Multisymplectic method for the Camassa-Holm equation
- Yu Zhang^{1},
- Zi-Chen Deng^{1}Email author and
- Wei-Peng Hu^{1}
https://doi.org/10.1186/s13662-015-0724-z
© Zhang et al. 2016
Received: 10 July 2015
Accepted: 19 November 2015
Published: 7 January 2016
Abstract
The Camassa-Holm equation, a completely integrable evolution equation, contains rich geometric structures. For the existence of the bi-Hamiltonian structure and the so-called peaked wave solutions, considerable interest has been aroused in the last several decades. Focusing on local geometric properties of the peaked wave solutions for the Camassa-Holm equation, we propose the multisymplectic method to simulate the propagation of the peaked wave in this paper. Based on the multisymplectic theory, we present a multisymplectic formulation of the Camassa-Holm equation and the multisymplectic conservation law. Then, we apply the Euler box scheme to construct the structure-preserving scheme of the multisymplectic form. Numerical results show the merits of the multisymplectic scheme constructed, especially the local conservative properties on the wave form in the propagation process.
Keywords
MSC
1 Introduction
This partial differential equation (PDE) is a one-dimensional version of the active fluid transport that has been proved to be integrable and is called an infinite bi-Hamiltonian system. In the early 1990s, the Camassa-Holm equation was proposed by Camassa and Holm [2] and possesses an infinite number of conservation laws in involution for the existence of the bi-Hamiltonian. In their work, the discontinuity in the first derivative at its peak of the soliton solution has been given as a limiting form. Constantin [3] proved that the Camassa-Holm equation is completely integrable. Fisher and Schiff [4] showed that the Camassa-Holm equation has an infinite number of local conserved quantities by a Miura-Gardner-Kruskal-type construction. In the late 1990s, based on the method of asymptotic integrability, Degasperis and Procesi derived the Degasperis-Procesi equation. Since then, there are many works reported on this equation: Degasperis et al. [5] proved that the Degasperis-Procesi equation is also integrable and admits multipeakon solutions. Lundmark [6] proposed an inverse scattering approach for seeking n-peakon solutions and presented the jump discontinuity of the shockpeakon solution for the Degasperis-Procesi equation firstly. Coclite et al. [7] constructed several numerical schemes for capturing discontinuous solutions of the Degasperis-Procesi equation and proved that the schemes converge to entropy solutions. Later, Coclite et al. [8] studied the process of singularity formation for the b-family equation and presented the numerical results of the Degasperis-Procesi equation.
For Eq. (2), there are some special solitons called peakon solutions, which are known to be rich in geometric structures and to have nonsmooth traveling wave solutions.
Many numerical methods have been used to solve the Camassa-Holm equation. For example, Coclite et al. [9, 10] derived a finite difference scheme for the Camassa-Holm equation that can handle general \(H^{1}\) initial data. Xu and Shu [11] developed a local discontinuous Galerkin method that enhances some nonlinear high-order derivatives to solve the Camassa-Holm equation. Matsuo [12] presented a new \(G_{2}\)-conserving Galerkin scheme for the Camassa-Holm equation. Kalisch and Lenells [13] used a pseudospectral scheme to investigate the traveling-wave solutions of the Camassa-Holm equation. Subsequently, Kalisch and Raynaud [14] investigated the convergence of the pseudospectral scheme of the Camassa-Holm equation. However, reports on the numerical analysis of the Camassa-Holm equation based on structure-preserving method are limited: Cohen et al. [15] presented two multisymplectic formulations for the Camassa-Holm equation to simulate the peakon and peakon-antipeakon collisions. Zhu et al. [16] developed a multisymplectic wavelet collocation method for solving multisymplectic Hamiltonian system with periodic boundary conditions and showed high accuracy and good conservation properties of the numerical method.
As a kind of structure-preserving methods, the multisymplectic integrator can preserve the intrinsic properties of the infinite-Hamiltonian systems [17–19] in each time step. It is well known that the peakon is a wave with local geometric property of the Camassa-Holm equation that can be investigated by the multisymplectic integrator because the multisymplectic method pays more attention to the local geometric properties of the systems.
The rest of this paper is organized as follows. In Section 2, we deduce the multisymplectic formulation of the Camassa-Holm equation. In Section 3, we construct the multisymplectic Euler box scheme to simulate the peaked wave of the Camassa-Holm equation. In Section 4, the numerical results are carried out to show that this scheme is of high accuracy and has good invariant-conserving properties. Finally, some conclusions are given in the last section.
2 The multisymplectic formulations of the Camassa-Holm equation
The multisymplectic integrator algorithm has aroused considerable interest since it was presented in 1997 [17]. As a very robust framework, this algorithm has many advantages in the process of dealing with some conservative PDE systems, such as high accuracy and good long-time numerical behavior. The basic idea of a multisymplectic integrator is that the numerical scheme is designed to preserve the multisymplectic form at each time step [17]. Because the multisymplectic idea presented a new requirement for a numerical algorithm, it has aroused an important revolution in numerical methods for PDEs. Based on the multisymplectic idea, multisymplectic methods have been applied to approximate some famous conservative PDEs, which are exactly in existence of multisymplectic forms, such as nonlinear wave equations [18, 20, 21], Schrödinger equations [22–25], KdV equations [26–28], Boussinesq equations [29, 30], Maxwell equations [31–33], b-family equations [15, 16, 34], Kawahara-type equation [35], etc. In this section, we give the multisymplectic form of the Camassa-Holm equation.
3 Multi-symplectic Euler box scheme for the Camassa-Holm equation
A numerical scheme is multisymplectic if and only if the numerical scheme preserves the discrete multisymplectic conservation law [19]. As a widely used numerical scheme, the Euler box scheme [36] for the PDEs (3) has been proved to be multisymplectic. An integrator form satisfying a discrete multisymplectic conservation law can be obtained by introducing the splitting of the two matrices M and K defined as \(\mathbf{M} = \mathbf{M}_{ +} + \mathbf{M}_{ -}\) and \(\mathbf{K} = \mathbf{K}_{ +} + \mathbf{K}_{ -}\), where \(\mathbf{M}_{ +}^{\mathrm{T}} = - \mathbf{M}_{ -}\) and \(\mathbf{K}_{ +}^{\mathrm{T}} = - \mathbf{K}_{ -}\).
4 Numerical results on the peaked wave solution of the Camassa-Holm equation
4.1 Numerical results on peaked wave solution
From Figure 1 we can conclude that the wave shape of the numerical solution is smooth enough and does not change in the process of propagation. Thus, the multisymplectic Euler box scheme for the Camassa-Holm equation can be successfully used to simulate the peakon solution.
The numerical and exact data on some grid points at \(\pmb{t = 1}\)
x | Numerical data | Exact data |
---|---|---|
−4.5 | 0.0040867762 | 0.0040867714 |
−4 | 0.0067379718 | 0.0067379470 |
−3.5 | 0.0111090695 | 0.0111089965 |
−3 | 0.0183158110 | 0.0183156389 |
−2.5 | 0.0301977431 | 0.0301973834 |
−2 | 0.0497877582 | 0.0497870684 |
−1.5 | 0.0820862237 | 0.0820849986 |
−1 | 0.1353372802 | 0.1353352832 |
−0.5 | 0.2231319769 | 0.2231301601 |
0.5 | 0.6065338144 | 0.6065306597 |
1 | 1.0000083119 | 1.0000000000 |
1.5 | 0.6065303657 | 0.6065306597 |
2 | 0.3678784135 | 0.3678794412 |
2.5 | 0.2231294390 | 0.2231301601 |
3 | 0.1353348944 | 0.1353352832 |
3.5 | 0.0820848236 | 0.0820849986 |
4 | 0.0497870068 | 0.0497870684 |
4.5 | 0.0301973713 | 0.0301973834 |
Comparing the numerical result with the exact solution, we can conclude that the multisymplectic Euler box scheme (10) can simulate the peaked traveling wave with small numerical errors. As shown in Figure 2, the maximum magnitude of the errors is up to 10^{−3}, which only appear at the points near the peakon, whereas on the grid points far away from the peakon, the magnitude of the errors are less than 10^{−6}, as shown in Table 1.
Numerical values of \(\pmb{u_{x}}\) at the peakon position
t = 0.2 | t = 0.6 | t = 1.0 | t = 1.4 | t = 1.8 | |
---|---|---|---|---|---|
\(\lim_{x \to 0^{ +}} u_{x}\) | 0.99657 | 0.97305 | 1.01050 | 0.98411 | 0.97879 |
\(\lim_{x \to 0^{ -}} u_{x}\) | −0.99751 | −0.99753 | −0.99755 | −0.99759 | −0.99765 |
From Table 2 we find that the value of \(\lim_{x \to 0^{ +}} u_{x}\) is approximately 1 and \(\lim_{x \to 0^{ -}} u_{x}\) is about −1 in the process of the peaked wave propagation. Similarly to the theoretical results shown in Eq. (14), the discontinuity phenomenon on the peakon of the soliton solution is reproduced numerically.
As an explicit scheme, such a high precision of the multisymplectic Euler box scheme is worth being further verified in the future. The results illustrate the structure-preserving properties of the multisymplectic integration.
4.2 Three global conserved quantities for the Camassa-Holm equation
As shown in Eq. (16), \(H_{0}\) and \(H_{1}\) represent the conservation laws for the mass and momentum with the process of the propagation, respectively. The third conserved quantity \(H_{2}\) is obtained by an action principle, which expressed in terms of a velocity potential [2]. The conservation laws for the Degasperis-Procesi equation are much weaker than the conservation laws for the Camassa-Holm equation. Thus, the orbital stability of Degasperis-Procesi peaked solitons is more subtle [37].
Under the periodic boundary conditions, the value of \([ D ]|_{l_{1}}^{l_{2}}\) is zero. Thus, the global conserved quantity \(H_{1}\) is just the global conserved momentum quantity associated with the Camassa-Holm equation. Similarly, integrating the local energy conservation law in Eq. (17) over the spatial domain, the global energy conservation law can be obtained, which is just the global conserved quantity \(H_{2}\) for the Camassa-Holm equation.
It is well known that the multisymplectic method can preserve the inherent local geometric properties of infinite-dimensional Hamiltonian systems, so we would like to know whether the quantities mentioned above are conserved. The relative errors in \(H_{0}\), \(H_{1}\), and \(H_{2}\) of the multisymplectic Euler box scheme (10) about the three conserved quantities (20) on each grid point are recorded, and the integration of which with respect to x is obtained to show the preserving status of the conserved quantities in the Camassa-Holm equation.
5 Conclusions
In this paper, the multisymplectic method, a numerical method that can preserve the local geometric properties of the Hamiltonian systems, is used to simulate the peakon of the Camassa-Holm equation.
- 1.
The multisymplectic Euler box method can well simulate the traveling peaked wave of the Camassa-Holm equation with small errors. The numerical simulation results also verified that the soliton has a discontinuity in the first derivative at its peak. These results illustrate the structure-preserving properties of the multisymplectic integration.
- 2.
The numerical results on the three conserved quantities of the peakon imply that the multisymplectic method can preserve the inherent geometric properties of the Camassa-Holm equation.
Declarations
Acknowledgements
The research is supported by the National Natural Science Foundation of China (11372252 and 11372253), the National Basic Research Program of China 973 (2011CB610300), the PhD Programs Foundation of Ministry of Education of China (20126102110023), and the Fundamental Research Funds for the Central Universities (3102014JCQ01035).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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