Discretization of second-order ordinary differential equations with symmetries

The one-dimensional modified shallow water equations in Lagrangian coordinates are considered.Relations between symmetries and conservation laws in Lagrangian coordinates, in mass Lagrangian variables and Eulerian coordinates are discussed. For equations in Lagrangian coordinates, invariant finite-difference schemes are constructed. These schemes admit all conservation laws, which are related to Lie symmetries admitted by the differential model via Noether's theorem: they possess the difference analogs of the conservation laws of mass, momentum, energy, the law of center-of-mass motion for horizontal, inclined, and parabolic bottom topographies. The invariant conservative difference schemes are tested numerically in comparison with an ad hoc approximation-invariant scheme.

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“…Nowadays the results were obtained for both ordinary and partial difference equations and systems (e.g., Refs. 38, 39, 49, 53–55).…”

Section: Introductionmentioning

confidence: 99%

Conservative invariant finite‐difference schemes for the modified shallow water equations in Lagrangian coordinates

2022

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“…The applications have certain peculiarities related with nonlocal character of difference operators and geometrical structure of difference mesh [39]. The results were obtained both for the ordinary and partial difference equations and systems (e. g., [28,29,39,[43][44][45]). The finite-difference analogues of Lagrangian [30,31,46] and the Hamiltonian [46,47] formalism were developed.…”

Section: Introductionmentioning

confidence: 99%

Conservative invariant finite-difference schemes for the modified shallow water equations in Lagrangian coordinates

Дородницын¹,

Kaptsov²,

Meleshko³

2021

Preprint

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The one-dimensional modified shallow water equations in Lagrangian coordinates are considered. It is shown the relationship between symmetries and conservation laws in Lagrangian coordinates, in mass Lagrangian variables, and Eulerian coordinates. For equations in Lagrangian coordinates an invariant finite-difference scheme is constructed for all cases for which conservation laws exist in the differential model. Such schemes possess the difference analogues of the conservation laws of mass, momentum, energy, the law of center of mass motion for horizontal, inclined and parabolic bottom topographies. Invariant conservative difference scheme is tested numerically in comparison with naive approximation invariant scheme.

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“…Symmetries, first integrals and conservation laws for differential equations have discrete analogs for difference equations (see, for example, Refs. [15,54,55], and references therein). Other related properties, such as discrete variational and Hamiltonian formulations and Noether's theorem, have also been established [56][57][58][59][60].…”

Section: Introductionmentioning

confidence: 99%

Invariant Conservation Law-Preserving Discretizations of Linear and Nonlinear Wave Equations

Cheviakov

Дородницын

Капцов

2020

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Symmetry-and conservation law-preserving finite difference discretizations are obtained for linear and nonlinear one-dimensional wave equations on five-and nine-point stencils, using the theory of Lie point symmetries of difference equations, and the discrete direct multiplier method of conservation law construction. In particular, for the linear wave equation, an explicit five-point scheme is presented that preserves the discrete analogs of its basic geometric point symmetries, and six of the corresponding conservation laws. For a class of nonlinear wave equations arising in hyperelasticity, a nine-point implicit scheme is constructed, preserving four point symmetries and three local conservation laws. Other discretization of the nonlinear wave equations preserving different subsets of conservation laws are discussed.

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Discretization of second-order ordinary differential equations with symmetries

Cited by 7 publications

References 10 publications

Conservative invariant finite‐difference schemes for the modified shallow water equations in Lagrangian coordinates

Conservative invariant finite‐difference schemes for the modified shallow water equations in Lagrangian coordinates

Conservative invariant finite-difference schemes for the modified shallow water equations in Lagrangian coordinates

Invariant Conservation Law-Preserving Discretizations of Linear and Nonlinear Wave Equations

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