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

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 O(h4 + τ2) in L2 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.


Introduction
We consider the following one-dimensional sine-Gordon equation: Here we require that ℎ 0 ( 0 ) = ( 0 ) and ℎ 1 ( 0 ) = ( 1 ) for consistency,  0 <  1 ∈ R. When (,   ) = sin() and (, ) = 0, (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 O( 2 + ℎ 2 ), where ℎ and  denote the spatial and temporal mesh sizes, respectively.Cui [9] gives a three-level implicit compact difference scheme with a convergence rate O( 2 +ℎ 4 ) 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 O( 2 + ℎ 4 ) in  2 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.
A family F of partitions is said to be quasi-uniform if there exists a finite positive number  such that max for every partition Δ in F. We assume that the partition Δ is a member of a quasi-uniform family F. Let {  }  =0 be a partition of [0, ], where   =  and  = /.
Let M be the space of piecewise Hermite cubics on Ω defined by where P  denotes the set of all polynomials of degree less than or equal to .
We introduce the following lemmas.

Accuracy and Stability of the Scheme
In this section, we study the accuracy and stability of the numerical method.
), then for  and ℎ 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 ℎ and  in the following proof.Substituting   () = (, ) into (21) and using Taylor expansion, we have where One may get from ( 21) and ( 23) that Computing the inner product ⟨⋅⟩ of (24) with  +1 −  −1 as in Section 2, we have where where   ,  = 1, . . ., 4, denote constants.It follows from Lemma 1 and (28)-(29) that If  > 1/2, by using similar arguments in the proof of Theorem 4.1 in [14], 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  5 ,  6 , and  7 denote constants.
Since  > 0, we conclude max This implies max 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   ℎ () and ũ ℎ =   ℎ () −   ().Then we have Computing the inner product of ( 40) with ( +1 −  −1 ), 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.
Example 1.We consider Dirichlet boundary conditions problem given in [9].We consider the problem Its theoretical solution is (, ) = (1 We define where   = (  ) −  ℎ (  ) and the corresponding relative error is ‖‖ G /‖ ℎ ()‖ G .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 ℎ = 0.2 and ℎ = 0.05 with  = 0.01.
The rate of convergence of the proposed method can be calculated from the formula where ℎ 1 , ℎ 2 are space steps and the value of  is called the rate of convergence.In Theorem 6, we prove that our proposed scheme is O( 2 + ℎ 4 ).In Figure 1, a comparison of the OSC scheme with the Cui scheme [9] has been made; the slope is 4. When the space grid size ℎ is reduced by 1/2 and the time grid size  is reduced by 1/4, the error between the analytic solution and the numerical solution is reduced by 1/16.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 ℎ =  = 0.01.(50) Its theoretical solution is (, ) =  − sin().Error comparison for ℎ = 0.01 and  = 0.01 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 ℎ =  = 0.01 are plotted in Figure 3.

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.

Figure 1 :Example 2 .
Figure 1: Convergence rate of two different schemes for Example 1 in  = 2.0.

Figure 2 :
Figure 2: Space-time graph of the solution for Example 1 up to  = 2.0 with  = 0.01 and ℎ = 0.01, analytical solution with blue color and estimated solution with red color.

Figure 3 :
Figure 3: Theoretical solution (red color) and numerical solution (blue color) for Example 2.

Table 1 :
Errors of the OSC scheme for Example 1 with  = 0.01.

Table 2 :
Relative errors comparison of the Cui scheme and the OSC scheme for Example 1 with  = 0.01.

Table 3 :
Errors comparison of the Cui scheme and the OSC scheme for Neumann problem.