Discrete Dynamics in Nature and Society

Discrete Dynamics in Nature and Society / 2015 / Article

Research Article | Open Access

Volume 2015 |Article ID 206264 | https://doi.org/10.1155/2015/206264

Xiaoquan Ding, Qing-Jiang Meng, Li-Ping Yin, "Discrete-Time Orthogonal Spline Collocation Method for One-Dimensional Sine-Gordon Equation", Discrete Dynamics in Nature and Society, vol. 2015, Article ID 206264, 8 pages, 2015. https://doi.org/10.1155/2015/206264

Discrete-Time Orthogonal Spline Collocation Method for One-Dimensional Sine-Gordon Equation

Academic Editor: Pilar R. Gordoa
Received13 Sep 2015
Revised23 Nov 2015
Accepted01 Dec 2015
Published30 Dec 2015

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 .



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