Noether-type theorems for difference equations

Дородницын, В. А.

doi:10.1016/s0168-9274(00)00041-6

Cited by 78 publications

(96 citation statements)

References 14 publications

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“…The infinitesimal invariance condition of the functional (2.15) on the lattice (2.16) is given by two equations [19,25,27]:…”

Section: Lagrangian Formalism For Second Order Difference Equationsmentioning

confidence: 99%

“…It has been shown elsewhere [19,25,27], that if the functional (2.15) is invariant under some group G, then the quasiextremal equations (2.21) are also invariant with respect to G. As in the continuous case, the quasiextremal equations can be invariant with respect to a larger group than the corresponding Lagrangian.…”

Section: Lagrangian Formalism For Second Order Difference Equationsmentioning

confidence: 99%

“…[19,25] On solutions of the quasiextremal equations (2.21) eq. (2.23) reduces to Let us compare the situation for second order ODEs and for three-point difference schemes.…”

Section: Lagrangian Formalism For Second Order Difference Equationsmentioning

confidence: 99%

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Continuous symmetries of Lagrangians and exact solutions of discrete equations

Дородницын

Kozlov

Winternitz

2003

Journal of Mathematical Physics

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One of the difficulties encountered when studying physical theories in discrete space-time is that of describing the underlying continuous symmetries (like Lorentz, or Galilei invariance). One of the ways of addressing this difficulty is to consider point transformations acting simultaneously on difference equations and lattices. In a previous article we have classified ordinary difference schemes invariant under Lie groups of point transformations. The present article is devoted to an invariant Lagrangian formalism for scalar single-variable difference schemes. The formalism is used to obtain first integrals and explicit exact solutions of the schemes. Equations invariant under two-and three-dimensional groups of Lagrangian symmetries are considered.

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“…The infinitesimal invariance condition of the functional (2.15) on the lattice (2.16) is given by two equations [19,25,27]:…”

Section: Lagrangian Formalism For Second Order Difference Equationsmentioning

confidence: 99%

Section: Lagrangian Formalism For Second Order Difference Equationsmentioning

confidence: 99%

See 1 more Smart Citation

Continuous symmetries of Lagrangians and exact solutions of discrete equations

Дородницын

Kozlov

Winternitz

2003

Journal of Mathematical Physics

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“…From the Noether theorem [2] it is known that invariance of a variational functional with respect to an r-parameter group leads to existence of r conservation laws for the corresponding Euler's equation. It was shown in [2] (see also [5,6] for details) that there exists a similar (although more complicated) construction for difference models. This discrete analog of the Noether theorem can be applied to analytic integration of difference equations.…”

Section: Introductionmentioning

confidence: 99%

On the Linearization of Second-Order Differential and Difference Equations

Дородницын

2006

SIGMA

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Abstract. This article complements recent results of the papers [J. Math. Phys. 41 (2000), 480; 45 (2004), 336] on the symmetry classification of second-order ordinary difference equations and meshes, as well as the Lagrangian formalism and Noether-type integration technique. It turned out that there exist nonlinear superposition principles for solutions of special second-order ordinary difference equations which possess Lie group symmetries. This superposition springs from the linearization of second-order ordinary difference equations by means of non-point transformations which act simultaneously on equations and meshes. These transformations become some sort of contact transformations in the continuous limit.

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“…In the discrete setting, Noether's Theorem has been studied in terms of difference equations [9,10], where it was shown that a discrete equivalent of the conservation law holds when a smooth symmetry was built into the discrete Lagrangian. In this work, we turn our attention to the finite element method (FEM).…”

Section: Introduction and Historical Backgroundmentioning

confidence: 99%

Noether-Type Discrete Conserved Quantities Arising from a Finite Element Approximation of a Variational Problem

Mansfield

Pryer

2015

Found Comput Math

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In this work, we prove a weak Noether-type Theorem for a class of variational problems that admit broken extremals. We use this result to prove discrete Noether-type conservation laws for a conforming finite element discretisation of a model elliptic problem. In addition, we study how well the finite element scheme satisfies the continuous conservation laws arising from the application of Noether's first theorem (1918). We summarise extensive numerical tests, illustrating the conservation of the discrete Noether law using the p-Laplacian as an example and derive a geometric-based adaptive algorithm where an appropriate Noether quantity is the goal functional.

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Noether-type theorems for difference equations

Cited by 78 publications

References 14 publications

Continuous symmetries of Lagrangians and exact solutions of discrete equations

Continuous symmetries of Lagrangians and exact solutions of discrete equations

On the Linearization of Second-Order Differential and Difference Equations

Noether-Type Discrete Conserved Quantities Arising from a Finite Element Approximation of a Variational Problem

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