Preservation of Bifurcations of Hamiltonian Boundary Value Problems Under Discretisation

McLachlan, Robert I.; Offen, Christian

doi:10.1007/s10208-020-09454-z

Cited by 10 publications

(14 citation statements)

References 23 publications

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“…However, multisymplecticity is known to be necessary to preserve traveling waves of hyperbolic equations [30], and compact multisymplectic methods can preserve dispersion relations much better than non-multisymplectic methods or noncompact finite difference methods [5,34,33]. For boundary value problems, multisymplecticity restricts the types of bifurcations that can occur [31,32]. Because it is a local property, multisymplecticity is a strictly stronger property than the symplecticity obtained by integrating over space.…”

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confidence: 99%

Multisymplecticity of Hybridizable Discontinuous Galerkin Methods

McLachlan

Stern

2019

Found Comput Math

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In this paper, we prove necessary and sufficient conditions for a hybridizable discontinuous Galerkin (HDG) method to satisfy a multisymplectic conservation law, when applied to a canonical Hamiltonian system of partial differential equations. We show that these conditions are satisfied by the "hybridized" versions of several of the most commonly-used finite element methods, including mixed, nonconforming, and discontinuous Galerkin methods. (Interestingly, for the continuous Galerkin method in dimension greater than one, we show that multisymplecticity only holds in a weaker sense.) Consequently, these general-purpose finite element methods may be used for structurepreserving discretization (or semidiscretization) of canonical Hamiltonian systems of ODEs or PDEs. This establishes multisymplecticity for a large class of arbitrarily-high-order methods on unstructured meshes.

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confidence: 99%

Multisymplecticity of Hybridizable Discontinuous Galerkin Methods

McLachlan

Stern

2019

Found Comput Math

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“…Indeed, the authors prove in [22] that the behaviour shown in figure 2 and 3 is universal. This means in any Hamiltonian boundary value problem with a generic hyperbolic or elliptic bifurcation any symplectic integrator will show a bifurcation diagram as on the left of figures 2 and 3 while any integrator which breaks the symplectic structure of the problem will show incorrect bifurcation diagrams which qualitatively look like those on the right of figures 2 and 3.…”

Section: Breaking Of Hyperbolic and Elliptic Umbilic Bifurcationsmentioning

confidence: 95%

“…If, additionally, the other integrals of motions are also captured, e.g. because they are of a simple form or arise from a simple symmetry, then a symplectic method captures the periodic pitchfork bifurcations exponentially well [22,section 4.5,Prop.7]. For a non-symplectic scheme for this to happen either all integrals must be of a special form or be coming from simple symmetries for the method to capture these automatically or we must enforce their preservation (e.g.…”

Section: Numerical Examplesmentioning

confidence: 99%

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Symplectic integration of boundary value problems

McLachlan

Offen

2018

Numer Algor

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Symplectic integrators can be excellent for Hamiltonian initial value problems. Reasons for this include their preservation of invariant sets like tori, good energy behaviour, nonexistence of attractors, and good behaviour of statistical properties. These all refer to long-time behaviour. They are directly connected to the dynamical behaviour of symplectic maps ϕ : M → M on the phase space under iteration. Boundary value problems, in contrast, are posed for fixed (and often quite short) times. Symplecticity manifests as a symplectic map ϕ : M → M which is not iterated. Is there any point, therefore, for a symplectic integrator to be used on a Hamiltonian boundary value problem? In this paper we announce results that symplectic integrators preserve bifurcations of Hamiltonian boundary value problems and that nonsymplectic integrators do not.

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“…As shown in (McLachlan and Offen, 2020), D-series bifurcations, such as hyperbolic umbilic bifurcations, are stable bifurcations in families of boundary value problems for symplectic maps but unstable in more general classes of boundary value problems. Other bifurcations, such as fold, cusp, which belong to the A-series, are also stable in wider classes of boundary value problems.…”

Section: Theoretical Considerationsmentioning

confidence: 99%

Bifurcation preserving discretisations of optimal control problems

Offen,

Ober-Blöbaum

2021

Preprint

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The first order optimality conditions of optimal control problems (OCPs) can be regarded as boundary value problems for Hamiltonian systems. Variational or symplectic discretisation methods are classically known for their excellent long term behaviour. As boundary value problems are posed on intervals of fixed, moderate length, it is not immediately clear whether methods can profit from structure preservation in this context. When parameters are present, solutions can undergo bifurcations, for instance, two solutions can merge and annihilate one another as parameters are varied. We will show that generic bifurcations of an OCP are preserved under discretisation when the OCP is either directly discretised to a discrete OCP (direct method) or translated into a Hamiltonian boundary value problem using first order necessary conditions of optimality which is then solved using a symplectic integrator (indirect method). Moreover, certain bifurcations break when a non-symplectic scheme is used. The general phenomenon is illustrated on the example of a cut locus of an ellipsoid.

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Preservation of Bifurcations of Hamiltonian Boundary Value Problems Under Discretisation

Cited by 10 publications

References 23 publications

Multisymplecticity of Hybridizable Discontinuous Galerkin Methods

Multisymplecticity of Hybridizable Discontinuous Galerkin Methods

Symplectic integration of boundary value problems

Bifurcation preserving discretisations of optimal control problems

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