30 research outputs found
Generating Functionals and Lagrangian PDEs
We introduce the concept of Type-I/II generating functionals defined on the
space of boundary data of a Lagrangian field theory. On the Lagrangian side, we
define an analogue of Jacobi's solution to the Hamilton-Jacobi equation for
field theories, and we show that by taking variational derivatives of this
functional, we obtain an isotropic submanifold of the space of Cauchy data,
described by the so-called multisymplectic form formula. We also define a
Hamiltonian analogue of Jacobi's solution, and we show that this functional is
a Type-II generating functional. We finish the paper by defining a similar
framework of generating functions for discrete field theories, and we show that
for the linear wave equation, we recover the multisymplectic conservation law
of Bridges.Comment: 31 pages; 1 figure -- v2: minor change
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
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
R-adaptive multisymplectic and variational integrators
Moving mesh methods (also called r-adaptive methods) are space-adaptive
strategies used for the numerical simulation of time-dependent partial
differential equations. These methods keep the total number of mesh points
fixed during the simulation, but redistribute them over time to follow the
areas where a higher mesh point density is required. There are a very limited
number of moving mesh methods designed for solving field-theoretic partial
differential equations, and the numerical analysis of the resulting schemes is
challenging. In this paper we present two ways to construct r-adaptive
variational and multisymplectic integrators for (1+1)-dimensional Lagrangian
field theories. The first method uses a variational discretization of the
physical equations and the mesh equations are then coupled in a way typical of
the existing r-adaptive schemes. The second method treats the mesh points as
pseudo-particles and incorporates their dynamics directly into the variational
principle. A user-specified adaptation strategy is then enforced through
Lagrange multipliers as a constraint on the dynamics of both the physical field
and the mesh points. We discuss the advantages and limitations of our methods.
Numerical results for the Sine-Gordon equation are also presented.Comment: 65 pages, 13 figure
Discrete Lagrangian field theories on Lie groupoids
We present a geometric framework for discrete classical field theories, where
fields are modeled as "morphisms" defined on a discrete grid in the base space,
and take values in a Lie groupoid. We describe the basic geometric setup and
derive the field equations from a variational principle. We also show that the
solutions of these equations are multisymplectic in the sense of Bridges and
Marsden. The groupoid framework employed here allows us to recover not only
some previously known results on discrete multisymplectic field theories, but
also to derive a number of new results, most notably a notion of discrete
Lie-Poisson equations and discrete reduction. In a final section, we establish
the connection with discrete differential geometry and gauge theories on a
lattice.Comment: 37 pages, 6 figures, uses xy-pic (v3: minor amendment to def. 3.5;
remark 3.7 added
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
Conservation of phase space properties using exponential integrators on the cubic Schrödinger equation
The cubic nonlinear Schrödinger (NLS) equation with periodic boundary conditions is solvable using Inverse Spectral Theory. The nonlinear spectrum of the associated Lax pair reveals topological properties of the NLS phase space that are difficult to assess by other means. In this paper we use the invariance of the nonlinear spectrum to examine the long time behavior of exponential and multisymplectic integrators as compared with the most commonly used split step approach. The initial condition used is a perturbation of the unstable plane wave solution, which is difficult to numerically resolve. Our findings indicate that the exponential integrators from the viewpoint of efficiency and speed have an edge over split step, while a lower order multisymplectic is not as accurate and too slow to compete. © 2006 Elsevier Inc. All rights reserved