80 research outputs found
New variational and multisymplectic formulations of the Euler-Poincar\'e equation on the Virasoro-Bott group using the inverse map
We derive a new variational principle, leading to a new momentum map and a
new multisymplectic formulation for a family of Euler--Poincar\'e equations
defined on the Virasoro-Bott group, by using the inverse map (also called
`back-to-labels' map). This family contains as special cases the well-known
Korteweg-de Vries, Camassa-Holm, and Hunter-Saxton soliton equations. In the
conclusion section, we sketch opportunities for future work that would apply
the new Clebsch momentum map with -cocycles derived here to investigate a
new type of interplay among nonlinearity, dispersion and noise.Comment: 19 page
Geometric numerical schemes for the KdV equation
Geometric discretizations that preserve certain Hamiltonian structures at the
discrete level has been proven to enhance the accuracy of numerical schemes. In
particular, numerous symplectic and multi-symplectic schemes have been proposed
to solve numerically the celebrated Korteweg-de Vries (KdV) equation. In this
work, we show that geometrical schemes are as much robust and accurate as
Fourier-type pseudo-spectral methods for computing the long-time KdV dynamics,
and thus more suitable to model complex nonlinear wave phenomena.Comment: 22 pages, 14 figures, 74 references. Other author's papers can be
downloaded at http://www.lama.univ-savoie.fr/~dutykh
Energy preserving integration of bi-Ham
The energy preserving average vector field (AVF) integrator is applied to evolutionary partial differential equations (PDEs) in bi-Hamiltonian form with nonconstant Poisson structures. Numerical results for the Korteweg de Vries (KdV) equation and for the Ito type coupled KdV equation confirm the long term preservation of the Hamiltonians and Casimir integrals, which is essential in simulating waves and solitons. Dispersive properties of the AVF integrator are investigated for the linearized equations to examine the nonlinear dynamics after discreization.Publisher's Versio
On the multi-symplectic structure of the Serre-Green-Naghdi equations
In this short note, we present a multi-symplectic structure of the
Serre-Green-Naghdi (SGN) equations modelling nonlinear long surface waves in
shallow water. This multi-symplectic structure allow the use of efficient
finite difference or pseudo-spectral numerical schemes preserving exactly the
multi-symplectic form at the discrete level.Comment: 10 pages, 1 figure, 30 references. Other author's papers can be
downloaded at http://www.denys-dutykh.com
Energy preserving integratıon of KDV-KDV systems
Coupled Korteweg de Vries (KdV) equations in Hamiltonian form are integrated by the energy preserving average vector field (AVF) method. Numerical results confirm long term preservation of the energy and the quadratic invariants. Produced generalized solitary waves are similar to those in the literature for larger mesh sizes and time steps. Numerical and continuous dispersion relations of the linearized equations are compared to analyze the behavior of the traveling waves and the interaction of the solitons.Publisher's Versio
Space-time FLAVORS: finite difference, multisymlectic, and pseudospectral integrators for multiscale PDEs
We present a new class of integrators for stiff PDEs. These integrators are
generalizations of FLow AVeraging integratORS (FLAVORS) for stiff ODEs and SDEs
introduced in [Tao, Owhadi and Marsden 2010] with the following properties: (i)
Multiscale: they are based on flow averaging and have a computational cost
determined by mesoscopic steps in space and time instead of microscopic steps
in space and time; (ii) Versatile: the method is based on averaging the flows
of the given PDEs (which may have hidden slow and fast processes). This
bypasses the need for identifying explicitly (or numerically) the slow
variables or reduced effective PDEs; (iii) Nonintrusive: A pre-existing
numerical scheme resolving the microscopic time scale can be used as a black
box and easily turned into one of the integrators in this paper by turning the
large coefficients on over a microscopic timescale and off during a mesoscopic
timescale; (iv) Convergent over two scales: strongly over slow processes and in
the sense of measures over fast ones; (v) Structure-preserving: for stiff
Hamiltonian PDEs (possibly on manifolds), they can be made to be
multi-symplectic, symmetry-preserving (symmetries are group actions that leave
the system invariant) in all variables and variational
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