Abstract
We introduce a new technique, a three-level average linear-implicit finite difference method, for solving the Rosenau-Burgers equation. A second-order accuracy on both space and time numerical solution of the Rosenau-Burgers equation is obtained using a five-point stencil. We prove the existence and uniqueness of the numerical solution. Moreover, the convergence and stability of the numerical solution are also shown. The numerical results show that our method improves the accuracy of the solution significantly.
1. Introduction
A nonlinear wave phenomenon is the important area of scientific research. There are mathematical models which describe the dynamic of wave behaviors such as the KdV equation, the Rosenau equation, and many others. The KdV equation cannot explain the wave-wave and wave-wall interactions for the model of the dynamics of compact discrete systems. Therefore, Rosenau [1, 2] presented the novel model, which is more suitable than the KdV equation, as follows: The existence and uniqueness of the solution for this equation were proved by Park [3]. Many methods have been applied to find a numerical solution of the Rosenau equation such as a discontinuous Galerkin method [4], a finite element Galerkin method [5], and a finite difference method [6–8]. Numerical solutions and error estimates in and norms were obtained for the Rosenau equation in one space variable [9].
As for Burgers’ equation, this equation has been studied in the evolution equation describing a wave propagation. Moreover, the simulation for Burgers’ equation was the very first step of conceptual understanding of the method for the computations of complex flow. The existence and uniqueness of the generalized Burgers’ equation have been shown with certain conditions.
In this paper, we consider the following initial-boundary value problem of the generalized Rosenau-Burgers equation: with an initial condition and boundary conditions where , , and is an integer.
This equation was proposed in 1989 to describe the wave in shallow water. It differs from Burgers’ equation by an additional strongly dissipative term . The behavior of the solution to the Rosenau-Burgers equation with the Cauchy problem has been well studied for the past years [10–13]. Several second-order accuracy finite difference methods in space were used for finding numerical solutions on both linear and nonlinear terms [14–20].
Hu et al. [14] have proposed the Crank-Nicolson difference scheme, nonlinear scheme, for the Rosenau-Burgers equation. Hu et al. [18] have proposed a three-level average implicit finite difference scheme for the Rosenau-Burgers equation. The schemes are obviously implicit and require a heavy calculation for each iteration. Pan and Zhang [20] have proposed a three-level linear-implicit difference scheme. The schemes, we have mentioned above, are second-order accuracy on both time and space.
In this paper, we propose a modified three-level average linear-implicit finite difference method for the Rosenau-Burgers equation. By comparing with the existence second-order accuracy finite difference scheme on a test problem, our new technique gives a better maximal error of the numerical solutions. A second-order accuracy on both space and time numerical solution of the equation is obtained using a five-point stencil.
This paper is organized into 7 sections. In Section 2, we describe our modified finite different scheme. In Section 3, we discuss the solvability of our scheme. The existence and uniqueness are also proven in this section. In Section 4, we give complete proofs on the convergence and stability of the finite difference scheme which is second-order accuracy on both space and time. The numerical results are given in Section 5 to confirm and illustrate our theoretical analysis. Then we finish our paper by concluding remarks.
2. Modified Finite Difference Scheme
In this section, we give a complete description of our modified finite difference scheme and an algorithm for the formulation of the problem (3)–(5). We first describe our solution domain and its grid. We define the solution domain to be , which is covered by a uniform grid , with spacings and . Denote and . Throughout this paper, we will denote as a generic constant independent of step sizes and . For nonnegative integer , let denote the usual Sobolev space of real-valued functions defined on . We define the following Sobolev space: We use the following notations for the simplicity: Since , the following finite difference scheme solves the problem (3)–(5): where
The following lemmas are some properties of the above finite difference scheme which can be obtained directly from the definition. They are essential for existence, uniqueness, convergence, and stability of our numerical solution.
Lemma 1 (Hu et al. [14]). For any two mesh functions , we have Furthermore, if , then
Lemma 2 (discrete Sobolev’s inequality [9]). There exist constants and such that
The following theorem guarantees that the numerical solution obtained from scheme (8)-(9) is bounded.
Theorem 3. Suppose . Then there is an estimation for the solution of the scheme (8)-(9) that satisfies which imply for some .
Proof. Consider the inner product between (8) and . According to Lemma 1, we have By a direct calculation and the boundary condition (9), we have Furthermore, by using the definition of the inner product From (17) and (18), (16) can be rewritten as Therefore, We now define Then inequality (20) can be rewritten as follows: If is sufficiently small which satisfies and , then Hence, By using Lemma 1 and the Cauchy-Schwarz inequality, we arrive at Then, we get If is sufficiently small in which , we arrive at that From (25), it follows that . By Lemma 2, it is obvious that and that completes the proof.
3. Solvability
In this section, we prove the solvability of a solution for scheme (8). This guarantees the existence and uniqueness of our numerical solution.
Theorem 4. The finite difference scheme (8)–(10) is uniquely solvable.
Proof. To prove the theorem, we proceed by the mathematical induction. We assume that satisfy the difference scheme (8). Indeed, can be computed by an available second-order accuracy method. Next we prove that there exists which satisfied (8). Consider where By taking the inner product of (28) with and using Lemma 1, we obtain Notice that Hence, Similar to the proof of inequality (25), (32) can be rewritten as For and are sufficiently small which satisfies , we obtain It follows that This implies that there uniquely exists a trivial solution satisfying (8)–(10). Hence, is uniquely solvable. This completes the proof.
4. Convergence and Stability
In this section, the second-order rate of convergence and stability of scheme (8)-(9) are guaranteed and explicitly proved. Let , where and are the solutions of the problem (3)–(5) and the problem (8)-(9), respectively. We arrive at the following error equation: where denotes the truncation error. By the Taylor expansion, we easily obtain that holds as . The following lemmas are well known and useful for the proofs of the convergence and stability.
Lemma 5 (Zheng and Hu [16]). Suppose that . Then the solution of the initial-boundary value problem (3)–(5) satisfies for a constant .
Lemma 6 (discrete Gronwall inequality [9]). Suppose , are nonnegative mesh functions and is a nondecreasing function. If and then
The following theorem guarantees the convergence of our scheme with the convergence rate of .
Theorem 7. Suppose . Then the solution of scheme (8)-(9) converges to the solution of the problem (3)–(5) in the sense of and the rate of convergence is .
Proof. By taking the inner product of (8) and and using the fact that , we get where According to Lemma 5, Theorem 3, and the Cauchy-Schwartz inequality, we have Similar to the proof of (42), we have also Furthermore, By substituting (42)–(45) into (40), we obtain Hence, Let From (44), then (47) can be rewritten as That is, If is sufficiently small which satisfies , then Summing up from 1 to , we have Then Using (44), we obtain that Equations (53) and (54) yield which is equivalent to Notice that Since we can approximate using any available second-order accuracy method, we have . Hence According to Lemma 6, implies It follows from (44) that By using Lemma 2, we have This completes the proof of Theorem 7.
Theorem 8. Under the conditions of Theorem 7, the solution of scheme (8)-(9) is stable with respect to .
5. Numerical Experiments
In this section, we present numerical experiments on a test problem with an initial condition and boundary conditions to confirm and illustrate the accuracy of our method. Since the exact solution is not known, the finest grid () is used as a reference solution (pseudoanalytical solution). We make comparisons between schemes proposed in [18, 20] with our scheme (8), which is also second-order in space and time. The errors from the three schemes are presented in Tables 1, 2, and 3. For , , and , it is clear that our scheme gives better approximation than both schemes proposed in [18, 20].
The corresponding errors with respect to and -norm are listed in Tables 4, 5, and 6 for , , and . The rate of convergence is computed using two grids according to the formula Since we have five grids, we can get four different estimations of the convergent rates. As shown in Tables 4, 5, and 6 for one particular choice of the parameters, the three numerically estimated rates are presented and they are close to the theoretically predicted ones which are second-order rates of convergence.
Figures 1, 2, 3, and 4 show the numerical solutions at with and . The graphs of the numerical solutions of Hu et al. [18], Pan and Zhang [20] schemes, and the proposed scheme are presented. It is to confirm that the approximated solutions are coinciding.
Figures 5, 6, 7, and 8 present the numerical solutions computed by the finite difference scheme (8) with , at , , , , and when and , respectively. It is clear that the amplitude of the numerical solution decreases over time. In Figures 9, 10, 11, and 12, numerical solutions are presented for a fixed set of , , and with different values of and , respectively. The graph shows that does not contribute to the height of the amplitude of the numerical solution. On the other hand, the larger the value of , the smaller the amplitude of the numerical solution.
Absolute error distributions for the three methods are plotted at in Figures 13, 14, 15, and 16. Clearly, our proposed method gives a better approximate solution than the schemes proposed in [18, 20], especially at the peak of the solution. As the results in some of the applications where the characterization of the solution at the peak needs to be precise our proposed method is highly recommended.
6. Concluding Remarks
We introduce a three-level average linear-implicit finite difference method for solving the Rosenau-Burgers equation. We prove the existence and uniqueness of the numerical solution. The convergence and stability of the numerical solution are also shown. The quantitative comparison of the numerical results from previously known methods shows that our method improves the accuracy of the solution significantly. In addition, our results provide the most precise peak amplitude.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
This research was supported by Chiang Mai University.