Abstract
In this work, the asymptotic stability result for Rosenau-Burgers equation is established, under appropriate assumptions on steady state eigenvalue problem and the forcing function. In addition, we propose and analyze a linearized numerical method for solving this nonlinear Rosenau-Burgers equation. We prove that the numerical scheme is unconditionally stable, and the error estimate shows that the numerical method is in the order of , where , , and are, respectively, step of time, polynomial degree, and regularity of . Numerical examples are illustrated to verify the theoretical results.
1. Introduction
In the research of dynamic dense discrete system, it was shown that the KdV equation can not completely describe the interaction between waves and waves, and in order to overcome the shortcoming of this KdV equation, Rosenau [1, 2] proposed the following Rosenau equation: For further consideration of dissipation in dynamic system, the term is included in the above equatoin. The resulting new equation is called the Rosenau-Burgers equation: Park [3] discussed existence and uniqueness properties for the Rosenau-Burgers equation. Mei & Liu [4–7] studied the large-time behavior of the solution for the Rosenau-Burgers equation, and they obtained some estimates for the asymptotic solution. Kinami [8] & Mei [9] considered the asymptotic behavior of solution for Benjamin-Bona-Mahony-Burgers equation. They proved that the solution asymptotically converges to 0.
Besides the theoretical analysis, some recent contributions focus on using numerical methods to approximate the solutions of Rosenau-Burgers equation. Chung [10] & Sank [11] introduced finite element Galerkin method for solving a Rosenau equation. They obtained the existence and uniqueness of solutions and the error estimates of the solutions are also discussed. Chung [12], Omrani [13], and Feng [14] presented a conservative finite difference scheme for solving Rosenau equation. Convergence, stability, and error estimate of full discrete scheme are also proved. Manickam et al. [15] applied orthogonal cubic spline collocation method to approximate Rosenau equation, and the error estimates are obtained in both norm and norm.
In this paper, we consider the following Rosenau-Burgers equation:where .
In this article, the convergence of the Rosenau-Burgers equation to its steady-state problem is discussed, under the assumption that the corresponding linearized steady-state eigenvalue problem has a positive minimum eigenvalue and appropriate condition on the forcing function. Moreover, we present numerical method to solve Rosenau-Burgers equation, and the proposed scheme is performed by combining Crank-Nicolson approach in time and Fourier-spectral in space. Our rigorous analysis result shows that the scheme is unconditionally stable, and the numerical method leads to second order in time and spectral accuracy in space.
The rest of the article is organized in the following way. Section 2 will study asymptotic behaviour of Rosenau-Burgers equation. In Section 3, we will discuss stability and error estimate for the full discrete scheme. In Section 4, we present some numerical experiments to illustrate the validity of the numerical method. The conclusions of this paper are given in Section 5.
2. Asymptotic Behaviour of Rosenau-Burgers Equation
In this section, we will investigate the asymptotic behavior of the solution as . Let , where is the steady state solution of (3)-(5), satisfyingwhere .
(A1) Assume the eigenvalue problem, has a minimum eigenvalue
Note that, for any , it holds that Then, we have Poincaré inequality [16]: For any , there holds , where is the minimum eigenvalue of the eigenvalue problem: , with .
Let , from (3)-(5) and (6)-(7), we havewhere .
The weak formulation of (12)-(14) is to find , such that
(A2) , and , .
Theorem 1. Under assumptions (A1) and (A2), there holds moreover, for given , there exists , such that
Proof. Set in (15), and we obtain Apply inequality (11), and Young’s inequality is to arrive at Now, integrating with respect to time from to , we obtain Applying the Poincaré inequalities, we obtain Substiuting (21) into (20), we have On the other hand, using Poincaré inequality again, we have where . Substituting (23) into (20) yields That is, Hence, we finish the proof of (17).
3. Time-Discrete and Error Analysis
In this section, we present the semidiscrete scheme for the solution of (3)-(5). First, we introduce a Crank-Nicholson method to discrete time. Let be a positive integer, is the time step, and are the mesh points.
Consider the following time-discrete scheme:
We have the following stability result.
Theorem 2. The semidiscrete scheme (26) is unconditionally stable, such that where
Proof. Taking the inner product of (26) with , we get That is, This concludes the proof.
We will consider Fourier-Galerkin spectral method for the discretization equations (27). We will present some error estimate for full-discretization schemes. First, let us define Denote to be the -projection operator which satisfies We also define the -projection operator by We have the following estimate [17]: Consider the full-discretization Fourier-Galerkin spectral method to (26) as follows: find , such that
Theorem 3. Let be the solutions of (26), and we derive that
We denote the truncation error ; here, From Taylor expansion, we have We also define the following error functions: We show the error estimate of the full-discretization problem (35) in the following theorem.
Theorem 4. Suppose is the exact solution of (3)-(5) and are the solutions of (38); then, we have
Proof. Subtracting (35) from a reformulation of (3) at , we obtain Let , andUsing Cauchy-Schwarz and Young’s inequality, we have where Applying Taylor expansion and Young’s inequality, Then, we obtain Adding up for and using the following inequality, we have For the first step, it is to verify that By using Gronwall inequality, we haveTherefore, estimate (42) is proved.
4. Numerical Results
This section presents several numerical examples to confirm the accuracy and applicability of schemes (35) for solving Rosenau-Burgers equation. Fisrt, we letAnd we obtain the following linear system: where or represents the th mode Fourier coefficient of the function .
4.1. Verification of the Temporal Convergence Order
We use the following quantity to compute the convergence rate in time direction [18]: The full discrete problem (52) is solved in with and . Tables 1–3 display the temporal convergence orders for different values of and initial condition . As shown in Tables 1–3, our numerical scheme (52) is of the order accuracy in time, which is confirmed with the theoretical result in Theorem 4.
4.2. Asymptotic Properties of Solutions
In order to verify the asymptotic behavior of , the effect of and initial condition will be investigated. Let and ; then, the exact solution is , and . The numerical result in Figure 1 shows the convergence of the solution to its steady-state solution . When , and in (3), the exaction solution is , the steady-state solution is , and . It is easy to verify that both and satisfy the assumptions of (A1) and (A2). Figure 2 indicates that the numerical solution of converges to as . In Figure 3, we plot the numerical solution at , we can observe that oscillates around 0. In Figures 4-5, we observed that the numerical solutions decay almost to zero when becomes larger. This implies that the impact of is significant in the long time behavior. The figures are for indicated in the result discussions.
The reference solutions were obtained by using the numerical scheme (35) with and . We present and errors for . The results are summarized in Figure 6, in which we conclude that the full-discrete scheme (35) reaches the second-order convergence rate in time direction. In order to test the spatial spectral accuracy, error functions for are given in Figure 7, which show that our numerical method has good convergence behavior. We observe it reaches spectral convergence in space, if the error in spatial direction is negligible.
5. Conclusion
This article studies the asymptotic stability of Rosenau-Burgers equation. Under suitable assumptions on an eigenvalue problem and the forcing function, the convergence order of the Rosenau-Burgers equation to its steady-state problem is derived. Then, we propose a linearized numerical method for solving this nonlinear Rosenau-Burgers equation. The numerical method is combined with a finite difference scheme in time and Fourier-spectral method in space. We have derived a discrete stability inequality and error estimate for the numerical scheme which leads to -order accuracy in time and spectral accuracy in space. Numerical examples are illustrated to verify the theoretical results.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The work of Jun Zhang is supported by the Chinese Postdoc Foundation Grant (no. 2019M653490) and the academic project of Guizhou University of Finance and Economics (no. 20185774-033). The work of Zixin Liu is supported by National Natural Science Foundation of China (no. 61472093), Guizhou Province University science and technology top talents project (no. KY2018047), and Guizhou University of Finance and Economics (no. 2018XZD01). This work of Fubiao Lin is supported by the Science Technology Foundation of Guizhou Education Department (no. QJK[KY] 20191051) and academic project of Guizhou University of Finance and Economics (no. 2017 5736-020). The work of Jianjun Jiao is supported by National Natural Science Foundation of China (nos. 11791019 and 11361014), the Joint Fund Project of Department of Commerce with GUFE (no. 2016SWBZD18), the Science Technology Foundation of Guizhou Education Department (nos. QJK[KY]2018019, 20191051), the Project of High Level Creative Talents in Guizhou Province (no. 20164035), and the Guizhou Province University science and technology top talents project (no. 2018-047).