Abstract
We introduce in this paper a new technique, a semiexplicit linearized Crank-Nicolson finite difference method, for solving the generalized Rosenau-Kawahara equation. We first prove the second-order convergence in -norm of the difference scheme by an induction argument and the discrete energy method, and then we obtain the prior estimate in -norm of the numerical solutions. Moreover, the existence, uniqueness, and satiability of the numerical solution are also shown. Finally, numerical examples show that the new scheme is more efficient in terms of not only accuracy but also CPU time in implementation.
1. Introduction
In the study of the dynamics of dense discrete systems, Rosenau in [1, 2] proposed the so-called Rosenau equation:as the KdV equation cannot describe the phenomena of wave-wave and wave-wall interactions. In [3], the existence and the uniqueness of the solution of (1) have been studied. For further consideration of nonlinear waves, adding the viscous terms and to (1), respectively, gives the following equations:where , , and are all real constants, while for the exponent, we assume that is an integer. Equations (2) and (3) are called the generalized Rosenau-RdV equation [4–6] and the generalized Rosenau-Kawahara equation [7, 8]. When , they can be considered as Rosenau-KdV equation and the Rosenau-Kawahara equation, respectively.
As we all know, most of the time, we need to think of the numerical solution of nonlinear evolution equations. Many scholars in this field have carried out good work. In [9–22], the nonlinear Crank-Nicolson scheme (see [9, 10, 13, 16, 19, 21]) and the linear finite difference scheme (see [11–15, 18, 20, 22]) have been used for the generalized Rosenau-type equations, including the Rosenau-Burgers equation, the Rosenau-RLW equation, the Rosenau-KdV equation, and the Rosenau-Kawahara equation. Those classical finite difference schemes often use the formula to construct second-order convergent linear finite difference scheme [11–15, 18, 20, 22]. In [23], the authors used the formula and proposed a linear finite difference scheme, but the proof of the prior estimate in -norm is not perfect.
In this paper, using technique , we propose a semiexplicit linearized Crank-Nicolson finite difference method for the generalized Rosenau-Kawahara equation (3) with an initial condition and boundary conditionsBy comparison with the classical second-order accuracy finite difference scheme on a test problem [15, 20–22], our new scheme improves the CPU time and gives a better maximal error of numerical solutions. But the prior estimate in -norm of the numerical solutions is very hard to obtain directly; the proofs of convergence and stability are difficult for our new schemes (7). So, the discrete energy analysis method [24] and an induction argument (see [25–27]) are used to prove the second-order convergence and stability. Furthermore, our new method has a wide range of applications for the generalized Rosenau-type equations including the Rosenau-Burgers equation, the Rosenau-KdV equation, and the Rosenau-RLW equation.
The content of this paper is organized as follows. In the following, we propose a semiexplicit linearized Crank-Nicolson finite difference scheme for initial boundary value problems (3)–(5). In Section 3, we prove the second-order convergence in -norm of the difference scheme by an induction argument and the discrete energy method, and then we obtain the prior estimate in -norm of the numerical solutions. Moreover, based on the prior estimate, the existence and uniqueness of the numerical solution are also shown. Section 4 is devoted to the numerical tests of the new scheme and shows that our scheme has reliable accuracy and spends less CPU time than the classical schemes in implementation. Finally, we finish our paper by concluding remarks in the last section.
2. Finite Difference Schemes
In this section, we give a complete description of our numerical method for the initial value problem (3)–(5). We first give some notation which will be used in this paper. As usual, we let , be any positive integers. Let , ; the domain of solution is defined to be , which is covered by uniform grid , . Suppose is a discrete function on and denote , , as a general positive constant independent of step sizes and which may have different values at different occurrences. Introduce the following notations:
In this paper, we propose a new linear finite difference scheme for the generalized Rosenau-Kawahara equation (3)–(5) which is written asThis scheme is a semiexplicit linearized Crank-Nicolson scheme; the truncation error of this scheme is of order . In this scheme, complicated nonlinear term is extrapolated by . Thus, we only need to solve a linear system of equations in computing . Hence, scheme (7) can be expected to be more efficient.
3. Convergence and Stability
In this section, we prove the convergence and stability of scheme (7). Let , where and are the solutions of (3)–(5) and (7), respectively. We then obtain the following error equation:whereand denotes the truncation error. By using Taylor expansion at , we easily obtain that the truncation error of scheme satisfies
The following lemma is a property of scheme (7); we can obtain that directly from the boundary conditions and notations. This is a well known result, which is essential for existence, uniqueness, convergence, and stability of our numerical solution.
Lemma 1 (see [18, 24]). For any two discrete functions , one has and then one has Furthermore, if , then
The following lemmas including Lemmas 2 and 3 are well known and useful for the proofs of the convergence and stability. Lemma 4 can be deduced directly from the Cauchy-Schwarz inequality and Sobolev inequality.
Lemma 2 (discrete Sobolev inequality [24]). For any discrete function on the finite interval , there is the inequality orHere, , , and are three constants independent of and , . can be any small and is a constant dependent on .
Lemma 3 (discrete Gronwall’s inequality [24]). Suppose that the discrete function satisfies the inequality where and are nonnegative constants. Then, where is sufficiently small, such that .
Lemma 4 (see [22]). Suppose that . Then, the solution of problems (3)–(5) satisfies
It should be pointed out that the induction argument is very useful for proving the convergence of a difference scheme whose prior estimate is difficult to obtain directly (see [25–27]). The following theorem shows the convergence of our scheme (7) with the convergence rate in the -norm by an induction argument.
Theorem 5. Suppose , and ; then the solution of the difference problem (7) converges to the solution of problem (3)–(5) with order in the -norm, if
Proof. We use the mathematical induction to prove it. First, from (10) and Lemma 4, we have where and are two constants independent of and . It follows from the initial conditions that We also can get by the C-N scheme. Hence, the following estimate holds:Now, assume that where is a constant independent of and . Using Lemma 2 and Cauchy-Schwarz inequality, we getNow, computing the inner product of error equation (8) with and using boundary condition (5) and Lemma 1, we obtainUsing Lemma 1, Lemma 4, Cauchy-Schwarz inequality, and (24) for we haveOn the other hand,Substituting (27)-(29) into (25), we obtainSumming up (30) from 0 to , we getTaking small and such that then using (22) and Lemma 3, for we have where is a constant independent of . By an induction argument, From Cauchy-Schwarz inequality, we can obtain By using Lemma 2, it is shown that This completes the proof of Theorem 5.
The following theorem guarantees that the numerical solution obtained from the difference scheme (7) is bounded. This can be proved from (24) of Theorem 5.
Theorem 6. Suppose , and ; if and are small enough, then, for , the inequality holds, where is a constant independent of and .
By a similar proof of Theorem 5, we can obtain the following theorems. Theorem 7 shows that the solution of difference scheme (7) is stable in -norm. Theorem 8 guarantees the existence and uniqueness of numerical solution.
Theorem 7. Under the conditions of Theorem 5, the solution of scheme (7) is stable in -norm for the initial values.
Theorem 8. There exists a unique solution for difference scheme (7).
Another similar semiexplicit scheme for the generalized Rosenau-Kawahara equation (3)–(5) is written as
Its second-order convergence in the -norm and stability can be proved by a similar proof to that of scheme (7) in this paper.
4. Numerical Examples
In this section, we compute three numerical examples to demonstrate and validate the effectiveness of our difference scheme. As a test problem for the scheme proposed here, we chose three test problems for which exact solution or numerical solutions have been reported previously. For the Rosenau-KdV, Rosenau-Kawahara, and generalized Rosenau-Kawahara equations, the parameters used by other researchers [15, 20, 22] to obtain their results were taken as a guiding principle for our computations. As scheme (7) is a semiexplicit scheme, which is a linear system about , we use the Thomas algorithm to solve the system. All the numerical experiments were executed on a 3.20 GHz computer, with 8 G RAM, running Matlab 2013a.
For convenience, we denote the new linear difference scheme (7) as New. In [15], we denote the linear difference scheme as Linear 1. In [20], we denote the linear difference scheme as Linear 2 when . In [22], we denote the linear difference scheme as Linear 3 when . We will measure the accuracy of the proposed scheme using the maximum norm errors defined by . The second-order convergence of the numerical solutions is verified directly from .
4.1. Example 1
Consider the Rosenau-KdV equation with the initial condition and the boundary conditions
It is known that the solitary wave solution [4] is
The results in terms of the maximal norm errors and CPU time at the time using , , , , and are reported in Table 1. It can be seen that the computational efficiency of the present new method is slightly better than that of the method in [15], in terms of grid point number. As shown in Table 2, the second-order convergence in -norm of the new schemes verifies the correction of the theoretical analysis. In Figure 1, plots of maximal errors from the four schemes are presented when , , , and . Clearly, our proposed new scheme gives smaller maximal error than the scheme in [15].
4.2. Example 2
According to [7, 8], when , , the solitary wave solution of the initial boundary problem (3)–(5) is and when , , the solitary wave solution [7] is
The results in terms of the maximal norm errors and CPU time at the time using , are reported in Tables 3 and 4. It is clear from Tables 3 and 4 that results by our new method show improvement over the previous one reported by [20, 22]. As shown in Table 5, the second-order convergence of the numerical solutions is verified. In Figures 2 and 3, the graphs of the maximal errors are presented; the graphs show that our method gives a better approximate solution than the scheme proposed in [20, 22]. Moreover, our proposed scheme gives less CPU time than [20, 22].
5. Conclusion
In brief, we first proposed a semiexplicit linearized Crank-Nicolson finite difference scheme for generalized Rosenau-Kawahara equation. We prove the second-order convergence in -norm of the difference scheme and then obtain the prior estimate in -norm of the numerical solutions. The stability, existence, and uniqueness of the numerical solution are also shown. Finally, some examples were given to show the efficiency of the new scheme. For future research, our new method has a wide range of applications for some nonlinear wave equations including the generalized Rosenau-type equations, Ginzburg-Landau equation, and a generalization of the BBM equation [28].
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This work was supported by the Fundamental Research Funds for the Central Universities (nos. JBK1803014, JBK170940, and JBK151126) and Applied Basic Research Programs of Sichuan Province Science and Technology Department (2016JY0133).