Lie-point symmetries of the discrete Liouville equation

Levi, Decio; Martina, L.; Winternitz, P.

doi:10.1088/1751-8113/48/2/025204

Cited by 20 publications

(60 citation statements)

References 35 publications

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“…As it is evident from (3.23) there is no way that we can get a Lie point symmetry in agreement with the results presented in [19,20]. The lowest possible symmetry is a generalized symmetry depending on three points (3.25).…”

Section: The Discrete Algebraic Liouville Equationssupporting

confidence: 80%

On Partial Differential and Difference Equations With Symmetries Depending on Arbitrary Functions

Gubbiotti¹,

Levi

Scimiterna³

2016

Acta Polytech

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Abstract. In this note we present some ideas on when Lie symmetries, both point and generalized, can depend on arbitrary functions. We show a few examples, both in partial differential and partial difference equations where this happens. Moreover we show that the infinitesimal generators of generalized symmetries depending on arbitrary functions, both for continuous and discrete equations, effectively play the role of master symmetries.

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Section: The Discrete Algebraic Liouville Equationssupporting

confidence: 80%

On Partial Differential and Difference Equations With Symmetries Depending on Arbitrary Functions

Gubbiotti¹,

Levi

Scimiterna³

2016

Acta Polytech

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“…The above normalization condition indicates that all terms in the base (non-invariant) compact scheme, Eq. (32), that include the spatial first derivative will be removed from the compact scheme in the transformed space leading to the following reduced form…”

Section: Viscous Burgers' Equationmentioning

confidence: 99%

“…= e 2s3 (t + s 4 ) λx = e s3 x + s 5 + s 2 (t + s 4 ) λ u = e −s3 (λu + s 1 (x + s 5 ) + s 2 ) (34) ux = e −2s3 (λ 2 u x + s 1 λ) uxx = e −3s3 λ 3 u xxwhere λ = 1 − s 1 (t + s 4 ). As similar to the inviscid Burgers' equation, the scaling symmetry parameter s 3 does not occur when these transformations are implemented to the compact scheme given in Eq (32)…”

mentioning

confidence: 99%

Construction of invariant compact finite-difference schemes

Ozbenli

Vedula

2020

Phys. Rev. E

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In this paper, we propose a method, that is based on equivariant moving frames, for development of high order accurate invariant compact finite difference schemes that preserve Lie symmetries of underlying partial differential equations. In this method, variable transformations that are obtained from the extended symmetry groups of PDEs are used to transform independent and dependent variables, and derivative terms of compact finite difference schemes (constructed for these PDEs) such that the resulting schemes are invariant under the chosen symmetry groups. The unknown symmetry parameters that arise from the application of these transformations are determined through selection of convenient moving frames. In some cases, owing to selection of convenient moving frames, numerical representation of invariant (or symmetry preserving) compact numerical schemes, that are developed through the proposed method, are found to be notably simpler than that of standard, noninvariant compact numerical schemes. Further, the order of accuracy of these invariant compact schemes can be arbitrarily set to a desired order by considering suitable compact finite difference algorithms. Application of the proposed method is demonstrated through construction of invariant compact finite difference schemes for some common linear and nonlinear PDEs (including the linear advection-diffusion equation in 1D/2D, the inviscid and viscous Burgers' equations in 1D). Results obtained from our numerical simulations indicate that invariant compact finite difference schemes not only inherit selected symmetry properties of underlying PDEs, but are also comparably more accurate than the standard, non-invariant base numerical schemes considered here.

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“…In two recent articles [35,36] we dealt with the real hyperbolic Liouville equation as part of a program of discretizing both ordinary and partial differential equations (ODEs and PDEs), while preserving their Lie point symmetries [21,40]. In particular we have shown that it is not possible to preserve the entire infinite-dimensional symmetry group as point symmetries in the discrete case.…”

Section: Introductionmentioning

confidence: 99%

“…For their discretization see [24]. The discretization of PDEs and DODEs preserving their Lie point symmetry groups started in [3,6,11,23,24,35,36,39] and the present article is an integral part of this research. As stated in previous articles there are several aspects to the program.…”

Section: Introductionmentioning

confidence: 99%

Conformally Invariant Elliptic Liouville Equation and Its Symmetry-Preserving Discretization

2018

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The symmetry algebra of the real elliptic Liouville equation is an infinite-dimensional loop algebra with the simple Lie algebra o(3, 1) as its maximal finite-dimensional subalgebra. The entire algebra generates the conformal group of the Euclidean plane E2. This infinite-dimensional algebra distinguishes the elliptic Liouville equation from the hyperbolic one with its symmetry algebra that is the direct sum of two Virasoro algebras. Following a discretisation procedure developed earlier, we present a difference scheme that is invariant under the group O(3, 1) and has the elliptic Liouville equation in polar coordinates as its continuous limit. The lattice is a solution of an equation invariant under O(3, 1) and is itself invariant under a subgroup of O(3, 1), namely the O(2) rotations of the Euclidean plane. *

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Lie-point symmetries of the discrete Liouville equation

Cited by 20 publications

References 35 publications

On Partial Differential and Difference Equations With Symmetries Depending on Arbitrary Functions

On Partial Differential and Difference Equations With Symmetries Depending on Arbitrary Functions

Construction of invariant compact finite-difference schemes

Conformally Invariant Elliptic Liouville Equation and Its Symmetry-Preserving Discretization

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