Some robust integrators for large time dynamics

Razafindralandy, Dina; Salnikov, Vladimir; Hamdouni, Aziz; Deeb, Ahmad

doi:10.1186/s40323-019-0130-2

Cited by 20 publications

(15 citation statements)

References 32 publications

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“…However, in his subsequent works, Courant focused only on their geometric properties. Applications to mechanics occurred later in the works on implicit Lagrangian [2][3][4][5] and port-Hamiltonian [6,7] systems, which are discussed in the next two sections.…”

Section: Courant Algebroid and Dirac Structuresmentioning

confidence: 99%

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Differential Geometry and Mechanics: A Source for Computer Algebra Problems

Salnikov

Hamdouni

2020

Program Comput Soft

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In this paper, we discuss the possibility of using computer algebra tools in the process of modeling and qualitative analysis of mechanical systems and problems from theoretical physics. We describe some constructions-Courant algebroids and Dirac structures-from the so-called generalized geometry. They prove to be a convenient language for studying the internal structure of the differential equations of port-Hamiltonian and implicit Lagrangian systems, which describe dissipative or coupled mechanical systems and systems with constraints, respectively. For both classes of systems, we formulate some open problems that can be solved using computer algebra tools and methods. We also recall the definitions of graded manifolds and Q-structures from graded geometry. On particular examples, we explain how classical differential geometry is described in the framework of the graded formalism and what related computational questions can arise. This direction of research is apparently an almost unexplored branch of computer algebra.

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Section: Courant Algebroid and Dirac Structuresmentioning

confidence: 99%

“…In more detail, this construction was described in [4,5]. In this paper, having omitted the technical details, we provide only the final equations that define :…”

Section: Implicit Lagrangian Formalismmentioning

confidence: 99%

Differential Geometry and Mechanics: A Source for Computer Algebra Problems

Salnikov

Hamdouni

2020

Program Comput Soft

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“…A natural question to ask oneself is about optimality of the choice of̃, and various arguments and consistency tests are possible in favour of one or another. For example, if a regression of Dirac-1 is considered without constraints, it is meaningful to ask whether it is symplecticthis question is addressed among others in [19]. We intend to study this choice in a separate (maybe rather short and technical) article, devoted exclusively to benchmarking and analysis of Dirac-based numerical method in different situations.…”

Section: Discretizationmentioning

confidence: 99%

From modelling of systems with constraints to generalized geometry and back to numerics

Salnikov

Hamdouni

2019

Z Angew Math Mech

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In this note we describe how some objects from generalized geometry appear in the qualitative analysis and numerical simulation of mechanical systems. In particular we discuss double vector bundles and Dirac structures. It turns out that those objects can be naturally associated to systems with constraints -we recall the mathematical construction in the context of so called implicit Lagrangian systems. We explain how they can be used to produce new numerical methods, that we call Dirac integrators.On a test example of a simple pendulum in a gravity field we compare the Dirac integrators with classical explicit and implicit methods, we pay special attention to conservation of constrains. Then, on a more advanced example of the Ziegler-type system we show that the choice of numerical methods can indeed affect the conclusions of qualitative analysis of the dynamics of mechanical systems. We also tell why we think that Dirac integrators are appropriate for this kind of systems by explaining the relation with the notions of geometric degree of non-conservativity and kinematic structural stability. K E Y W O R D SDirac structures, geometric integrators, systems with constraints

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“…The precise question we ask ourselves in this paper is mostly motivated by the results of [1], where the appearance of (almost) Dirac structures for mechanical systems with constraints is discussed. It can be vaguely formulated as: "given a Dirac structure, what else do we need to know to define meaningful dynamics on it".…”

Section: Introduction / Motivationmentioning

confidence: 99%