Asynchronous Variational Integrators

Lew, Adrián J.; Marsden, Jerrold E.; Ortiz, Michael; West, Matthew

doi:10.1007/s00205-002-0212-y

Cited by 209 publications

(127 citation statements)

References 30 publications

(14 reference statements)

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“…One approach, which has yielded fruitful results, is to discretize a variational principle and perform variations on the discrete action to derive an integration scheme. Some examples of field theoretic integrators constructed in this way are those for elastomechanics 29,30 electromagnetism 31 , fluids and magnetohydrodynamics 32,33 and a particle-in-cell (PIC) scheme for the Vlasov-Maxwell system 34 .…”

Section: Introductionmentioning

confidence: 99%

The Hamiltonian Structure and Euler-Poincare Formulation of the Valsov-Maxwell and Gyrokinetic System

Squire¹,

Qin²,

Tang³

2012

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We present a new variational principle for the gyrokinetic system, similar to the MaxwellVlasov action presented in Ref. 1. The variational principle is in the Eulerian frame and based on constrained variations of the phase space fluid velocity and particle distribution function. Using a Legendre transform, we explicitly derive the field theoretic Hamiltonian structure of the system. This is carried out with the Dirac theory of constraints, which is used to construct meaningful brackets from those obtained directly from Euler-Poincaré theory. Possible applications of these formulations include continuum geometric integration techniques, large-eddy simulation models and Casimir type stability methods.

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Section: Introductionmentioning

confidence: 99%

The Hamiltonian Structure and Euler-Poincare Formulation of the Valsov-Maxwell and Gyrokinetic System

Squire¹,

Qin²,

Tang³

2012

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“…In other words, integrators that respect the basic geometry underlying the problem seem to play an key role. It would be interesting to pursue this aspect further and also incorporate discrete exterior calculus and variational multisymplectic integration methods (see Desbrun, Hirani, Leok and Marsden [2003] as well as Marsden, Patrick and Shkoller [1998] and Lew, Marsden, Ortiz and West [2003]). …”

Section: The Geometry Of the Momentum Mapmentioning

confidence: 99%

Momentum maps and measure-valued solutions (peakons, filaments, and sheets) for the EPDiff equation

Holm

Marsden

Progress in Mathematics

115

253

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We study the dynamics of measure-valued solutions of what we call the EPDiff equations, standing for the Euler-Poincaré equations associated with the diffeomorphism group (of R n or an n-dimensional manifold M ). Our main focus will be on the case of quadratic Lagrangians; that is, on geodesic motion on the diffeomorphism group with respect to the right invariant Sobolev H 1 metric. The corresponding Euler-Poincaré (EP) equations are the EPDiff equations, which coincide with the averaged template matching equations (ATME) from computer vision and agree with the Camassa-Holm (CH) equations in one dimension. The corresponding equations for the volume preserving diffeomorphism group are the well-known LAE (Lagrangian averaged Euler) equations for incompressible fluids.We first show that the EPDiff equations are generated by a smooth vector field on the diffeomorphism group for sufficiently smooth solutions. This is analogous to known results for incompressible fluids-both the Euler equations and the LAE equations-and it shows that for sufficiently smooth solutions, the equations are well-posed for short time. In fact, numerical evidence suggests 1 CONTENTS 2 that, as time progresses, these smooth solutions break up into singular solutions which, at least in one dimension, exhibit soliton behavior.With regard to these non-smooth solutions, we study measure-valued solutions that generalize to higher dimensions the peakon solutions of the (CH) equation in one dimension. One of the main purposes of this paper is to show that many of the properties of these measure-valued solutions may be understood through the fact that their solution ansatz is a momentum map. Some additional geometry is also pointed out, for example, that this momentum map is one leg of a natural dual pair.

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“…In other words, integrators that respect the basic geometry underlying the problem obtained accurate singular solutions in numerical simulations. It would be interesting to pursue this aspect further and also incorporate discrete exterior calculus and variational multisymplectic integration methods (see Desbrun, Hirani, Leok and Marsden [2003] as well as Marsden, Patrick and Shkoller [1998] and Lew, Marsden, Ortiz and West [2003]). …”

Section: Challenges Future Directions and Speculationsmentioning

confidence: 99%

The Breadth of Symplectic and Poisson Geometry

Marsden¹,

Raţiu²

2007

Progress in Mathematics

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This paper concerns the dynamics of measure-valued solutions of the EPDiff equations, standing for the Euler-Poincaré equations associated with the diffeomorphism group (of R n or of an n-dimensional manifold M ). The paper focuses on Lagrangians that are quadratic in the velocity fields and their first derivatives; that is, on geodesic motion on the diffeomorphism group with respect to a right invariant Sobolev H 1 metric. The corresponding Euler-Poincaré (EP) equations are the EPDiff equations, which coincide with the averaged template matching equations (ATME) from computer vision and agree with the Camassa-Holm (CH) equations for shallow water waves in one dimension. The corresponding equations for the volume preserving diffeomorphism group are the LAE (Lagrangian averaged Euler) equations for incompressible fluids.The paper shows that the EPDiff equations are generated by a smooth vector field on the diffeomorphism group for sufficiently smooth solutions. This is analogous to known results for incompressible fluids-both the Euler equations and the LAE equations-and it shows that for sufficiently smooth solutions, the equations are well-posed for short time. Numerical evidence suggests that, as time progresses, these smooth solutions break up into singular solutions which, at least in one dimension, exhibit soliton behavior.These non-smooth, or measure-valued, solutions are higher dimensional generalizations of the peakon solutions of the CH equation in one dimension. One of the main purposes of the paper is to show that many of the properties of these measure-valued solutions can be understood through the fact that their solution Ansatz is a momentum map. Some additional geometry is also pointed out, for example, that this momentum map is one part of a dual pair.

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Asynchronous Variational Integrators

Cited by 209 publications

References 30 publications

The Hamiltonian Structure and Euler-Poincare Formulation of the Valsov-Maxwell and Gyrokinetic System

The Hamiltonian Structure and Euler-Poincare Formulation of the Valsov-Maxwell and Gyrokinetic System

Momentum maps and measure-valued solutions (peakons, filaments, and sheets) for the EPDiff equation

The Breadth of Symplectic and Poisson Geometry

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