On the convergence of a high-accuracy compact conservative scheme for the modified regularized long-wave equation

In this article, we develop a high-order efficient numerical scheme to solve the initial-boundary problem of the MRLW equation. The method is based on a combination between the requirement to have a discrete counterpart of the conservation of the physical “energy” of the system and finite difference method. The scheme consists of a fourth-order compact finite difference approximation in space and a version of the leap-frog scheme in time. The unique solvability of numerical solutions is shown. A priori estimate and fourth-order convergence of the finite difference approximate solution are discussed by using discrete energy method and some techniques of matrix theory. Numerical results are given to show the validity and the accuracy of the proposed method.

In recent years, the MRLW equation has attracted much attention of many researchers. Many mathematical and numerical studies have been developed for the MRLW equation in the literatures. Along the mathematical front, for the exact solutions via double reduction theory and Lie symmetries, the bifurcation and travelling wave solutions as well as some explicit analytic solutions obtained from dynamical systems theory, numerical solutions with high degree of accuracy by the variational iteration method and the Adomian decomposition method, we refer readers to Naz et al. (2013), Yan et al. (2012), Labidi and Omrani (2011), Khalifa et al. (2008a).
In recent works (Dehghan et al. 2009;Xie et al. 2009;Wang and Guo 2011;Wang 2014Wang , 2015, the fourth-order compact finite difference approximation solutions to solve the Klein-Gordon equation, the Schrödinger equation and Klein-Gordon-Schrödinger equation were shown, respectively. The numerical results are encouraging. Motivated by the techniques of these works, in this paper, we propose a linearized compact conservative difference scheme with high accuracy to solve the MRLW equation (2) numerically. The presented compact difference scheme is three-level, linear-implicit and secondorder accuracy in time and fourth-order accuracy in space. By means of the matrix theory, we convert the proposed scheme into the vector difference one. The coefficient matrices of the present scheme are symmetric and tridiagonal, and Thomas algorithm can be employed to solve them effectively. Numerical example on the model problem shows that the present scheme is of high accuracy and good stability, which preserves the original conservative properties at the same time.
The rest of this paper is organized as follows. In "The high-accuracy compact conservative vector difference scheme" section, a linearized compact finite difference scheme for the MRLW equation is described. In "Discrete conservative property, estimate and solvability" section, we discuss the solvability of the scheme and the estimate of the difference solution. In "Convergence and stability of the difference scheme" section, convergence and stability of the scheme are proved by using energy method. In "Numerical experiments" section, numerical experiments are reported to test the theoretical results.

The high-accuracy compact conservative vector difference scheme
In this section, we describe a high-order linear-compact conservative difference scheme for the Eq. (2). Consider the MRLW equation with an initial condition

and the boundary conditions
where u 0 (x) is a known smooth function.
The IBV problem (3)-(5) is known to possess the following conservative property: Let h = x r −x l J and τ = T N be the uniform step size in the spatial and temporal direction, respectively. Denote In the paper, C denotes a general positive constant which may have different values in different occurrences.
For the one-order derivative u x and two-order derivative u xx , we have the following formulas: Omitting the high-order terms O(h 4 ) in the formulas above, we consider the following three-level linear compact scheme for the IBV problem (3)-(5).
The scheme (7) is three-level and linear-implicit, so it can be easily implemented and suitable for parallel computing.

Define
Notice that M and K are two real-value symmetric positive definite matrices. Hence there exist two real-value symmetric positive definite matrices G and H, such that G = M −1 , H = K −1 . Then (7)-(10) can be rewritten into the vector form as follows: For convenience, the last term of (11) is defined by

Discrete conservative property, estimate and solvability
In this section, we shall discuss the estimate for the difference solution and the solvability of the difference scheme (11). For ∀v n , w n ∈ Z 0 h , we define the discrete inner products and norms on Z 0 h via: .

Lemma 1 (Wang and Guo 2011) For any real value symmetric positive definite matrix
where R is obtained from G by Cholesky decomposition (Zhang 2004). where C 0 = 1, C 1 = 3 2 , C 2 = 3, R and S are obtained from G, H by Cholesky decomposition (Zhang 2004) respectively.

Proof It follows from Lemma 2 that
This implies that Notice that G and H are also real value symmetric positive definite matrices. From Cholesky decomposition, we obtain Then Gδ x δxu n , u n = −||Rδ x u n || 2 , (15) C 0 ||u n || 2 ≤ Gu n , u n = ||Ru n || 2 ≤ C 1 ||u n || 2 , (16) C 0 ||u n || 2 ≤ Hu n , u n = ||Su n || 2 ≤ C 2 ||u n || 2 , This together with the definition of matrix norm and (18) gives that Similarly, we can also obtain Remark 1 On the above real value symmetric positive definite matrices G and H, according to Lemmas 2 and 3, for C is big enough, we can have ||Su n || 2 ≤ C||Ru n || 2 .
We also use the following Lemma.
Lemma 4 (Discrete Sobolev's inequality Zhou 1990) There exist two positive constants C 1 and C 2 such that , then the scheme (11)-(14) admits the following invariant Proof Taking the inner product of (11) with u n+1 + u n−1 (i.e. 2ū n ) and using Lemma 1 yield Computing the third term of the left-hand side in (24), we get (20) Gu n , u n = Ru n , Ru n = ||Ru n || 2 .
Proof By the mathematical induction. It is obvious that u 0 is uniquely determined by (13). We can choose a fourth-order method to compute u 1 [such as C-N scheme (12)]. Assuming that u 1 , . . . , u n are uniquely solvable, consider u n+1 in (11) which satisfies Doing in (33) the inner product of with u n+1 and using Lemma 1 yield Similarly to the proof of (26), we obtain This together with (34) gives that This implies that there uniquely exists trivial solution satisfying Eq. (33). Hence, u n+1 in (11) is uniquely solvable. This completes the proof of Theorem 3.

Convergence and stability of the difference scheme
First, we shall consider the truncation error of the difference scheme (11)-(14). Let v n j = u(x j , t n ). We define the truncation error as follows: (32) ||u n || ∞ ≤ C.
Next, we shall discuss the convergence and stability of the scheme (11)-(14). Theorem 4 Assume that u 0 is sufficiently smooth and u(x, t) ∈ C 5,3 x,t , then the solution u n of the scheme (10)-(12) converges to the solution of the IBV problem (3)-(5) and the rate of convergence is O(τ 2 + h 4 ) by the || · || ∞ norm.
Taking the inner product in (42) with e 1 , we have
. Pan and Zhang SpringerPlus (2016) 5:474 This implies that It follows from (52) that which together with Lemmas 3 and 4, and the definition of B n gives that This completes the proof of Theorem 4. Similarly, we can prove stability of the difference solution.
Theorem 5 Under the conditions of Theorem 4, the solution u n of the scheme (11)- (14) 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. In order to test whether the present scheme (11)- (14) exhibits the expect convergence rates in time and in space, we will measure the accuracy of the proposed scheme using the square norm errors and the maximum norm errors defined by The exact solution of the IBV problem (3)-(5) has the following form (Gardner et al. 1997): where x 0 , c are arbitrary constants.
The initial condition of the studied model is obtained from (64) with the parameters x 0 , c, α and μ: In computations, we always choose the parameter x 0 = 0. Take the parameters c = α = µ = 1. To verify the accuracy O(τ 2 + h 4 ) in the spatial direction, we take τ = h 2 . And we choose h small enough to verify the second-order accuracy in the temporal direction. The convergence order figure of log(e n )-log(h) with τ = h 2 and the one of log(e n )-log(τ ) with h small enough are given in Figs. 1 and 2 under various mesh steps h and τ at t = 10. From Figs. 1 and 2, it is obvious that the scheme (11)-(14) is convergent in maximum norm, and the convergence order is O(τ 2 + h 4 ).
The errors in the sense of L ∞ -norm and L 2 -norm of the numerical solutions u n of the scheme (11) are listed on Tables 1 and 2. Tables 1 and 2 shows good stability of the numerical solutions and also verify the scheme in present paper is efficient and of high accuracy.
We have shown in Theorem 1 that the numerical solution u n of the scheme (11) satisfies the conservative property (23). The values of E n , Q Ẽ for the scheme (11) are presented in Table 3 under steps h = 0.1 and τ = 0.01. It is easy to see from Table 3 that the scheme (11) preserves the discrete conservative properties very well, thus it can be used to computing for a long time.
The curves of the solitary wave with time computed by scheme (11) with h = 0.05 and τ = 0.0025 are given in Fig. 3; the waves at t = 5 and 10 agree with the ones at t = 0 quite well, which also demonstrate the accuracy and efficiency of the scheme in present paper.
To compare the numerical results with other results shown in previous studies, we denote the proposed scheme in Akbari and Mokhtari (2014) as Scheme I with p = 2, µ = ε = 1 and d = 1 3 . Denote the present scheme (11) with c = 1 3 , α = µ = 1 as Scheme II. The corresponding errors in the sense of L ∞ -norm and CPU time are listed on Table 4 under different mesh steps h and τ. From Table 4, we get that a fourth-order three-level linear scheme as accurate as Scheme I which is a two-level one.