A heat transfer with a source: the complete set of invariant difference schemes

In this letter we present the set of invariant difference equations and meshes which preserve the Lie group symmetries of the equation u_{t}=(K(u)u_{x})_{x}+Q(u). All special cases of K(u) and Q(u) that extend the symmetry group admitted by the differential equation are considered. This paper completes the paper [J. Phys. A: Math. Gen. 30, no. 23 (1997) 8139-8155], where a few invariant models for heat transfer equations were presented.Comment: arxiv version is already officia

Symmetry-preserving discrete schemes for some heat transfer equations

Lie group analysis of differential equations is a generally recognized method, which provides invariant solutions, integrability, conservation laws etc. In this paper we present three characteristic examples of the construction of invariant difference equations and meshes, where the original continuous symmetries are preserved in discrete models. Conservation of symmetries in difference modeling helps to retain qualitative properties of the differential equations in their difference counterparts.Comment: 21 pages, 4 ps figure

Lie point symmetries of difference equations and lattices

A method is presented for finding the Lie point symmetry transformations acting simultaneously on difference equations and lattices, while leaving the solution set of the corresponding difference scheme invariant. The method is applied to several examples. The found symmetry groups are used to obtain particular solutions of differential-difference equations

Extensions of Noether's Second Theorem: from continuous to discrete systems

A simple local proof of Noether's Second Theorem is given. This proof immediately leads to a generalization of the theorem, yielding conservation laws and/or explicit relationships between the Euler--Lagrange equations of any variational problem whose symmetries depend upon a set of free or partly-constrained functions. Our approach extends further to deal with finite difference systems. The results are easy to apply; several well-known continuous and discrete systems are used as illustrations

Star-Jet Interactions and Gamma-Ray Outbursts from 3C454.3

We propose a model to explain the ultra-bright GeV gamma-ray flares observed from the blazar 3C454.3. The model is based on the concept of a relativistic jet interacting with compact gas condensations produced when a star (red giant) crosses the jet close to the central black hole. The study includes an analytical treatment of the evolution of the envelop lost by the star within the jet, and calculations of the related high-energy radiation. The model readily explains the day-long, variable on timescales of hours, GeV gamma-ray flare from 3C454.3, observed during November 2010 on top of a weeks-long plateau. In the proposed scenario, the plateau state is caused by a strong wind generated by the heating of the star atmosphere by nonthermal particles accelerated at the jet-star interaction region. The flare itself could be produced by a few clouds of matter lost by the red giant after the initial impact of the jet. In the framework of the proposed scenario, the observations constrain the key model parameters of the source, including the mass of the central black hole: $M_{\rm BH}\simeq 10^9 M_{\odot}$, the total jet power: $L_{\rm j}\simeq 10^{48}\,\rm erg\,s^{-1}$, and the Doppler factor of the gamma-ray emitting clouds, $\delta\simeq 20$. Whereas we do not specify the particle acceleration mechanisms, the potential gamma-ray production processes are discussed and compared in the context of the proposed model. We argue that synchrotron radiation of protons has certain advantages compared to other radiation channels of directly accelerated electrons.Comment: 16 pages, 5 figures, submitted to Ap

Difference schemes with point symmetries and their numerical tests

Symmetry preserving difference schemes approximating second and third order ordinary differential equations are presented. They have the same three or four-dimensional symmetry groups as the original differential equations. The new difference schemes are tested as numerical methods. The obtained numerical solutions are shown to be much more accurate than those obtained by standard methods without an increase in cost. For an example involving a solution with a singularity in the integration region the symmetry preserving scheme, contrary to standard ones, provides solutions valid beyond the singular point.Comment: 26 pages 7 figure