180 research outputs found
Invariant Finite-Difference Schemes for Cylindrical One-Dimensional MHD Flows with Conservation Laws Preservation
On the basis of the recent group classification of the one-dimensional
magnetohydrodynamics (MHD) equations in cylindrical geometry, the construction
of symmetry-preserving finite-difference schemes with conservation laws is
carried out. New schemes are constructed starting from the classical completely
conservative Samarsky-Popov schemes. In the case of finite conductivity,
schemes are derived that admit all the symmetries and possess all the
conservation laws of the original differential model, including previously
unknown conservation laws. In the case of a frozen-in magnetic field (when the
conductivity is infinite), various schemes are constructed that possess
conservation laws, including those preserving entropy along trajectories of
motion. The peculiarities of constructing schemes with an extended set of
conservation laws for specific forms of entropy and magnetic fluxes are
discussed.Comment: 29 pages; some minor fixes and generalizations + Appendix containing
an additional numerical schem
Group analysis of a class of nonlinear Kolmogorov equations
A class of (1+2)-dimensional diffusion-convection equations (nonlinear
Kolmogorov equations) with time-dependent coefficients is studied with Lie
symmetry point of view. The complete group classification is achieved using a
gauging of arbitrary elements (i.e. via reducing the number of variable
coefficients) with the application of equivalence transformations. Two possible
gaugings are discussed in detail in order to show how equivalence groups serve
in making the optimal choice.Comment: 12 pages, 4 table
Group classification of the two-dimensional magnetogasdynamics equations in Lagrangian coordinates
The present paper is devoted to the group classification of
magnetogasdynamics equations in which dependent variables in Euler coordinates
depend on time and two spatial coordinates. It is assumed that the continuum is
inviscid and nonthermal polytropic gas with infinite electrical conductivity.
The equations are considered in mass Lagrangian coordinates. Use of Lagrangian
coordinates allows reducing number of dependent variables. The analysis
presented in this article gives complete group classification of the studied
equations. This analysis is necessary for constructing invariant solutions and
conservation laws on the base of Noether's theorem
Lie group analysis of a generalized Krichever-Novikov differential-difference equation
The symmetry algebra of the differential--difference equation
where , and are arbitrary analytic functions is shown to have the
dimension 1 \le \mbox{dim}L \le 5. When , and are specific second
order polynomials in (depending on 6 constants) this is the integrable
discretization of the Krichever--Novikov equation. We find 3 cases when the
arbitrary functions are not polynomials and the symmetry algebra satisfies
\mbox{dim}L=2. These cases are shown not to be integrable. The symmetry
algebras are used to reduce the equations to purely difference ones. The
symmetry group is also used to impose periodicity and thus to
reduce the differential--difference equation to a system of coupled
ordinary three points difference equations
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