Multi-symplectic method for peakon–antipeakon collision of quasi-Degasperis–Procesi equation

doi:10.1016/j.cpc.2014.04.006

Computer Physics Communications

Volume 185, Issue 7, July 2014, Pages 2020-2028

https://doi.org/10.1016/j.cpc.2014.04.006 Get rights and content

Abstract

Focusing on the local geometric properties of the shockpeakon for the Degasperis–Procesi equation, a multi-symplectic method for the quasi-Degasperis–Procesi equation is proposed to reveal the jump discontinuity of the shockpeakon for the Degasperis–Procesi equation numerically in this paper. The main contribution of this paper lies in the following: (1) the uniform multi-symplectic structure of the b-family equation is constructed; (2) the stable jump discontinuity of the shockpeakon for the Degasperis–Procesi equation is reproduced by simulating the peakon–antipeakon collision process of the quasi-Degasperis–Procesi equation. First, the multi-symplectic structure and several local conservation laws are presented for the b-family equation with two exceptions ( $b = 3$ and $b = 4$ ). And then, the Preissman Box multi-symplectic scheme for the multi-symplectic structure is constructed and the mathematical proofs for the discrete local conservation laws of the multi-symplectic structure are given. Finally, the numerical experiments on the peakon–antipeakon collision of the quasi-Degasperis–Procesi equation are reported to investigate the jump discontinuity of shockpeakon of the Degasperis–Procesi equation. From the numerical results, it can be concluded that the peakon–antipeakon collision of the quasi-Degasperis–Procesi equation can be simulated well by the multi-symplectic method and the simulation results can reveal the jump discontinuity of shockpeakon of the Degasperis–Procesi equation approximately.

Introduction

The Degasperis–Procesi equation [1], $u_{t} - u_{x x t} + 4 u u_{x} = 3 u_{x} u_{x x} + u u_{x x x}$ an approximation of the incompressible Euler equation for the modeling of shallow water propagation in conditions of small amplitude and long wavelength, considered as a special case ( $b = 3$ ) of the so-called b-family equation [2], $m_{t} + u m_{x} + b m u_{x} = 0$ where $u = u (x, t)$ is the fluid velocity, $b$ is a real parameter that characterizes the b-family equation as the ratio of stretching to convective transport and the number of covariant dimensions associated with the momentum density $m = u - u_{x x}$ , has aroused considerable interest due to its integrable structure as an infinite bi-Hamiltonian system.

The b-family equation (Eq. (2)) is a one-dimensional version of active fluid transport, in which the second term $u m_{x}$ is a convection term and the third term $b m u_{x}$ is a stretching term. The competition between the convection and the stretching results in the existence of the peakon solution for the b-family equation, which has been studied widely in the last decade: Camassa and Holm [3] derived the Camassa–Holm equation ( $b = 2$ ) with an infinite number of conservation laws in involution and proved that the soliton solutions of the Camassa–Holm equation have a discontinuity in the first derivative at its peak, which is named as the peakon solutions; Degasperis and Procesi [1] found the Degasperis–Procesi equation using the method of asymptotic integrability; subsequently, Degasperis, Holm and Hone [2] proved that the Degasperis–Procesi equation is integrable and the Degasperis–Procesi equation admits exact solutions as a superposition of multipeakons; Feng and Liu [4] proposed an operator splitting method for the Degasperis–Procesi equation and constructed a numerical scheme to simulate the propagations of the multipeakons as well as the shockpeakons of the Degasperis–Procesi equation; recently, based on the construction of suitable smooth periodic solutions with small amplitude, Christov and his colleagues [5] have shown that the solution map of the b-family equation is not uniformly continuous.

The above reports on the peakon solution for the b-family equation are mainly on the two special situations ( $b = 2$ and $b = 3$ ) because that b-family equation (2) is integrable only for $b = 2$ [3] and $b = 3$ [6], which implies that the Camassa–Holm equation and the Degasperis–Procesi equation may contain many interesting geometric properties. One remarkable geometric property of both equations is the existence of the breaking phenomenon during the peakon–antipeakon collision process. To investigate this local behavior, the multi-symplectic method, a high-accuracy structure-preserving numerical method with excellent long-time behaviors [7], [8] has been used to investigate the peakon–antipeakon collisions for the Camassa–Holm equation [9], [10], [11]; the numerical results in these Refs. agree with the theoretical results: a solution of the Camassa–Holm equation after breaking can be continued as either a global conservative solution [12] or a global dissipative solution [13]. But the situation of the Degasperis–Procesi equation may be contrary: a solution of the Degasperis–Procesi equation after breaking may be discontinued at the wave crest, which is named as a type of shockpeakon solution featured by the jump discontinuity phenomenon during the peakon–antipeakon collision [14].

The jump discontinuity of the shockpeakon solution for the Degasperis–Procesi equation was first reported by Lundmark in 2007 [14]. And then, Guo [15] constructed the periodic peakon and shockpeakon solutions of the Degasperis–Procesi equation and Yu [16] developed a symplecticity-preserving time-stepping scheme to resolve the computational difficulty at the wave crest where the first-order derivative may diverge and the shockpeakon solution may form recently. It is well known that the jump discontinuity is a local geometric property of the shockpeakon associated with the energy and the momentum and the multi-symplectic method is a structure-preserving numerical method paying more attention on the local geometric property of the infinite-Hamiltonian systems [7], [8]. It is a pity that the Degasperis–Procesi equation could not be converted to multi-symplectic form, as shown in Section 2. Thus, in this paper, the multi-symplectic structure for the b-family equation is deduced and a Preissman Box multi-symplectic scheme is presented to investigate the peakon–antipeakon collision of the quasi-Degasperis–Procesi equation ( $b \to 3$ ). From the collision results of $b \to 3$ , the jump discontinuity of the shockpeakon for the Degasperis–Procesi equation is illustrated.

Section snippets

The multi-symplectic structure of the b-family equation

Assuming $b \neq 3$ and $b \neq 4$ , and introducing the following canonical momenta, $\begin{matrix} u_{t} = - w_{x}, & φ_{x} = u, \\ u_{x} = v, & \frac{b - 3}{b - 4} u_{t} = (3 - b) u v + \frac{1}{3 - b} ψ \end{matrix}$ the b-family equation (2) can be rewritten as the first-order form $\frac{1}{2} φ_{t} - \frac{b - 3}{b - 4} v_{t} + \frac{1}{b - 3} ψ_{x} = \frac{1}{2} w - \frac{b + 1}{2} u^{2} + \frac{3 - b}{2} v^{2} .$

Introducing the state variable $z = {[u, φ, w, ψ, v]}^{T} \in R^{5}$ and connecting Eqs. (3), (4), a multi-symplectic structure of the b-family equation is obtained, ${Mz}_{t} + {Kz}_{x} = \nabla_{z} S (z)$ where $M = [\begin{matrix} 0 & \frac{1}{2} & 0 & 0 & - \frac{b - 3}{b - 4} \\ - \frac{1}{2} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ \frac{b - 3}{b - 4} & 0 & 0 & 0 & 0 \end{matrix}], K = [\begin{matrix} 0 & 0 & 0 & \frac{1}{b - 3} & 0 \\ 0 & 0 & - \frac{1}{2} & 0 & 0 \\ 0 & \frac{1}{2} & 0 & 0 & 0 \\ \frac{1}{3 - b} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{matrix}]$ as well as the Hamiltonian

The multi-symplectic Preissman Box scheme of the b-family equation

The Preissman Box scheme, a space–time version of the implicit midpoint rule for ordinary differential equations, has been proved as one of the typical discrete methods that results in multi-symplectic [17], [18], [19]. In this section, the Preissman Box scheme for the b-family equation and the necessary proofs for the discrete conservation laws will be presented in detail.

An expanded form of the Preissman Box scheme for the multi-symplectic structure (5) is ${\begin{matrix} δ_{t}^{+} φ_{j + 1 / 2}^{i} - \frac{2 (b - 3)}{b - 4} δ_{t}^{+} v_{j + 1 / 2}^{i} + \frac{2}{b - 3} \end{matrix}$

Numerical experiments on the peakon–antipeakon collision of the quasi-Degasperis–Procesi equation

It has been mentioned that the Degasperis–Procesi equation ( $b = 3$ ) is not included in the multi-symplectic structure (5). Thus, in this section, we simulate the peakon–antipeakon collision of the quasi-Degasperis–Procesi equation ( $b \to 3^{-}$ and $b \to 3^{+}$ ) to investigate the jump discontinuity of shockpeakon of the Degasperis–Procesi equation.

In the following experiments, we let the time step length $Δ t = 0.001$ and the spatial step length $Δ x = 0.002$ , and consider the following initial conditions, $u (0, x) = e^{- | x - 1 |} - e$

Conclusions

Capturing the jump discontinuity of shockpeakon for the Degasperis–Procesi equation numerically is a challenge of modern numerical method. In this paper, the multi-symplectic method, a numerical method that can exactly preserve the local geometric properties of the Hamiltonian system is used to capture the peakon–antipeakon collision processes of the quasi-Degasperis–Procesi equation ( $b \to 3^{-}$ or $b \to 3^{+}$ ) numerically, which can reveal the jump discontinuity of shockpeakon for the Degasperis–Procesi

Acknowledgments

The first author wishes to thank Professor Thomas J. Bridges of Surrey University for giving him several good suggestions. The research was supported by the National Natural Science Foundation of China (11372253, 11372252, 11172239), the Science Foundation of Aviation of China (2013ZB53020), 111 project (B07050) to the Northwestern Polytechnical University, the NPU Foundation for Fundamental Research (JC20110259), the Open Project of Guangxi Key Laboratory of Disaster Prevention and Structural

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Multi-symplectic method for peakon–antipeakon collision of quasi-Degasperis–Procesi equation

Abstract

Introduction

Section snippets

The multi-symplectic structure of the b-family equation

The multi-symplectic Preissman Box scheme of the b-family equation

Numerical experiments on the peakon–antipeakon collision of the quasi-Degasperis–Procesi equation

Conclusions

Acknowledgments

J. Comput. Phys.

Nonlinear Anal.: TMA

Phys. Lett. A

J. Comput. Phys.

Comput. Phys. Comm.

Nonlinear Anal.: TMA

J. Comput. Phys.

Comput. Phys. Comm.

Comput. Phys. Comm.

Future Gener. Comput. Syst.