Multisymplecticity of Hybridizable Discontinuous Galerkin Methods

McLachlan, Robert I.; Stern, Ari

doi:10.1007/s10208-019-09415-1

Cited by 12 publications

(16 citation statements)

References 40 publications

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“…We show that multisymplectic semidiscretization in space, followed by a symplectic integrator in time, yields a multisymplectic method in spacetime. We also show that hybrid finite elements may be used for multisymplectic semidiscretization, generalizing the results of McLachlan and Stern [22] to time-evolution problems. • Finally, Section 5 extends these results from B-series methods to additive and partitioned methods, including additive/partitioned Runge-Kutta methods.…”

Section: Introductionsupporting

confidence: 62%

“…Canonical Hamiltonian PDEs. Before discussing the canonical Hamiltonian formalism for time-evolution PDEs, we first quickly recall the stationary (time-independent) case, following the treatment in McLachlan and Stern [22].…”

Section: Multisymplectic Integratorsmentioning

confidence: 99%

“…Multisymplectic semidiscretization with hybrid finite element methods. In McLachlan and Stern [22], we developed a framework for multisymplectic discretization of time-independent Hamiltonian PDEs by hybrid finite element methods, including hybridizable discontinuous Galerkin methods (cf. Cockburn, Gopalakrishnan, and Lazarov [10]).…”

Section: Multisymplectic Semidiscretization On Rectangular Gridsmentioning

confidence: 99%

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Functional equivariance and conservation laws in numerical integration

McLachlan¹,

Stern²

2021

Preprint

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Preservation of linear and quadratic invariants by numerical integrators has been well studied. However, many systems have linear or quadratic observables that are not invariant, but which satisfy evolution equations expressing important properties of the system. For example, a time-evolution PDE may have an observable that satisfies a local conservation law, such as the multisymplectic conservation law for Hamiltonian PDEs.We introduce the concept of functional equivariance, a natural sense in which a numerical integrator may preserve the dynamics satisfied by certain classes of observables, whether or not they are invariant. After developing the general framework, we use it to obtain results on methods preserving local conservation laws in PDEs. In particular, integrators preserving quadratic invariants also preserve local conservation laws for quadratic observables, and symplectic integrators are multisymplectic.

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Section: Introductionsupporting

confidence: 62%

Section: Multisymplectic Integratorsmentioning

confidence: 99%

Section: Multisymplectic Semidiscretization On Rectangular Gridsmentioning

confidence: 99%

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Functional equivariance and conservation laws in numerical integration

McLachlan¹,

Stern²

2021

Preprint

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“…Cockburn developed a unifying framework for Galerkin methods [12] that multiple authors have extended. McLachlan extends Cockburn's discontinuous Galerkin (DG) methods into a formulation that allows proof of multi-symplecticity for elliptic equations [29]. Sanchez uses a hybridizable discontinuous Galerkin approach to obtain a wave equation discretisation similar to our method, that is proven to be Hamiltonian-conserving and symplectic in time [36].…”

Section: Introductionmentioning

confidence: 99%

Theory and Implementation of Coupled Port-Hamiltonian Continuum and Lumped Parameter Models

Argus

Bradley

Hunter

2021

J Elast

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A continuous Galerkin finite element method that allows mixed boundary conditions without the need for Lagrange multipliers or user-defined parameters is developed. A mixed coupling of Lagrange and Raviart-Thomas basis functions are used. The method is proven to have a Hamiltonian-conserving spatial discretisation and a symplectic time discretisation. The energy residual is therefore guaranteed to be bounded for general problems and exactly conserved for linear problems. The linear 2D wave equation is discretised and modelled by making use of a port-Hamiltonian framework. This model is verified against an analytic solution and shown to have standard order of convergence for the temporal and spatial discretisation. The error growth over time is shown to grow linearly for this symplectic method, which agrees with theoretical results. A modal analysis is performed which verifies that the eigenvalues of the model accurately converge to the exact eigenvalues, as the mesh is refined. The port-Hamiltonian framework allows boundary coupling with bondgraph or, more generally, lumped parameter models, therefore unifying the two fields of lumped parameter modelling and continuum modelling of Hamiltonian systems. The wave domain discretisation is shown to be equivalent to a coupling of canonical port-Hamiltonian forms. This feature allows the model to have mixed boundary conditions as well as to have mixed causality interconnections with other port-Hamiltonian models. A model of the 2D wave equation is coupled, in a monolithic manner, with a lumped parameter model of an electromechanical linear actuator. The combined model is also verified to conserve energy exactly.

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“…To correct the dissipative nature of the method analysed in [93], an energy-conservative HDG formulation with a two-step Stormer-Numerov time-marching is proposed in [86]. Moreover, symplectic [232] and multisymplectic [190] HDG schemes preserving the Hamiltonian structure of the PDEs under analysis were developed to achieve energy conservation.…”

Section: Wave Propagation Phenomenamentioning

confidence: 99%

HDGlab: An Open-Source Implementation of the Hybridisable Discontinuous Galerkin Method in MATLAB

Giacomini

Sevilla

Huerta

2020

Arch Computat Methods Eng

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This paper presents , an open source MATLAB implementation of the hybridisable discontinuous Galerkin (HDG) method. The main goal is to provide a detailed description of both the HDG method for elliptic problems and its implementation available in . Ultimately, this is expected to make this relatively new advanced discretisation method more accessible to the computational engineering community. presents some features not available in other implementations of the HDG method that can be found in the free domain. First, it implements high-order polynomial shape functions up to degree nine, with both equally-spaced and Fekete nodal distributions. Second, it supports curved isoparametric simplicial elements in two and three dimensions. Third, it supports non-uniform degree polynomial approximations and it provides a flexible structure to devise degree adaptivity strategies. Finally, an interface with the open-source high-order mesh generator is provided to facilitate its application to practical engineering problems.

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Multisymplecticity of Hybridizable Discontinuous Galerkin Methods

Cited by 12 publications

References 40 publications

Functional equivariance and conservation laws in numerical integration

Functional equivariance and conservation laws in numerical integration

Theory and Implementation of Coupled Port-Hamiltonian Continuum and Lumped Parameter Models

HDGlab: An Open-Source Implementation of the Hybridisable Discontinuous Galerkin Method in MATLAB

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