Applications of Lie Groups to Difference Equations

Дородницын, В. А.

doi:10.1201/b10363

Cited by 84 publications

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“…Arguably, the most important observation established in the field of invariant finite difference schemes is that it is generally not possible to maintain all symmetries of a system of differential equations if the discretization scheme is constructed on a fixed, orthogonal discretization mesh [11,15,19]. In the case of Burgers equation, it is the presence of the Galilean transformations Γ 3 that prohibits the use of a fixed discretization mesh.…”

Section: Invariant Finite Difference Schemes For Burgers Equationsmentioning

confidence: 99%

“…Invariant numerical schemes for Eq. (2) have already been investigated in the literature [9][10][11]21]. The schemes constructed in these references preserve the entire five-parameter symmetry group G of Burgers equation.…”

Section: Invariant Finite Difference Schemes For Burgers Equationsmentioning

confidence: 99%

“…In other words, introducing a moving mesh into the discretization of Burgers equation restores the Galilean invariance of the finite difference scheme. See [11,14,18,19] for further details on the systematic construction of invariant finite difference discretization schemes.…”

Section: Invariant Finite Difference Schemes For Burgers Equationsmentioning

confidence: 99%

“…[2,3,10,11,14,15,19] and references therein for some of the most recent results. In particular, it was established by Dorodnitsyn and collaborators [6,11,12] that it is not possible to maintain the Galilean invariance of partial differential equations in a finite difference model when the mesh does not move in the course of the numerical integration. This result qualitatively explains why the method proposed in [1] may work from the geometrical point of view.…”

Section: Introductionmentioning

confidence: 99%

“…t n i+1 = t n i = t n . It can be checked that this assumption does not violate the invariance of most of the equations of hydrodynamics, see also [11] for more details.…”

Section: Introductionmentioning

confidence: 99%

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Convecting reference frames and invariant numerical models

Bihlo

Nave

2014

Journal of Computational Physics

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In the recent paper by Bernardini et al. [J. Comput. Phys. 232 (2013), 1-6] the discrepancy in the performance of finite difference and spectral models for simulations of flows with a preferential direction of propagation was studied. In a simplified investigation carried out using the viscous Burgers equation the authors attributed the poorer numerical results of finite difference models to a violation of Galilean invariance in the discretization and propose to carry out the computations in a reference frame moving with the bulk velocity of the flow. Here we further discuss this problem and relate it to known results on invariant discretization schemes. Non-invariant and invariant finite difference discretizations of Burgers equation are proposed and compared with the discretization using the remedy proposed by Bernardini et al..

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Section: Invariant Finite Difference Schemes For Burgers Equationsmentioning

confidence: 99%

Section: Invariant Finite Difference Schemes For Burgers Equationsmentioning

confidence: 99%

Section: Invariant Finite Difference Schemes For Burgers Equationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…t n i+1 = t n i = t n . It can be checked that this assumption does not violate the invariance of most of the equations of hydrodynamics, see also [11] for more details.…”

Section: Introductionmentioning

confidence: 99%

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Convecting reference frames and invariant numerical models

Bihlo

Nave

2014

Journal of Computational Physics

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Structures and Waves in a Nonlinear Heat-Conducting Medium

Dimova

Vasileva

2013

Numerical Solution of Partial Differential Equations: Theory, Algorithms, and Their Applications

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The paper is an overview of the main contributions of a Bulgarian team of researchers to the problem of finding the possible structures and waves in the open nonlinear heat conducting medium, described by a reaction-diffusion equation. Being posed and actively worked out by the Russian school of A. A. Samarskii and S.P. Kurdyumov since the seventies of the last century, this problem still contains open and challenging questions.of A.A. Samarskii and M.I. Sobol in 1963 [51]. The problem of localization for quasilinear equations with a source is posed by S.P. Kurdyumov [42] in 1974. The works of I.M. Gelfand, A.S. Kalashnikov, the scientists of the school of A.A. Samarskii and S.P. Kurdyumov are devoted to the challenging physical and mathematical problems, related with this model and its generalizations. Among them are: localization in space of the process of burning, different types of blow-up, arising of structurestraveling and standing waves, complex structures with varying degrees of symmetry. The combination of the computational experiment with the progress in the qualitative and analytical methods of the theory of ordinary and partial differential equations, the Lie and the Lie-Bäcklund group theory, has been crucial for the success of these investigations. The book [50] contains many of these results, achieved to 1986, in the review [34] there are citations of later works.A special part of these investigations is devoted to finding and studying different kinds of selfsimilar and invariant solutions of equation (1) with power nonlinearities :This choice is suggested by the following reasoning.First, such temperature dependencies are usual for many real processes [5], [54], [57]. For example, when σ i = σ = 2.5, β ≤ 5.2, equation (1) describes thermo-nuclear combustion in plasma in the case of electron heat-conductivity; the parameters σ = 0, 2 ≤ β ≤ 3 correspond to the models of autocatalytic processes with diffusion in the chemical reactors; σ≈6.5 corresponds to the radiation heat-conductivity of the high-temperature plasma in the stars, and so on.Second, it is shown in [25], that in the class of power functions the symmetry of equation (1) is maximal in some sense -the equation admits a rich variety of invariant solutions. In general, almost all of the dissipative structures known so far are invariant or partially invariant solutions of nonlinear equations. The investigations of the dissipative structures provide reasons to believe that the invariant solutions describe the attractors of the dissipative structures' evolution and thus they characterize important internal properties of the nonlinear dissipative medium.Third, this rich set of invariant solutions of equation (1) with power nonlinearities is necessary for the successful application of the methods for investigating the same equation in the case of more general dependencies k i (u), Q(u). By using the methods of operator comparison [29] and stationary states [32] it is possible to analyze the properties of the solutions (such as localization, blo...

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Symmetry-Preserving Numerical Schemes

Bihlo

Valiquette

2017

Symmetries and Integrability of Difference Equations

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In these lectures we review two procedures for constructing finite difference numerical schemes that preserve symmetries of differential equations. The first approach is based on Lie's infinitesimal symmetry generators, while the second method uses the novel theory of equivariant moving frames. The advantages of both techniques are discussed and illustrated with the Schwarzian differential equation, the Korteweg-de Vries equation and Burgers' equation. Numerical simulations are presented and innovative techniques for obtaining better invariant numerical schemes are introduced. New research directions and open problems are indicated at the end of these notes.

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Applications of Lie Groups to Difference Equations

Cited by 84 publications

References 0 publications

Convecting reference frames and invariant numerical models

Convecting reference frames and invariant numerical models

Structures and Waves in a Nonlinear Heat-Conducting Medium

Symmetry-Preserving Numerical Schemes

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