Multisymplectic Geometry, Variational Integrators, and Nonlinear PDEs

Marsden, Jerrold E.; Patrick, George W.; Shkoller, Steve

doi:10.1007/s002200050505

Cited by 497 publications

(602 citation statements)

References 29 publications

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“…The theory and implementation for procedures of this sort are yet to be worked out. These ideas are also consistent with parallel efforts to develop space-time adaptive numerical codes such as those that are built around multisymplectic integrators [23].…”

Section: Improvementssupporting

confidence: 59%

“…It is very important to understand to what extent one has to model the small scale dynamics to achieve accurate models of the large-scale motions. Recent work on large eddy simulation models and averaged fluid equations [10,23,25,26] suggests that indeed one can do this with considerable savings in computational cost. In general, multiscale phenomena, both temporal and spatial, are of great importance as well as the source of many of the difficulties.…”

Section: Problem Descriptionmentioning

confidence: 99%

See 1 more Smart Citation

Structure-preserving model reduction for mechanical systems

Lall¹,

Krysl²,

Marsden³

2003

Physica D: Nonlinear Phenomena

141

112

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This paper focuses on methods of constructing of reduced-order models of mechanical systems which preserve the Lagrangian structure of the original system. These methods may be used in combination with standard spatial decomposition methods, such as the Karhunen-Loève expansion, balancing, and wavelet decompositions. The model reduction procedure is implemented for three-dimensional finite-element models of elasticity, and we show that using the standard Newmark implicit integrator, significant savings are obtained in the computational costs of simulation. In particular simulation of the reduced model scales linearly in the number of degrees of freedom, and parallelizes well.

show abstract

Section: Improvementssupporting

confidence: 59%

Section: Problem Descriptionmentioning

confidence: 99%

Structure-preserving model reduction for mechanical systems

Lall¹,

Krysl²,

Marsden³

2003

Physica D: Nonlinear Phenomena

141

112

View full text Add to dashboard Cite

show abstract

“…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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“…How to deal with the case with an even number? In reference [7], the numerical experiments presented by Marsden et. al.…”

Section: We Get the Iterative Form Of The Nonlinear Equationsmentioning

confidence: 99%

Numerical implementation of the multisymplectic Preissman scheme and its equivalent schemes

Wang

Qin

2004

Applied Mathematics and Computation

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We analyze the multisymplectic Preissman scheme for the KdV equation with the periodic boundary condition and show that the unconvergence of the widely-used iterative methods to solve the resulting nonlinear algebra system of the Preissman scheme is due to the introduced potential function. A artificial numerical condition is added to the periodic boundary condition. The added boundary condition makes the numerical implementation of the multisymplectic Preissman scheme practical and is proved not to change the numerical solutions of the KdV equation. Based on our analysis, we derive some new schemes which are not restricted by the artificial boundary condition and more efficient than the Preissman scheme because of less computing cost and less computer storages. By eliminating the auxiliary variables, we also derive two schemes for the KdV equation, one is a 12-point scheme and the other is an 8-point scheme. As the byproducts, we present two new explicit schemes which are not multisymplectic but still have remarkable numerical stable property. Numerical experiments on soliton collisions are also provided to confirm our conclusion and to show the benefits of the multisymplectic schemes with comparison of the spectral method and Zabusky-Kruskal scheme.

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Multisymplectic Geometry, Variational Integrators, and Nonlinear PDEs

Cited by 497 publications

References 29 publications

Structure-preserving model reduction for mechanical systems

Structure-preserving model reduction for mechanical systems

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

Numerical implementation of the multisymplectic Preissman scheme and its equivalent schemes

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