Two Conservative Difference Schemes for Rosenau-Kawahara Equation

Two conservative finite difference schemes for the numerical solution of the initialboundary value problem of Rosenau-Kawahara equation are proposed. The difference schemes simulate two conservative quantities of the problem well. The existence and uniqueness of the difference solution are proved. It is shown that the finite difference schemes are of second-order convergence and unconditionally stable. Numerical experiments verify the theoretical results.


Introduction
In the study of compact discrete systems, the wave-wave and wave-wall interactions cannot be described by the wellknown KdV equation.To overcome this shortcoming of KdV equation, Rosenau proposed the following so-called Rosenau equation in [1,2]: which is usually used to describe the dense discrete system and simulate the long-chain transmission model through an L-C flow in radio and computer fields.Park proved the existence and uniqueness of solution to (1) in [3].However, it is difficult to find its analytical solution.Therefore, the numerical study of (1) is very significant and attract many scholars (see, e.g., [4][5][6][7][8][9][10]).
It is well known that a reasonable difference scheme has not only high-accuracy but also can maintain some physical properties of original problem.Lots of numerical experiments show that conservative difference scheme can simulate the conservative law of initial problem well since it could avoid the nonlinear blow-up [13][14][15][16][17][18][19][20][21][22][23][24].Moreover, it is suitable to compute for long-time.Li and Vu-Quoc pointed in [14] that 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.Therefore, constructing the conservative difference scheme is a significant job.
The rest of this paper is organized as follows.We respectively propose a two-level nonlinear Crank-Nicolson difference scheme and three-level linear difference scheme for initial boundary value problems ( 2)-( 4) in Sections 2 and 3. We analyze its two discrete conservative laws.The existence and uniqueness of the difference solution are proved.It is shown that the finite difference schemes are of second-order convergence and unconditionally stable.In Section 4, we verify our theoretical analysis by numerical experiments.

Nonlinear Crank-Nicolson Conservative Difference Scheme
In this section, we propose a two-level nonlinear Crank-Nicolson difference scheme and give the theoretical analysis.
In the rest of this paper,  denotes a general positive constant which may denote different value in different occurrence.

Solvability of the Difference Scheme.
In order to prove the solvability of difference scheme, we present the following Brouwer fixed point theorem [25].
Proof.We use the mathematical induction to prove our theorem.
Therefore, from Lemma 6, we obtain Remark 9. Theorem 8 implies that the solution of difference schemes ( 9)-( 11) is unconditionally stable in the sense of norm ‖ ⋅ ‖ ∞ .

A Linear Conservative Difference Scheme
In this section, we propose a three-level linear conservative difference scheme for (2)-( 4) and give the theoretical analysis.
The following theorem shows how the difference schemes (63)-(65) simulate the conservative law as follows.
Taking the inner product of (63) with 2  , from boundary (65) and Lemma 1, we obtain where On the other hand,

Numerical Simulations
In We denote the nonlinear Crank-Nicolson conservative difference schemes ( 9)- (11) as Scheme I and the linear threelevel conservative difference schemes (63)-(65) as Scheme II.For some different values of  and ℎ, we list errors of Scheme I and Scheme II at several different time in Tables 1  and 2, respectively.We verify the two-order accuracy of the difference scheme in Tables 3 and 4 by using the method of [26,27].The numerical simulation of two conservative quantities ( 6) and ( 7) is listed in Tables 5 and 6.Finally, a numerical simulation figure comparison of (, ) at various time steps with  = ℎ = 0.05 is as follows.(see Figures 1 and 2).Numerical simulations show that the finite difference Schemes I and II in this paper are efficient.The calculation results of Scheme I are slightly better than Scheme II.But iterative numerical calculation is not required, Scheme II can save computing time.

Table 1 :
The errors of Scheme I at various time steps with various ℎ and .

Table 2 :
The errors of Scheme II at various time steps with various ℎ and .

Table 5 :
When  = 1, numerical simulations on the conservation invariant   .

Table 6 :
When  = 1, numerical simulations on the conservation invariant   .