Complete integrable particle methods and the recurrence of initial states for a nonlinear shallow-water wave equation

Camassa, Roberto; Lee, Long

doi:10.1016/j.jcp.2008.04.011

Cited by 19 publications

(27 citation statements)

References 13 publications

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“…For the dispersive case in Lagrangian form, an equivalent formulation, more convenient for numerical purposes, can be provided in analogy with that for the one-dimensional SW equation presented in [8]. Appending the Lagrangian form of the evolution equation (10) for the determinant J(ξ, t),…”

Section: Dispersive Deformationmentioning

confidence: 99%

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Solitary Waves and N‐Particle Algorithms for a Class of Euler–Poincaré Equations

Camassa¹,

Kuang

Lee

2016

Stud Appl Math

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Abstract. We study a class of partial differential equations (PDEs) in the family of the so-called Euler-Poincaré differential systems, with the aim of developing a foundation for numerical algorithms of their solutions. This requires particular attention to the mathematical properties of this system when the associated class of elliptic operators possesses non-smooth kernels. By casting the system in its Lagrangian (or characteristics) form, we first formulate a particles system algorithm in free space with homogeneous Dirichlet boundary conditions for the evolving fields. We next examine the deformation of the system when non-homogeneous "constant stream" boundary conditions are assumed. We show how this simple change at the boundary deeply affects the nature of the evolution, from hyperbolic-like to dispersive with a non-trivial dispersion relation, and examine the potentially regularizing properties of singular kernels offered by this deformation. From the particle algorithm viewpoint, kernel singularities affect the existence and uniqueness of solutions to the corresponding ordinary differential equations systems. We illustrate this with the case when the operator kernel assumes a conical shape over the spatial variables, and examine in detail two-particle dynamics under the resulting lack of Lipschitz-continuity. Curiously, we find that for the conically-shaped kernels the motion of the related two-dimensional waves can become completely integrable under appropriate initial data. This reduction projects the two-dimensional system to the one-dimensional completely integrable Shallow-Water equation [Camassa, R. and Holm, D. D., Phys. Rev. Lett., 71, 1961-1964, 1993, while retaining the full dependence on two spatial dimensions for the single channel solutions. Finally, by comparing with an operator-splitting pseudospectral method we illustrate the performance of the particle algorithms with respect to their Eulerian counterpart for this class of non-smooth kernels.

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Section: Dispersive Deformationmentioning

confidence: 99%

“…Just as the particle algorithms developed for the SW equation [4,[7][8][9][10], the N -particle system in this paper can be seen as a Lagrangian numerical algorithm for solving the model PDEs (2). The peakon of the particle method for the SW equation behaves like a member of a functional basis.…”

Section: Smooth Initial Datamentioning

confidence: 99%

Solitary Waves and N‐Particle Algorithms for a Class of Euler–Poincaré Equations

Camassa¹,

Kuang

Lee

2016

Stud Appl Math

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“…The first example is the traveling wave solution in periodic domains considered in [8,9]. The periodic travelling wave solution is given by uðx; tÞ ¼ Uðx À c tÞ, provided that the minima of u are located at u ¼ 0 and the wave elevation is positive.…”

Section: Travelling Wave Solution In Periodic Domainsmentioning

confidence: 99%

“…. ; 3Þ, c, and T are discussed in [8,9]. The parameters used in the test problem are c ¼ 2; j ¼ 1=2, and the integration constant C ¼ 1, which altogether yield the wavelength (period) of L % 6:3019 according to Eq.…”

Section: Travelling Wave Solution In Periodic Domainsmentioning

confidence: 99%

“…The convergence rates are found approximately 5.4 in space and 2 in time, respectively, for the proposed scheme. Table 6.3 shows the comparison between the proposed method and the particle method developed in [8,9] at the final time t ¼ 3:1509 (over one period). A fixed small time step Dt ¼ 0:12308 Â 10 À3 is used for both methods.…”

Section: Travelling Wave Solution In Periodic Domainsmentioning

confidence: 99%

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A dispersion-relation-preserving algorithm for a nonlinear shallow-water wave equation

Chiu

Lee

Sheu

2009

Journal of Computational Physics

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Development of a numerical phase optimized upwinding combined compact difference scheme for solving the Camassa-Holm equation with different initial solitary waves

Sheu

Chang

et al. 2015

Numer. Methods Partial Differential Eq.

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In this article, the solution of Camassa-Holm (CH) equation is solved by the proposed two-step method. In the first step, the sixth-order spatially accurate upwinding combined compact difference scheme with minimized phase error is developed in a stencil of four points to approximate the first-order derivative term. For the purpose of retaining both of the long-term accurate Hamiltonian property and the geometric structure inherited in the CH equation, the time integrator used in this study should be able to conserve symplecticity. In the second step, the Helmholtz equation governing the pressure-like variable is approximated by the sixth-order accurate three-point centered compact difference scheme. Through the fundamental and numerical verification studies, the integrity of the proposed high-order scheme is demonstrated. Another aim of this study is to reveal the wave propagation nature for the investigated shallow water equation subject to different initial wave profiles, whose peaks take the smooth, peakon, and cuspon forms. The transport phenomena for the cases with/without inclusion of the linear first-order advection term κu x in the CH equation will be addressed.

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Complete integrable particle methods and the recurrence of initial states for a nonlinear shallow-water wave equation

Cited by 19 publications

References 13 publications

Solitary Waves and N‐Particle Algorithms for a Class of Euler–Poincaré Equations

Solitary Waves and N‐Particle Algorithms for a Class of Euler–Poincaré Equations

A dispersion-relation-preserving algorithm for a nonlinear shallow-water wave equation

Development of a numerical phase optimized upwinding combined compact difference scheme for solving the Camassa-Holm equation with different initial solitary waves

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