110 research outputs found
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
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
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
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
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
Multisymplectic Geometry and Multisymplectic Preissman Scheme for the KP Equation
The multisymplectic structure of the KP equation is obtained directly from
the variational principal. Using the covariant De Donder-Weyl Hamilton function
theories, we reformulate the KP equation to the multisymplectic form which
proposed by Bridges. From the multisymplectic equation, we can derive a
multisymplectic numerical scheme of the KP equation which can be simplified to
multisymplectic forty-five points scheme.Comment: 17 papges, 8 figure
Variational Structures in Cochain Projection Based Variational Discretizations of Lagrangian PDEs
Compatible discretizations, such as finite element exterior calculus, provide
a discretization framework that respect the cohomological structure of the de
Rham complex, which can be used to systematically construct stable mixed finite
element methods. Multisymplectic variational integrators are a class of
geometric numerical integrators for Lagrangian and Hamiltonian field theories,
and they yield methods that preserve the multisymplectic structure and
momentum-conservation properties of the continuous system. In this paper, we
investigate the synthesis of these two approaches, by constructing
discretization of the variational principle for Lagrangian field theories
utilizing structure-preserving finite element projections. In our
investigation, compatible discretization by cochain projections plays a pivotal
role in the preservation of the variational structure at the discrete level,
allowing the discrete variational structure to essentially be the restriction
of the continuum variational structure to a finite-dimensional subspace. The
preservation of the variational structure at the discrete level will allow us
to construct a discrete Cartan form, which encodes the variational structure of
the discrete theory, and subsequently, we utilize the discrete Cartan form to
naturally state discrete analogues of Noether's theorem and multisymplecticity,
which generalize those introduced in the discrete Lagrangian variational
framework by Marsden et al. [29]. We will study both covariant spacetime
discretization and canonical spatial semi-discretization, and subsequently
relate the two in the case of spacetime tensor product finite element spaces.Comment: 44 pages, 1 figur
Some applications of semi-discrete variational integrators to classical field theories
We develop a semi-discrete version of discrete variational mechanics with
applications to numerical integration of classical field theories. The
geometric preservation properties are studied.Comment: 14 page
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