237 research outputs found
An explicit finite difference scheme for the Camassa-Holm equation
We put forward and analyze an explicit finite difference scheme for the
Camassa-Holm shallow water equation that can handle general initial data
and thus peakon-antipeakon interactions. Assuming a specified condition
restricting the time step in terms of the spatial discretization parameter, we
prove that the difference scheme converges strongly in towards a
dissipative weak solution of Camassa-Holm equation.Comment: 45 pages, 6 figure
Discrete Variational Derivative Methods for the EPDiff equation
The aim of this paper is the derivation of structure preserving schemes for
the solution of the EPDiff equation, with particular emphasis on the two
dimensional case. We develop three different schemes based on the Discrete
Variational Derivative Method (DVDM) on a rectangular domain discretized with a
regular, structured, orthogonal grid.
We present numerical experiments to support our claims: we investigate the
preservation of energy and linear momenta, the reversibility, and the empirical
convergence of the schemes. The quality of our schemes is finally tested by
simulating the interaction of singular wave fronts.Comment: 41 pages, 41 figure
Decoupled and unidirectional asymptotic models for the propagation of internal waves
We study the relevance of various scalar equations, such as inviscid
Burgers', Korteweg-de Vries (KdV), extended KdV, and higher order equations (of
Camassa-Holm type), as asymptotic models for the propagation of internal waves
in a two-fluid system. These scalar evolution equations may be justified with
two approaches. The first method consists in approximating the flow with two
decoupled, counterpropagating waves, each one satisfying such an equation. One
also recovers homologous equations when focusing on a given direction of
propagation, and seeking unidirectional approximate solutions. This second
justification is more restrictive as for the admissible initial data, but
yields greater accuracy. Additionally, we present several new coupled
asymptotic models: a Green-Naghdi type model, its simplified version in the
so-called Camassa-Holm regime, and a weakly decoupled model. All of the models
are rigorously justified in the sense of consistency
Integration of the EPDiff equation by particle methods
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
On the isospectral problem of the dispersionless Camassa-Holm equation
We discuss direct and inverse spectral theory for the isospectral problem of
the dispersionless Camassa--Holm equation, where the weight is allowed to be a
finite signed measure. In particular, we prove that this weight is uniquely
determined by the spectral data and solve the inverse spectral problem for the
class of measures which are sign definite. The results are applied to deduce
several facts for the dispersionless Camassa--Holm equation. In particular, we
show that initial conditions with integrable momentum asymptotically split into
a sum of peakons as conjectured by McKean.Comment: 26 page
On error estimates for Galerkin finite element methods for the Camassa-Holm equation
We consider the Camassa-Holm (CH) equation, a nonlinear dispersive wave
equation that models one-way propagation of long waves of moderately small
amplitude. We discretize in space the periodic initial-value problem for CH
(written in its original and in system form), using the standard Galerkin
finite element method with smooth splines on a uniform mesh, and prove
optimal-order -error estimates for the semidiscrete approximation. We
also consider an initial-boundary-value problem on a finite interval for the
system form of CH and analyze the convergence of its standard Galerkin
semidiscretization. Using the fourth-order accurate, explicit, "classical"
Runge-Kutta scheme for time-stepping, we construct a highly accurate, stable,
fully discrete scheme that we employ in numerical experiments to approximate
solutions of CH, mainly smooth travelling waves and nonsmooth solitons of the
`peakon' type
A numerical study of variational discretizations of the Camassa-Holm equation
We present two semidiscretizations of the Camassa-Holm equation in periodic
domains based on variational formulations and energy conservation. The first is
a periodic version of an existing conservative multipeakon method on the real
line, for which we propose efficient computation algorithms inspired by works
of Camassa and collaborators. The second method, and of primary interest, is
the periodic counterpart of a novel discretization of a two-component
Camassa-Holm system based on variational principles in Lagrangian variables.
Applying explicit ODE solvers to integrate in time, we compare the variational
discretizations to existing methods over several numerical examples.Comment: 45 pages, 14 figure
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