Conservative Difference Scheme for Generalized Rosenau-KdV Equation

Zuo [1] discussed the solitary wave solutions and periodic solutions for Rosenau-KdV equation. In [2], a conservative nonlinear finite difference scheme for an initial-boundary value problem of Rosenau-Kdv equation is considered. In [3, 4] the solitary solution and invariant for generalized Rosenau-KdV equation are given. In [4] the singular 1soliton solution is derived by the ansatz method, and the adiabatic parameter dynamics of the water waves is obtained by perturbation theory. In [5, 6], the ansatz method is applied to obtain the topological soliton solution of the generalized Rosenau-KdV equation. The method as well as the exp-function method is also applied to extract a few more solutions to this equation. In [7], Zheng and Zhou give an average linear scheme for the generalized Rosenau-KdV equation. In this paper, we propose a conservative CrankNicolson finite difference scheme for an initial-boundary value problem of the generalized Rosenau-Kdv equation. The initial-boundary value problem (1)–(3) possesses the following conservative property [3, 4]:

In [3,4] the solitary solution and invariant for generalized Rosenau-KdV equation are given.In [4] the singular 1soliton solution is derived by the ansatz method, and the adiabatic parameter dynamics of the water waves is obtained by perturbation theory.In [5,6], the ansatz method is applied to obtain the topological soliton solution of the generalized Rosenau-KdV equation.The method as well as the exp-function method is also applied to extract a few more solutions to this equation.In [7], Zheng and Zhou give an average linear scheme for the generalized Rosenau-KdV equation.In this paper, we propose a conservative Crank-Nicolson finite difference scheme for an initial-boundary value problem of the generalized Rosenau-Kdv equation.
It is known that the conservative scheme is better than the nonconservative ones.The nonconservative scheme may 2 Advances in Mathematical Physics easily show nonlinear blow-up.A lot of numerical experiments show that the conservative scheme can possess some invariant properties of the original differential equation [7][8][9][10][11][12][13][14][15][16][17][18].The conservative scheme is more suitable for long-time calculations.In [18] Pan and Zhang said ". . . in some areas, the ability to preserve some invariant properties of the original differential equation is a criterion to judge the success of a numerical simulation".
The rest of this paper is organized as follows.In Section 2, we propose a Crank-Nicolson implicit nonlinear finite difference scheme for the generalized Rosenau-KdV equation and discuss the property of its solution.In Section 3, we prove that the finite difference scheme is of second order convergence.Finally, some numerical tests are given in Section 4 to verify our theoretical analysis.
Lemma 1.It follows from summation by parts that, for any two mesh functions , V ∈  0 ℎ , Then one has Furthermore, if (  0 )  = (   )  = 0, then To show the existence of the solution for ( 7)- (10), the following Brouwer fixed point theorem should be introduced.For the proof, see [19].
In order to prove the bounded quality of the difference solution, we introduce the following lemma.

𝑗
) . (36) Computing the inner product of (36) with 2 +(1/2) , we obtain Similar to the proof of ( 19), we have where Noting that In Table 1 we give the error at various time step.We denote the C-N scheme in this paper as scheme I and the difference scheme in [7] as scheme II.In Table 2 we give the error comparison between scheme I and scheme II.It is easy to see that the calculation results of scheme I are slightly better than scheme II.Using the method in [20,21], we verified the second convergence of the difference scheme in Table 3. Numerical simulations on the conservation invariant   are given in Table 4.
The wave graph comparison of (, ) at various times is given in Figures 1 and 2 when  = 3 and  = 5.

Conclusions
In this paper, we propose a conservative Crank-Nicolson finite difference scheme for the initial-boundary value problem of the generalized Rosenau-KdV equation.The twolevel finite difference scheme is of second order convergence and unconditionally stable, which can start by itself.From Table 2 we conclude that the C-N scheme is more efficient than scheme II in [7].From Table 3 we conclude that the C-N scheme is of second order convergence obviously.Numerical simulations on the conservation invariant   are given in Table 4. Figures 1 and 2 show that the height of the wave graph at different time is almost identical.Table 4 and Figures 1 and 2 imply that the finite difference scheme is conservative and efficient.

Table 1 :
The error at various time step.

Table 3 :
The verification of the second convergence.

Table 4 :
Numerical simulations on conservation invariant   .