_{1}=N_{2}=N\), then \(h=\frac{\pi}{2N}\), \(x_{i}=ih\), \(y_{j}=jh\), \(i,j\in\Lambda\), where \(\Lambda=\{ 1,\ldots,2N-1\}\). Denote
For the purpose of this article, we introduce the inner product in \(C^{\infty}(\Omega)\) as
the associated \(H^{m}\) norm is \(\|f\|_{m}=\sqrt{\sum_{\alpha\leq m}\| D^{\alpha}f\|^{2}}\), and the \(H^{m}\) semi-norm is defined as \(|f|_{m}=\sqrt {\sum_{\alpha= m}\|D^{\alpha}f\|^{2}}\).
For any integer \(2N>0\), we introduce the finite dimensional subspace of \(H_{0}^{2}(\Omega)\)
Let \(P_{N}:L^{2}(\Omega)\rightarrow S_{N}\) be an orthogonal projecting operator which satisfies
For the operator \(P_{N}\), we have the following results (see [14–16]).
Lemma 2.1
\(P_{N}\) commutes with derivation on \(H_{E}^{2}(\Omega)\), i.e.,
Using the same methods [14, 16] as previously, we can obtain the following result for problem (1.1)-(1.3).
Lemma 2.2
For any real \(0\leq\mu\leq\delta\), there is a constant C such that
where \(| u|_{\delta}^{2}=\sum_{\alpha=\delta}\|D^{\alpha}f\|_{0}^{2}\).
We introduce the discrete \(L^{2}\) inner product
For \(u\in S_{N}\), the discrete \(L^{2}\)-norm is defined as
Demonstrating the discrete inner product is equivalent to the continuous case; we introduce the following lemma first.
Lemma 2.3
For \(k, l\in\Lambda\), we have
Proof
Note that
Hence, the proof is complete. □
Lemma 2.4
For \(u_{h}, v_{h}\in S_{N}\), we have
Proof
For any \(k_{1}, k_{2}, l_{1}, l_{2}\in\Lambda\), the following equalities hold:
By using Lemma 2.3, we can get the following equality:
Hence, the proof is complete. □
Define \(I_{N}:L^{2}(\Omega)\rightarrow S_{N}\) be an interpolation operator which satisfies
From [16–18], we have the following results on \(I_{N}\).
Lemma 2.5
For the operator \(I_{N}\), we have
-
\(I_{N}u=u\), \(u\in S_{N}\),
-
\((I_{N}u,I_{N}v)=(u,v)_{h}\), \(u,v\in C^{0}(\Omega)\),
-
\(\| I_{N}(uv)\|\leq\| u\|_{\infty}\| I_{N}v\|\), \(u,v\in C^{0}(\Omega)\),
-
\((I_{N}u,v)_{h}=(u,v)_{h}\), \(\forall v\in S_{N}\), \(\forall u\in C^{0}(\Omega)\),
-
for any real \(0\leq\mu\leq\delta\), \(\delta>\frac{1}{2}\), there is a constant C such that
$$\| u-I_{N}u\|_{\mu}\leq CN^{\mu-\delta}| u|_{\delta}, \quad \forall u\in H^{\delta}(\Omega). $$
The following lemma is useful in our analysis.
Lemma 2.6
see [19]
There exist two positive constants \(C_{1}\) and \(C_{2}\), which may depend on Ω such that
3 Semi-discrete approximation
In this section, we consider the semi-discrete approximation for problem (1.1)-(1.3). We first give the weak formulation of problem (1.1)-(1.3)
Define the Fourier pseudo-spectral approximation: Find
such that
According to the definition of discrete inner product and properties of \(I_{N}\), we can rewrite (3.2) in the form of
Furthermore, we get
For the solution of scheme (3.2), we have the following prior bounds.
Theorem 3.1
Let \(u(0)\in H_{E}^{2}(\Omega)\), then there exists a unique solution \(u_{h}(t)\in S_{N}\) for problem (3.2), such that
Proof
The equations of problem (3.2) are ordinary differential equations. According to ODE theory, there exists a unique local solution for problem (3.2) in the temporal interval \([0,t_{n})\). If (3.5) holds, using the extension theorem, we can obtain the existence of a global solution. So, we only need to prove (3.2).
Setting \(v = u_{h}\) in (3.3), we can obtain
Using Lemma 2.3, we have
Note that
Hence,
We can obtain
We may define a Lyapunov function
where \(H(u_{h})=\frac{1}{4}(1-u_{h}^{2})^{2}\). Noticing that \(H'(u_{h})=f(u_{h})\). It is also seen that
Therefore,
that is,
Since \(H(u_{h})\geq0\), we have
We also have
Combining (3.6), (3.8) and (3.9), we complete the proof. □
Now, we consider the error estimates for the semi-discrete pseudo-spectral solution. According to the properties of \(P_{N}\), we only need to analyze the error between \(P_{N}u\) and \(u_{h}\). Denote \(\eta (t)=u-P_{N}u\) and \(\theta(t)=P_{N}u-u_{h}\). Therefore
By Lemma 2.1, we can rewrite (3.1) as
Let u be the solution of (3.10), \(u_{h}\) be the solution of (3.3). Subtracting (3.3) from (3.10), we find
To estimate θ, we give the following lemma.
Lemma 3.2
Let u be the solution of (3.1), \(u_{h}\) be the solution of (3.3). If \(u\in L^{\infty}([0,T]; W^{2,\infty})\), then there exists a constant \(C=C(\|u\|_{L^{\infty}([0,T],W^{2,\infty})},\gamma)\) such that
Proof
Taking in (3.12) the inner product with θ, we have
Furthermore
So in the next step, we need to estimate \(\| P_{N}u^{3}-I_{N}u_{h}^{3}\|\). Note that
Applying the properties of \(P_{N}\) and \(I_{N}\), on the one hand,
and on the other hand,
Combining the above two results gives
Plugging (3.16) into (3.14) and using Gronwall’s inequality, we complete the proof. □
Applying Lemma 3.2, we have the following theorem.
Theorem 3.3
Let u be the solution of (3.1), \(u_{h}\) be the solution of (3.3). If \(u\in L^{\infty}([0,T]; W^{2,\infty})\), then there exists a constant \(C=C(\gamma,\| u\|_{L^{\infty}([0,T];W^{2,\infty})})\) such that
Proof
Using the equality (3.10) and Lemma 3.2, we obtain
This completes the proof. □
4 Full-discrete approximation
In this section, we discretize the semi-discrete equation (3.2) in the temporal direction using Crank-Nicolson scheme. For any given positive integer M, let \(\tau=T/M\) be the time step.
Define the net function \(U_{i,j}^{n} = U(x_{i},y_{j}, t_{n})\) for \(t_{n} = n\tau\), \(n = 1,\ldots,M\), and \(U^{n+1/2}=\frac{U^{n}+U^{n+1}}{2}\). The full-discrete pseudo-spectral scheme for problem (1.1)-(1.3) reads: find \(U^{n}\in S_{N}\), \(n=1,\ldots,M\) such that
Here
and
Observe that
According to Lemma 2.3, (4.1) is equivalent to
For the solution of scheme (4.3), we have the following prior bounds.
Theorem 4.1
Let \(U^{0}\in S_{N}\), then there exists a unique solution \(U^{0}(t)\in S_{N}\) for problem (4.3), such that
Proof
Taking in (4.3) the inner product with \(U^{n+1/2}\), we obtain
Applying the Cauchy inequality to the last term yields
It follows from the above result that
Then we can get
As in (3.7), we define the Lyapunov function
Setting \(v=U^{n+1}-U^{n}\) in (4.2) gives
According to the definition of \(E(U^{n})\) and (4.7), we obtain
For Theorem 3.1, we complete the proof. □
We now consider the error estimate for the full-discrete pseudo-spectral solution. It follows from (3.1) and (4.1) that
Let \(u^{n+1}=u(\cdot,t_{n+1})\) be the solution of (4.8), and \(U^{n+1}\) be the solution of (4.9). Denote \(\eta^{n}=u^{n}-P_{N}u^{n}\) and \(\theta^{n}=P_{N}u^{n}-U^{n}\), we have
To estimate \(\theta^{n}\), we need the following lemmas.
Lemma 4.2
Let \(u^{n}\) be the solution of (4.8), and \(U^{n}\) be the solution of (4.8). If \(u\in L^{\infty}([0,T];W^{2,\infty})\). we have
where \(t_{n}\leq\xi^{n}\leq t_{n+1}\).
Proof
Using the definition of \(P_{N}\), we have
Making use of the Taylor expansion, we find
Combing the above two results, we complete this proof. □
Lemma 4.3
Let \(u^{n}\) be the solution of (4.8), \(U^{n}\) be the solution of (4.9). If \(u\in L^{\infty}([0,T]; W^{2,\infty})\) and \(u_{t},u_{tt},u_{ttt}\in L^{\infty}([0,T];H^{2})\), then there exists a constant \(C=(C(u)_{L^{\infty}([0,T];W^{2,\infty})}, \gamma)\) such that
Proof
Note that
By Lemma 4.2 and Sobolev’s embedding theorem, we get
So we just need to estimate \(\| f(u(t_{n+1/2}))-\tilde {f}(U^{n},U^{n+1})\|\).,
We now estimate them term by term. Using the smoothness of f, and the boundedness of \(\| u\|_{L^{\infty}}\), we estimate \(E_{1}\) as
According to the definition of \(f(\cdot)\) and \(\tilde{f}(\cdot,\cdot)\), we derive the bound for \(E_{2}\):
Similarly, we easily derive the following estimates:
and
Combining the above estimates gives the result. □
Theorem 4.4
Let u be the solution of (4.8), \(U^{n}\) be the solution of (4.9). If \(u_{0}\in H^{2}\), \(u\in L^{\infty}([0,T];W^{2,\infty})\), \(u_{t},u_{tt},u_{ttt}\in L^{\infty}([0,T];H^{2})\), then there exists a constant C such that
Proof
Combining (4.8) and (4.9), setting \(v=\theta^{n+1/2}\), we obtain
Using Lemma 4.3 to the above equality gives
Using Lemma 4.3, we have
By applying the discrete Gronwall inequality to the above result gives
This completes the proof. □
Theorem 4.5
Let u be the solution of (3.1), \(U^{n}\) be the solution of (4.1). If \(u_{0}\in H^{2}\), \(u\in L^{\infty}([0,T];W^{2,\infty})\), \(u_{t},u_{tt},u_{ttt}\in L^{\infty}([0,T];H^{2})\), then there exists a constant \(C=C(\gamma, \| u\|_{L^{\infty}([0,T];W^{2,\infty })})\) for τ small enough such that
Proof
Using (4.12) and Theorem 4.1, we obtain
□
5 Numerical results
In this section, two examples are presented to illustrate the effectiveness of the pseudo-spectral scheme (4.1). The Crank-Nicolson scheme is a typical difference scheme, we get nonlinear equations during each iteration in t. To compute the approximate solution, we use the Newton iterative method in each iteration. Set \(\Omega=(0,\pi )\times(0,\pi)\) and \(T=1\) in this section.
5.1 Example 1
We consider the following nonlinear inhomogeneous extended Fisher-Kolmogorov (EFK) equation:
with the initial condition
subject to the initial boundary conditions
where \(f(u)=u^{3}-u\), \(g(x,y,t)=4\gamma \sin(x)\sin(y)e^{-t}+[\sin(x)\sin(y)e^{-t}]^{3}\). The exact solution of the problem is
The profile of the exact solution is given in Figure 1, the behavior of the approximate solution is shown in Figure 2. The variation of the \(L_{\infty}\)-errors for different times is presented in Figure 3.
It shows the \(L_{\infty}\)-errors to be increasing along with the increase of time, and the increasing rates decline gradually with time increasing.
5.2 Example 2
We consider the following problem for the EFK equation:
with the initial condition
subject to the initial boundary conditions
where \(f(u)=u^{3}-u\). The profile of the approximate solution of the equation is given in Figure 4. In addition, Figure 5 represents the Lyapunov functional as it decreases.
Now, we consider the \(L_{\infty}\)-errors for different τ with fixed \(h=\frac{\pi}{20}\). Since there is no exact solution for this problem to the best of our knowledge, we take the numerical solution u with \(h=\frac{\pi}{20}\), \(\tau=\frac{1}{200}\) as the ‘exact solution’. \(L_{\infty}\)-errors are showed in Table 1 for different time steps at \(x=\frac{\pi}{4}\), \(y=\frac{\pi}{4}\).
It is easy to see that the fourth column \(\frac{\operatorname{err}(\frac {\pi}{4}, \frac{\pi}{4}, \tau)}{\tau}\) and the fifth column \(\frac {\operatorname{err}(\frac{\pi}{4}, \frac{\pi}{4}, \tau)}{\tau^{2}}\) is monotone decreasing along with the time step’s waning. That means the order of convergence for time is better than \(O(\tau^{2})\), so the numerical result is better than the theoretical result. The reason may be the existence of a nonlinear term or the limit of the theoretical proof tool.
We also consider the \(L_{\infty}\)-errors for the different h with fixed \(\tau=0.005\). In this situation, we take the numerical solution u with \(h=\frac{\pi}{60}\), \(\tau=\frac{1}{100}\) as the ‘exact solution’. \(L_{\infty}\)-errors are showed in Table 2 for different space steps at \(t=0.3\).
In Table 2, the fifth column is not monotone increasing along with the space step’s waning all the time, but it tends to be monotone decreasing when the space subdivision is small enough. That means the accuracy in space is better than the theoretical result.
In summary, the spatial accuracy of the Fourier pseudo-spectral scheme is of second order, and the time accuracy is better than the second order. The results indicate this numerical scheme is efficient and its computational accuracy is slightly better than the theoretical precision.
Remark 5.1
At the reviewer’s suggestion, we considered the relations between \(L_{\infty}\)-error and γ with numerical experiments. We find the \(L_{\infty}\)-error is decreasing along with the decrease of γ, which is the coefficient of the fourth order term. For the general fourth order nonlinear parabolic equations, the \(L_{\infty}\)-errors should have become bigger with the decrease of γ. Because we need to balance the nonlinear term with the fourth order term in the estimation of the errors, the errors is proportional to the inverse of γ. Compared with the general case, the numerical results are reversed in EFK equation. One of the reasons maybe is that we get the theoretical results by the Lyapunov function, so the estimated coefficients are irrelevant to γ. As for the lack of theoretical basis, we mention it as an open problem and will continue to study it.
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We warmly thank the anonymous referee for his/her pertinent comments and suggestions, which greatly improved the earlier manuscript.
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Liu, F., Zhao, X. & Liu, B. Fourier pseudo-spectral method for the extended Fisher-Kolmogorov equation in two dimensions. Adv Differ Equ 2017, 94 (2017). https://doi.org/10.1186/s13662-017-1154-x
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DOI: https://doi.org/10.1186/s13662-017-1154-x