Abstract
We present a discrete-time orthogonal spline collocation scheme for the one-dimensional sine-Gordon equation. This scheme uses Hermite basis functions to approximate the solution throughout the spatial domain on each time level. The convergence rate with order in norm and stability of the scheme are proved. Numerical results are presented and compared with analytical solutions to confirm the accuracy of the presented scheme.
1. Introduction
We consider the following one-dimensional sine-Gordon equation:with initial conditions and Dirichlet boundary conditions or Neumann boundary conditionsHere we require that and for consistency, . When and , (1) is a classical sine-Gordon equation. The sine-Gordon equation has applications in various research areas such as the Lie group of methods [1] and the inverse scattering transform [2]. It also appears in a number of other physical applications, including the propagation of fluxons in Josephson junctions between two superconductors, the motion of rigid pendulums attached to a stretched wire, and dislocations in crystals [3, 4].
The numerical solution to the sine-Gordon equation has received considerable attention in the literature. Among others Khaliq et al. [5] use a predictor-corrector scheme to solve the finite difference scheme using the methods of line. Bratsos [6] applies a predictor-corrector scheme from the use of rational approximation to the matrix-exponential term. Mohebbi and Dehghan [7] propose a high-order and accurate method for solving sine-Gordon equation using compact finite difference and DIRKN methods. Xu and Chang [8] present an implicit scheme and a compact scheme for the solution of an initial-boundary value problem of the generalized nonlinear sine-Gordon equation with a convergence rate , where and denote the spatial and temporal mesh sizes, respectively. Cui [9] gives a three-level implicit compact difference scheme with a convergence rate by using the Padé approximant.
The purpose of this paper is to investigate the use of the orthogonal spline collocation (OSC) method with piecewise Hermite cubic polynomials for the spatial discretization of (1). The accuracy and stability of solutions with order in norm are verified. This method has evolved as a valuable technique for the solution of many types of partial differential equations. See [10] for a comprehensive survey. The popularity of such a method is due in part to its conceptual simplicity and ease of implementation. One obvious advantage of the OSC method over the finite element method is that the calculation of the coefficient matrices is very efficient since no integral calculation is required. Another advantage of this method is that it systematically incorporates boundary conditions and interface conditions.
The paper is organized as follows. In Section 2, we briefly review the OSC method and give the discretization scheme of the sine-Gordon equation. In Section 3, we demonstrate the accuracy and stability of the scheme. Numerical results are presented in Section 4.
2. The OSC Method for Sine-Gordon Equation
With a positive integer , let be a partition of : Let , and . A family of partitions is said to be quasi-uniform if there exists a finite positive number such that for every partition in . We assume that the partition is a member of a quasi-uniform family . Let be a partition of , where and .
Let be the space of piecewise Hermite cubics on defined by where denotes the set of all polynomials of degree less than or equal to .
Let denote the roots of the Legendre polynomial of degree 2, where and . To apply the collocation method, we introduce a set of collocation points taken as For , we define a discrete inner product and its induced norm by We always use the following difference quotient notations: Let be a nonnegative integer; we have We denote by the Banach space of all integrable functions from into with norm for and the standard modification for . In this paper, we take .
Let be basis functions of . So one may write where are unknown coefficients which should be worked out.
We introduce the following lemmas.
Lemma 1 (Lemma in [11], Equation in [12]). For , there exist positive constants and such that
Lemma 2 (Lemma , Lemma in [11]). For , one has
Lemma 3 (Theorem in [11]). Let and suppose that satisfies Then one has
Lemma 4 (Lemma in [13]). Suppose that discrete function satisfies the recurrence formula where , , and are nonnegative constants. Then where is small, such that .
Lemma 5 (Inequality in [8]). Let , , and be constants. Suppose that the following conditions are satisfied: (i) ; (ii) , . Then one has
We use finite difference scheme and construct the discrete-time OSC scheme as follows:for , , and .
3. Accuracy and Stability of the Scheme
In this section, we study the accuracy and stability of the numerical method.
Theorem 6. Suppose is the solution of (21), , and is the solution of (16). If is defined by (16), , and are , then for and sufficiently small one has where is the exact solution of (21) when .
Proof.
We use to denote a generic positive constant that is independent of and in the following proof. Substituting into (21) and using Taylor expansion, we have where . Let and , then .
One may get from (21) and (23) that Computing the inner product of (24) with as in Section 2, we have whereUsing Lemma 5, we can get According to the definition of , one can easily obtain . Thus, Applying Lemma 3, we have where , , denote constants. It follows from Lemma 1 and (28)-(29) thatIf , by using similar arguments in the proof of Theorem in [14], we have If , then If , by using Sobolev’s inequality and Theorem in [15], we have where is a positive constant.
Thus, one can obtain from (30)–(33) that where Apply Lemma 4; after simple calculation we get the following inequality: where , , and denote constants.
Since , we conclude This implies These all together yield the following inequality:
In the following theorem, we give the stability of the numerical method.
Theorem 7. If the conditions of Theorem 6 are satisfied, then scheme (21) is unconditionally stable.
Proof. Let be the error of and . Then we have Computing the inner product of (40) with , we obtain by a similar proof as that of Theorem 6: where According to [16] and references therein, this theorem expresses the generalized stability of the numerical scheme.
4. Numerical Experiments
In this section, we present some numerical results of our scheme for sine-Gordon equations. We adopt the following form of (1) for Examples 1 and 2: According to (21), the corresponding OSC scheme might be written as for , , and .
Setting and substituting (13) into (44), one can obtainwhere , , , and are matrices with special structures commonly known as almost block diagonal, so the system of algebraic equations (45) could be solved by using the COLROW algorithm [17].
Applying Taylor’s theorem, one can get from (2) and (43) Consequently, and can be prescribed by approximating and using piecewise Hermite cubic interpolations, respectively. In all of the following experiments, we choose .
Example 1. We consider Dirichlet boundary conditions problem given in [9]. We consider the problem Its theoretical solution is . We define where and the corresponding relative error is . The numerical results for the OSC scheme are given in Table 1. In order to discuss the accuracy of the method at long time level, we give relative errors in the brackets. In [9], Cui approximates the second-order derivative in the space variable by compact finite difference. Table 2 gives error comparison of the Cui scheme [9] and the OSC scheme for and with .
The rate of convergence of the proposed method can be calculated from the formula where , are space steps and the value of is called the rate of convergence. In Theorem 6, we prove that our proposed scheme is . In Figure 1, a comparison of the OSC scheme with the Cui scheme [9] has been made; the slope is . When the space grid size is reduced by and the time grid size is reduced by , the error between the analytic solution and the numerical solution is reduced by . Thus the scheme is of fourth-order accuracy in space and second-order accuracy in time. From Figure 1, Tables 1 and 2, we can see that the OSC method is more efficient and accurate than the Cui scheme [9] though they have the same fourth order in space and second order in time, and the OSC method has conceptual simplicity. The space-time graphs of analytical and estimated functions are given in Figure 2 with .
Example 2. We consider the problem with Neumann boundary conditions Its theoretical solution is . Error comparison for and between the OSC scheme and the Cui scheme is given in Table 3. From Table 3, we can see that the OSC scheme is more accurate than the Cui scheme [9] according to their absolute error and relative error. The theoretical solution and the numerical solution with are plotted in Figure 3.
5. Conclusion
In this paper, we discuss the generalized nonlinear sine-Gordon equation. We propose the OSC method to solve this nonlinear equation. The implementation of the method is as simple as finite difference methods. The numerical results given in the previous section demonstrate the accuracy of this scheme.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work is partially supported by the National Natural Science Foundation of China under Grant 11271110, the Key Programs for Science and Technology of the Education Department of Henan Province under Grant 12A110007, and the Scientific Research Funds of Henan University of Science and Technology.