Integral and integrable algorithms for a nonlinear shallow-water wave equation

Camassa, Roberto; Huang, Jingfang; Lee, Long

doi:10.1016/j.jcp.2005.12.013

Cited by 43 publications

(66 citation statements)

References 26 publications

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“…N for some constant ǫ, then the solution of system (3.5) exists uniquely for all times. In particular, no two particles can occupy the same position q i (t) = q j (t), for some i = j, at any finite time t [7]. The global existence result can be easily modified to include the case κ = 0, as it is evident from the integrable formulation (2.2), which shows that the p-components are simply shifted by the constant κ.…”

Section: Discrete Integrable Systemmentioning

confidence: 99%

On a Completely Integrable Numerical Scheme for a Nonlinear Shallow-Water Wave Equation

Camassa¹,

Huang²,

Lee³

2005

JNMP

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An algorithm for an asymptotic model of wave propagation in shallow-water is proposed and analyzed. The algorithm is based on the Hamiltonian structure of the equation, and corresponds to a completely integrable particle lattice. Each "particle" in this method travels along a characteristic curve of the shallow water equation. The resulting system of nonlinear ordinary differential equations can have solutions that blow up in finite time. Conditions for global existence are isolated and convergence of the method is proved in the limit of zero spatial step size and infinite number of particles. A fast summation algorithm is introduced to evaluate integrals in the particle method so as to reduce computational cost from O(N 2 ) to O(N ), where N is the number of particles. Accuracy tests based on exact solutions and invariants of motion assess the global properties of the method. Finally, results on the study of the nonlinear equation posed in the quarter (space-time) plane are presented. The minimum number of boundary conditions required for solution uniqueness and the complete integrability are discussed in this case, while a modified particle scheme illustrates the evolution of solutions with numerical examples.

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Section: Discrete Integrable Systemmentioning

confidence: 99%

On a Completely Integrable Numerical Scheme for a Nonlinear Shallow-Water Wave Equation

Camassa¹,

Huang²,

Lee³

2005

JNMP

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“…In [38], the existence and uniqueness of strong solutions to the IBV problem with boundary condition u(0, t) ≡ 0 and decaying initial data from certain functional classes has been studied for the case ω = 0. In [15] the authors demonstrated that the IBV problem with boundary data u(0, t) = v 0 (t) is well-posed in the case v 0 (t) ≤ 0, in the sense that a strong solution, if it exists up to a certain time, is unique. The CH equation is known to be formally integrable: it has a Lax pair representation as a compatibility condition for the system of linear equations ψ xx = 1 4 ψ + λ(m + ω)ψ,…”

Section: Introductionmentioning

confidence: 99%

Long time asymptotics of the Camassa–Holm equation on the half-line

Monvel¹,

Shepelsky²

2009

Ann. inst. Fourier

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Abstract. We study the long-time behavior of solutions of the initial-boundary value (IBV) problem for the Camassa-Holm (CH) equation ut − utxx + 2ux + 3uux = 2uxuxx + uuxxx on the half-line x ≥ 0. The paper continues our study of the IBV problems for the CH equation [9], the key tool of which is the formulation and analysis of the associated RiemannHilbert factorization problem. We specify the regions in the quarter space-time plane x > 0, t > 0 having qualitatively different asymptotic pictures, and give the main terms of the asymptotics in terms of the spectral data associated with the initial and boundary values.

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“…We would like to point out that the system (3.3), obtained using the weak formulation of the problem, is equivalent to the one derived in [6,7] by considering the Hamiltonian structure of equation (3.1). One can verify that the functions x i (t) and p i (t) in (3.3) satisfy the canonical Hamiltonian equations:…”

Section: Properties Of the Particle Systemmentioning

confidence: 99%

“…Particle methods have also recently been applied to numerical solutions of a 1-D nonlinear shallow water equation in [6,7], and to the study of the dynamics of N point particles ("blobs") governed by the Euler equations, [34]. The methods presented in [6,7,34] have been derived using a discretization of a variational principle, while here the equivalent representation of the particle system is obtained by considering a weak formulation of the problem.…”

Section: Introductionmentioning

confidence: 99%

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Integration of the EPDiff equation by particle methods

2012

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Abstract. The purpose of this paper is to apply particle methods to the numerical solution of the EPDiff equation. The weak solutions of EPDiff are contact discontinuities that carry momentum so that wavefront interactions represent collisions in which momentum is exchanged. This behavior allows for the description of many rich physical applications, but also introduces difficult numerical challenges. We present a particle method for the EPDiff equation that is well-suited for this class of solutions and for simulating collisions between wavefronts. Discretization by means of the particle method is shown to preserve the basic Hamiltonian, the weak and variational structure of the original problem, and to respect the conservation laws associated with symmetry under the Euclidean group. Numerical results illustrate that the particle method has superior features in both one and two dimensions, and can also be effectively implemented when the initial data of interest lies on a submanifold.

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Integral and integrable algorithms for a nonlinear shallow-water wave equation

Cited by 43 publications

References 26 publications

On a Completely Integrable Numerical Scheme for a Nonlinear Shallow-Water Wave Equation

On a Completely Integrable Numerical Scheme for a Nonlinear Shallow-Water Wave Equation

Long time asymptotics of the Camassa–Holm equation on the half-line

Integration of the EPDiff equation by particle methods

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