Numerical Simulation for General Rosenau-RLW Equation : An Average Linearized Conservative Scheme

Numerical solutions for the general Rosenau-RLW equation are considered and 
an energy conservative linearized finite difference scheme is proposed. Existence of the solutions 
for the difference scheme has been shown. Stability, convergence, and a priori error estimate of the 
scheme are proved using energy method. Numerical results demonstrate that the scheme is efficient 
and reliable.


Introduction
In this paper, we examine the use of the finite difference method for the general Rosenau-RLW equation x 0, x ∈ x l , x r , t ∈ 0, T , 1.1 with an initial condition u x, 0 u 0 x , x ∈ x l , x r , 1.2 and boundary conditions u x l , t u x r , t 0, u xx x l , t u xx x r , t 0, t ∈ 0, T , 1.3 where p ≥ 2 is a integer and u 0 x is a known smooth function.When p 2, the equation 1.1 is called usual Rosenau-RLW equation.When p 3, 1.1 is called modified Rosenau-RLW equation.
It can be proved easily that the problem 1.1 -1.3 possesses the following conservative laws: As already pointed out by Fei et al. 1 , the nonconservative difference schemes may easily show nonlinear blow-up, and the conservative difference schemes perform better than the non-conservative ones.In 2-15 , some conservative finite difference schemes were used for Sine-Gordon equation, Cahn-Hilliard equation, Klein-Gordon equation, a system of Schr ödinger equation, Zakharov equations, Rosenau equation, GRLW equation, Klein-Gordon-Schr ödinger equation, respectively.Numerical results of all the schemes are very good.
As far as computational studies are concerned, Zuo et al. 16 have proposed a Crank-Nicolson difference scheme for the Rosenau-RLW equation.The difference scheme in 16 is nonlinear implicit, so it requires heavy iterative calculations and is not suitable for parallel computation.In a recent work 14 , we have made some preliminary computation by proposing a conservative linearized difference scheme for GRLW equation which is unconditionally stable and reduces the computational work, and the numerical results are encouraging.In this paper, we continue our work and propose a conservative linearized difference scheme for the general Rosenau-RLW equation which is unconditionally stable and secondorder convergent and simulates conservative laws 1.4 -1.5 at the same time.
The remainder of this paper is organized as follows.In Section 2, an energy conservative linearized difference scheme for the general Rosenau-RLW equation is described and the discrete conservative laws of the difference scheme are discussed.In Section 3, we show that the scheme is uniquely solvable.In Section 4, convergence and stability of the scheme are proved.In Section 5, numerical experiments are reported.

An Average Linearized Conservative Scheme and Its Discrete Conservative Law
In this section, we describe a new conservative difference scheme for the problems of 1.1 -1.3 .Let h and τ be the uniform step size in the spatial and temporal direction, respectively.Denote x j jh 0 ≤ j ≤ J , t n nτ 0 ≤ n ≤ N , u n j ≈ u x j , t n and Z 0 h {u u j | u 0 u j 0, j 0, 1, 2, . . ., J}. Define 2.1 and in the paper, C denotes a general positive constant which may have different values in different occurrences.Notice that u p x p/ p 1 u p−1 u x u p x .We consider the following three-level average linearized conservative scheme for the IBV problems 1.1 -1.3 : where 0 ≤ θ ≤ 1 is a real constant.The scheme 2.2 -2.4 is three level and linear implicit, so it can be easily implemented.It should be pointed out that we need another suitable twolevel scheme such as C-N scheme to compute u 1 .For convenience, the last term of 2.2 is defined by

2.5
Lemma 2.1 see 17 .For any two mesh functions: u, v ∈ Z 0 h , one has

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Furthermore, if u n Theorem 2.2.Suppose u 0 ∈ H 2 0 x l , x r and u x, t ∈ C 5,3 .Then the scheme 2.2 -2.4 admits the following invariant: 2.9 Proof.Multiplying 2.2 with h, according to the boundary conditions 2.4 , then summing up for j from 1 to J − 1, we obtain
Taking the inner product of 2.2 with 2u n , according to Lemma 2.1, we have

2.12
Now, computing the last term of the left-hand side in 2.12 , we have

2.13
Substitute 2.13 into 2.12 , and we let u n 1 j .

2.14
By the definition of E n , 2.9 holds.

Solvability
In this section, we will prove the solvability of the difference scheme 2.2 .
Proof.By the mathematical induction.It is obvious that u 0 is uniquely determined by 2.3 .We can choose a second-order method to compute u 1 such as C-N scheme 16 .Assuming that u 1 , . . ., u n are uniquely solvable, consider u n 1 in 2.2 which satisfies

3.1
Taking the inner product of 3.1 with u n 1 , we obtain

Mathematical Problems in Engineering
Notice that

3.3
It follows from 3.2 that That is, there uniquely exists trivial solution satisfying 3.1 .Hence, u n 1 j in 2.2 is uniquely solvable.This completes the proof of Theorem 3.1.Remark 3.2.All results above in this paper are correct for IBV problem of the general Rosenau-RLW equation with finite or infinite boundary.

Convergence and Stability of Finite Difference Scheme
First we will consider the truncation error of the difference scheme of 2.2 -2.4 .Denote v n j u x j , t n .We define the truncation error as follows:

4.1
Using Taylor expansion, we obtain that Er n j O τ 2 h 2 holds if τ, h → 0. This is that.Lemma 4.1.Assume u x, t is smooth enough, then the local truncation error of difference scheme Next, we will discuss the convergence and stability of finite difference scheme Proof.It follows from 2.9 that

4.6
This implies for small τ which satisfies 1 − θτ > 0, we get        According to Lemma 4.4, the fifth term of right-hand side of 4.10 is estimated as follows: In addition, it is obvious that Er n j , 2e n ≤ Er n 2 1 2 e n 1 2 e n−1 2 , 4.14 4.17  Hence, for τ sufficiently small, such that 1 − Cτ > 0, we obtain

4.19
Summing up 4.19 from 1 to n yields

4.20
Choose a second-order method to compute u 1 such as C-N scheme and notice that From the discrete initial conditions, we know that e 0 is of second-order accuracy, then Then we obtain

4.26
This completes the proof of Theorem 4.6.
Similarly, we can prove stability of the difference solution.
Theorem 4.7.Under the conditions of Theorem 4.6, the solution of the scheme 3.1 -2.4 is unconditionally stable by the • ∞ norm.

Numerical Experiments
In this section, we conduct some numerical experiments to verify our theoretical results obtained in the previous sections.

Figure 1 :
Figure 1: Errors in the sense of e n ∞ computed by the scheme 2.2 when h τ 0.1 and p 4.

Figure 2 :
Figure 2: Errors in the sense of e n 2 computed by the scheme 2.2 when h τ 0.1 and p 4.

Table 1 :
The errors of numerical solutions at t 10 with p 2 and τ 0.1.

Table 2 :
The errors of numerical solutions at t 10 with p 4 and τ 0.1.

Table 3 :
The errors of numerical solutions at t 10 with p 8 and τ 0.1.

Table 4 :
Discrete mass and energy of scheme 2.2 for a few of θ values at different time t with h τ 0.1 and p 2.

Table 5 :
Discrete mass and energy of scheme 2.2 for a few of θ values at different time t with h τ 0.1 and p 4.

Table 6 :
Discrete mass and energy of scheme 2.2 for a few of θ values at different time t with h τ 0.1 and p 8.