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On the convergence of a high-accuracy compact conservative scheme for the modified regularized long-wave equation
SpringerPlus volume 5, Article number: 474 (2016)
Abstract
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.
Background
The generalized regularized long-wave (GRLW) equation reads (Peregrine 1996):
where \(\alpha \) and \(\mu \) are positive constants, \(p\ge 1\) is an integer. When \(p=1\), the Eq. (1) is usually called as the regularized long-wave (RLW) equation proposed by Peregrine (1996) and Benjamin et al. (1972) to describe nonlinear dispersive waves. Various numerical techniques have been developed to solve this equation. These partly include finite difference method, finite element methods, least squares method and collocation method with quadratic B-splines, cubic B-splines and septic splines, multisymplectic numerical method, in this respect, we refer readers to Kutluay and Esen (2006), Zhang (2005), Avilez-Valente and Seabra-Santos (2004), Esen and Kutluay (2006), Guo and Chen (2006); Gu and Chen 2008), Saka and Dag (2008), Dag (2000), Dag and Ozer (2001), Dag et al. (2004), Soliman and Raslan (2001), Soliman and Hussien (2005), Cai (2009, 2011), and references therein. Another special case of Eq. (1) for \(p=2\), the modified regularized long-wave (MRLW) equation is given by
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).
Along the numerical front, many efficient numerical methods have been developed for the MRLW equation, such as a new ten-point multisymplectic explicit numerical method (Cai 2010), Sinc-collocation method (Mokhtari and Mohammadi 2010), split least-squares mixed finite element method (Gao et al. 2015), B-spline finite element method (Gardner et al. 1997), finite difference method (Khalifa et al. 2007; Akbari and Mokhtari 2014), cubic B-spline collocation method (Khalifa et al. 2008b), quadratic B-spline collocation method (Tirmizi 2010), quadratic B-spline collocation method (Raslan 2009).
In recent works (Dehghan et al. 2009; Xie et al. 2009; Wang and Guo 2011; Wang 2014, 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 second-order 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 \(h=\frac{x_r-x_l}{J}\) and \(\tau =\frac{T}{N}\) be the uniform step size in the spatial and temporal direction, respectively. Denote \(x_j=jh\,(0\le j\le J), t_n=n\tau \,(0\le n\le N)\), \(u^n_j\approx u(x_j,t_n)\) and \(Z^0_h=\{u=(u_j)|u_0=u_J=0, j=0,1,2,\ldots , J\}\). Define
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 \({\varvec{M}}\) and \({\varvec{K}}\) are two real-value symmetric positive definite matrices. Hence there exist two real-value symmetric positive definite matrices \({\varvec{G}}\) and \({\varvec{H}}\), such that \({\varvec{G}}= {\varvec{M}}^{-1}\), \({\varvec{H}}= {\varvec{K}}^{-1}\). 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 \(\forall v^n, w^n\in Z^0_h\), we define the discrete inner products and norms on \(Z^0_h\) via:
To analyze the discrete conservative property and estimates of difference solution for the scheme (11)–(14), the following lemmas should be introduced.
Lemma 1
(Wang and Guo 2011) For any real value symmetric positive definite matrix \({{\varvec{G}}}_{(J-1)\times (J-1)}\) , then we have
where \({\varvec{R}}\) is obtained from \({\varvec{G}}\) by Cholesky decomposition (Zhang 2004).
Lemma 2
(Wang 2015) On the matrices \({\varvec{M}}\) and \({\varvec{K}}\) , there are the following results:
-
(1)
The eigenvalues of the matrices \({\varvec{M}}\) and \({\varvec{K}}\) are
$$ \lambda _{M,j}=\frac{1}{12}\left( 10+2\cos \frac{j\pi }{J}\right) ,\quad \lambda _{K,j}=\frac{1}{6}\left( 4+2\cos \frac{j\pi }{J}\right) ,\quad j=1,2,\ldots ,J-1. $$ -
(2)
The two matrices have the same eigenvectors
$$\begin{aligned} \mathbf{v }_k= \left( \sin \frac{k\pi }{J},\sin \frac{2k\pi }{J},\ldots ,\sin \frac{(J-1)k\pi }{J}\right) ^{\mathrm{T}}, \quad k=1,2,\ldots ,J-1. \end{aligned}$$
Lemma 3
For real value symmetric positive definite matrices \({\varvec{G}}_{(J-1)\times (J-1)}={\varvec{M}}^{-1}\) and \({\varvec{H}}_{(J-1)\times (J-1)}={\varvec{K}}^{-1}\) , then there exist three positive constants \(C_0\) , \(C_1\) and \(C_2\) , such that
where \(C_0=1,C_1=\frac{3}{2},C_2=3,\) \({\varvec{R}}\) and \({\varvec{S}}\) are obtained from \({\varvec{G}},\) \({\varvec{H}}\) by Cholesky decomposition (Zhang 2004) respectively.
Proof
It follows from Lemma 2 that
This implies that
Notice that \({\varvec{G}}\) and \({\varvec{H}}\) are also real value symmetric positive definite matrices. From Cholesky decomposition, we obtain
Then
This together with the definition of matrix norm and (18) gives that
where \(C_0=1,C_1=\frac{3}{2}\). Similarly, we can also obtain
where \(C_0=1,C_2=3\).
Remark 1
On the above real value symmetric positive definite matrices \({\varvec{G}}\) and \({\varvec{H}}\), according to Lemmas 2 and 3, for C is big enough, we can have \(||{\varvec{S}}{} {\mathbf{u}} ^n||^2\le C||{\varvec{R}}{} {\mathbf{u}} ^n||^2\).
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
Theorem 1
Suppose \(u_0\in H^1_0[x_l,x_r]\) , then the scheme (11)–(14) admits the following invariant
Proof
Taking the inner product of (11) with \({\mathbf{u}} ^{n+1}+{\mathbf{u}} ^{n-1}\) (i.e. \(2\bar{{\mathbf{u}}}^n\)) and using Lemma 1 yield
Computing the third term of the left-hand side in (24), we get
Computing the fourth term of the left-hand side in (24) yields
It follows from (24) to (26) that
Thus
where \(E^n=\frac{1}{2}(||{\mathbf{u}} ^{n+1}||^2+||{\mathbf{u}} ^{n}||^2)+\frac{1}{2}\mu (||{\varvec{R}}\delta _x{\mathbf{u}} ^{n+1}||^2+||{\varvec{R}}\delta _x{\mathbf{u}} ^n||^2)+ h\tau \sum \nolimits _{j=1}^{J-1}({\varvec{S}}\delta _{\hat{x}}u_j^{n}){\varvec{S}}u^{n+1}_j\). This completes the proof of Theorem 1.
Theorem 2
Assume that \(u_0\) is sufficiently smooth, then there is the estimation for the solution \({\mathbf{u}} ^n\) of the scheme (11)–(14): \(||{\mathbf{u}} ^n||\le C, ||\delta _x{\mathbf{u}} ^n||\le C\), which yield \(||{\mathbf{u}} ^n||_{\infty }\le C\).
Proof
It follows from (23) that
This together with Lemma 3 gives that
Let \(\tau \) be small enough, such that \(1-C_2\tau>0, C_0\mu -C_2\beta \tau >0\), then we obtain from (30) that
An application of Lemma 4 yields
Remark 2
Theorem 2 implies that scheme (11)–(14) is unconditionally stable.
Theorem 3
The difference scheme (11) is uniquely solvable.
Proof
By the mathematical induction. It is obvious that \({\mathbf{u}} ^0\) is uniquely determined by (13). We can choose a fourth-order method to compute \({\mathbf{u}} ^1\) [such as C–N scheme (12)]. Assuming that \({\mathbf{u}} ^1,\ldots ,{\mathbf{u}} ^n\) are uniquely solvable, consider \({\mathbf{u}} ^{n+1}\) in (11) which satisfies
Doing in (33) the inner product of with \({\mathbf{u}} ^{n+1}\) and using Lemma 1 yield
where \(\phi ({\mathbf{u}} ^n,{\mathbf{u}} ^{n+1})=\frac{1}{8}\alpha {\varvec{H}}[\delta _{\hat{x}}{} {\mathbf{u}} ^{n+1}({\mathbf{u}} ^{n})^2+\delta _{\hat{x}}(({\mathbf{u}} ^n)^2{\mathbf{u}} ^{n+1})]\).
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, \({\mathbf{u}} ^{n+1}\) 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:
Using Taylor expansion and considering the construction of the difference scheme of (7)–(10), we know the accuracy of (7)–(10) is \(O(\tau ^2+h^4)\). The scheme (11)–(14) is equivalent to (7)–(10). Then we have that \(|{\varvec{r}}^n|=O(\tau ^2+h^4),|\varvec{\sigma }^0|=O(\tau ^2+h^4)\) hold if \(\tau ,h\rightarrow 0\).
Next, we shall discuss the convergence and stability of the scheme (11)–(14).
Lemma 5
(Discrete Gronwall inequality Zhou 1990) Suppose that the discrete mesh function \(\{w^n|n=1,2,\ldots ,N; N\tau =T\}\) satisfies recurrence formula
where A,B and \(C_n\) (\(n=1,\ldots ,N\)) are nonnegative constants. Then
where \(\tau \) is small, such that \((A+B)\tau \le \frac{N-1}{2N} (N > 1)\).
Theorem 4
Assume that \(u_0\) is sufficiently smooth and \(u(x,t)\in C^{5,3}_{x,t}\) , then the solution \({\mathbf{u}} ^n\) of the scheme (10)–(12) converges to the solution of the IBV problem (3)–(5) and the rate of convergence is \(O(\tau ^2+h^4)\) by the \(||\cdot ||_{\infty }\) norm.
Proof
Let \({\mathbf{e}} ^n={\mathbf{v}} ^n-{\mathbf{u}} ^n\). From (37) to (40) and (11) to (14), we have
Doing in (41) the inner product with \(2\bar{\mathbf{e }}^{n}\) (i.e. \({\mathbf{e}} ^{n+1}+{\mathbf{e}} ^{n-1}\)) and using Lemma 1, we obtain
where \(P=\frac{1}{4}\alpha {\varvec{H}}[(\delta _{\hat{x}}\bar{\mathbf{v }}^n)({\mathbf{v}} ^n)^2-(\delta _{\hat{x}}\bar{{\mathbf{u}}}^n)({\mathbf{u}} ^n)^2]\), \(Q=\frac{1}{4}\alpha {\varvec{H}}\{[\delta _{\hat{x}}[({\mathbf{v}} ^n)^2\bar{\mathbf{v }}^n]-\delta _{\hat{x}}[({\mathbf{u}} ^n)^2\bar{{\mathbf{u}}}^n]\}\).
Computing the fourth term of right-hand side of (45) and using Theorem 2, Lemma 3 yield
Similarly, we have that
In addition, it is obvious that
It follows from (45) to (49) that
Let \(B^n=\frac{1}{2}(||{\mathbf{e}} ^{n+1}||^2+||{\mathbf{e}} ^{n}||^2)+\frac{1}{2}(||{\varvec{R}}\delta _x{\mathbf{e}} ^{n+1}||^2+||{\varvec{R}}\delta _x{\mathbf{e}} ^{n}||^2)\), then (50) can be written as follows:
By Lemma 5, it can immediately be obtained that
Taking the inner product in (42) with \({\mathbf{e}} ^1\), we have
where \(I=\frac{1}{4}\alpha {\varvec{H}}[(\delta _{\hat{x}}{} {\mathbf{v}} ^{0+\frac{1}{2}})({\mathbf{v}} ^{0+\frac{1}{2}})^2-(\delta _{\hat{x}}{} {\mathbf{u}} ^{0+\frac{1}{2}})({\mathbf{u}} ^{0+\frac{1}{2}})^2]\), \(II=\frac{1}{4}\alpha {\varvec{H}}\{\delta _{\hat{x}}[({\mathbf{v}} ^{0+\frac{1}{2}})^3]-\delta _{\hat{x}}[({\mathbf{u}} ^{0+\frac{1}{2}})^3]\}\).
Notice that
From the discrete initial condition (43), we know that \({\mathbf{e}} ^0=0\). Computing the fourth term of right-hand side in (53) and using Theorem 2, Lemma 3 yield
Similarly, we get
It follows from (53) to (56) that
This together with \((\varvec{\sigma }^0,{\mathbf{e}} ^1)\le \frac{1}{2}(||\varvec{\sigma }^0||^2+||{\mathbf{e}} ^1||^2)\) and Lemma 3 gives that
Let \(\tau \) be small enough, such that \(\frac{1}{\tau }-\frac{1}{2}-C> 0, \frac{1}{\tau }C_0\mu -C> 0\). Then we obtain from (58) that
where \(K_1=\frac{1}{\frac{2}{\tau }-1-2C}, K_2=\frac{1}{\frac{2}{\tau }C_0\mu -2C}\).
It follows from Lemma 3 that
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 \({\mathbf{u}} ^n\) of the scheme (11)–(14) is unconditionally stable by the \(||\cdot ||_{\infty }\) 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.
It follows from (64) that the initial-boundary value problem (3)–(5) is consistent to the initial value problem (3) and (4) for \(-x_l\gg 0\), \(x_r\gg 0\). In the following numerical experiments, we take \(x_l\) = −40, \(x_r=60\), \(T=10.\)
The IBV problem (3)–(5) has another two invariants (Gardner et al. 1997):
The initial condition of the studied model is obtained from (64) with the parameters \(x_0\), c, \(\alpha \) and μ:
In computations, we always choose the parameter \(x_0=0\). Take the parameters \(c=\alpha =\mu =1\). To verify the accuracy \(O(\tau ^2+h^4)\) in the spatial direction, we take \(\tau =h^2\). 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 \(\tau =h^2\) and the one of \(\log (e^n)\)–\(\log (\tau )\) with h small enough are given in Figs. 1 and 2 under various mesh steps h and \(\tau \) 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 \(O(\tau ^2+h^4).\)
The errors in the sense of \(L_\infty \)-norm and \(L_2\)-norm of the numerical solutions \({\mathbf{u}} ^n\) 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 \({\mathbf{u}} ^n\) of the scheme (11) satisfies the conservative property (23). The values of \(E^n\), Q \(\tilde{E}\) for the scheme (11) are presented in Table 3 under steps \(h=0.1\) and \(\tau =0.01\). 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 \(\tau =0.0025\) 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 \(p=2, \mu =\varepsilon =1\) and \(d=\frac{1}{3}\). Denote the present scheme (11) with \(c=\frac{1}{3}, \alpha =\mu =1\) as Scheme II. The corresponding errors in the sense of \(L_\infty \)-norm and CPU time are listed on Table 4 under different mesh steps h and \(\tau \). From Table 4, we get that a fourth-order three-level linear scheme as accurate as Scheme I which is a two-level one.
Conclusion
In this paper, an attempt has been made to construct a new numerical scheme to solve the initial-boundary problem of the MRLW equation, which has the following advantages: Coupling with the matrix theory, we convert the proposed scheme into the vector difference one. The new scheme is high-accuracy which has the accuracy of \(O(\tau ^2+h^4)\); The new scheme is conservative and preserves the original conservative properties; The coefficient matrices of the scheme is symmetric and tridiagonal, so Thomas algorithm can be employed to solve it effectively. Some numerical results are reported to show the efficiency and accuracy of the scheme.
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Authors' contributions
The article was carried out in collaboration between all authors. Both authors have contributed to, read and approved the final manuscript.
Acknowledgements
This work is supported by the Natural Science Foundation of China (Nos. 11201343, 11401438), Natural Science Foundation of Shandong Province (ZR2012AM017, ZR2013FL032), A Project of Shandong Province Higher Educational Science and Technology Program (No. J14LI52), the Youth Research Foundation of WFU (No. 2013Z11) and the Project of Science and Technology Program of Weifang (Grant No. 201301006).
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Pan, X., Zhang, L. On the convergence of a high-accuracy compact conservative scheme for the modified regularized long-wave equation. SpringerPlus 5, 474 (2016). https://doi.org/10.1186/s40064-016-2085-9
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DOI: https://doi.org/10.1186/s40064-016-2085-9
Keywords
- The MRLW equation
- Compact conservative method
- Solvability
- Convergence
- Stability