231 research outputs found
Asynchronous Variational Integrators
We describe a new class of asynchronous variational integrators (AVI) for nonlinear
elastodynamics. The AVIs are distinguished by the following attributes: (i)
The algorithms permit the selection of independent time steps in each element, and
the local time steps need not bear an integral relation to each other; (ii) the algorithms
derive from a spacetime form of a discrete version of Hamilton’s variational
principle. As a consequence of this variational structure, the algorithms conserve
local momenta and a local discrete multisymplectic structure exactly.
To guide the development of the discretizations, a spacetime multisymplectic
formulation of elastodynamics is presented. The variational principle used incorporates
both configuration and spacetime reference variations. This allows a unified
treatment of all the conservation properties of the system.A discrete version of reference
configuration is also considered, providing a natural definition of a discrete
energy. The possibilities for discrete energy conservation are evaluated.
Numerical tests reveal that, even when local energy balance is not enforced
exactly, the global and local energy behavior of the AVIs is quite remarkable, a
property which can probably be traced to the symplectic nature of the algorith
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
Backward error analysis for multisymplectic discretizations of Hamiltonian PDEs
Several recently developed multisymplectic schemes for Hamiltonian PDEs have
been shown to preserve associated local conservation laws and constraints very
well in long time numerical simulations. Backward error analysis for PDEs, or
the method of modified equations, is a useful technique for studying the
qualitative behavior of a discretization and provides insight into the
preservation properties of the scheme. In this paper we initiate a backward
error analysis for PDE discretizations, in particular of multisymplectic box
schemes for the nonlinear Schrodinger equation. We show that the associated
modified differential equations are also multisymplectic and derive the
modified conservation laws which are satisfied to higher order by the numerical
solution. Higher order preservation of the modified local conservation laws is
verified numerically.Comment: 12 pages, 6 figures, accepted Math. and Comp. Simul., May 200
Multisymplectic geometry, variational integrators, and nonlinear PDEs
This paper presents a geometric-variational approach to continuous and
discrete mechanics and field theories. Using multisymplectic geometry, we show
that the existence of the fundamental geometric structures as well as their
preservation along solutions can be obtained directly from the variational
principle. In particular, we prove that a unique multisymplectic structure is
obtained by taking the derivative of an action function, and use this structure
to prove covariant generalizations of conservation of symplecticity and
Noether's theorem. Natural discretization schemes for PDEs, which have these
important preservation properties, then follow by choosing a discrete action
functional. In the case of mechanics, we recover the variational symplectic
integrators of Veselov type, while for PDEs we obtain covariant spacetime
integrators which conserve the corresponding discrete multisymplectic form as
well as the discrete momentum mappings corresponding to symmetries. We show
that the usual notion of symplecticity along an infinite-dimensional space of
fields can be naturally obtained by making a spacetime split. All of the
aspects of our method are demonstrated with a nonlinear sine-Gordon equation,
including computational results and a comparison with other discretization
schemes.Comment: LaTeX2E, 52 pages, 11 figures, to appear in Comm. Math. Phy
Multisymplectic formulation of fluid dynamics using the inverse map
We construct multisymplectic formulations of fluid dynamics using the inverse of the Lagrangian path map. This inverse map, the ‘back-to-labels’ map, gives the initial Lagrangian label of the fluid particle that currently occupies each Eulerian position. Explicitly enforcing the condition that the fluid particles carry their labels with the flow in Hamilton's principle leads to our multisymplectic formulation. We use the multisymplectic one-form to obtain conservation laws for energy, momentum and an infinite set of conservation laws arising from the particle relabelling symmetry and leading to Kelvin's circulation theorem. We discuss how multisymplectic numerical integrators naturally arise in this approach.</p
Multisymplectic Lie group variational integrator for a geometrically exact beam in R3
In this paper we develop, study, and test a Lie group multisymplectic
integra- tor for geometrically exact beams based on the covariant Lagrangian
formulation. We exploit the multisymplectic character of the integrator to
analyze the energy and momentum map conservations associated to the temporal
and spatial discrete evolutions.Comment: Article in press. 22 pages, 18 figures. Received 20 November 2013,
Received in revised form 26 February 2014, Accepted 27 February 2014.
Communications in Nonlinear Science and Numerical Simulation. 201
An Overview of Variational Integrators
The purpose of this paper is to survey some recent advances in variational
integrators for both finite dimensional mechanical systems as well as continuum
mechanics. These advances include the general development of discrete
mechanics, applications to dissipative systems, collisions, spacetime integration algorithms,
AVI’s (Asynchronous Variational Integrators), as well as reduction for
discrete mechanical systems. To keep the article within the set limits, we will only
treat each topic briefly and will not attempt to develop any particular topic in
any depth. We hope, nonetheless, that this paper serves as a useful guide to the
literature as well as to future directions and open problems in the subject
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