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

Lie group analysis of a generalized Krichever-Novikov differential-difference equation

The symmetry algebra of the differential--difference equation $$\dot u_n = [P(u_n)u_{n+1}u_{n-1} + Q(u_n)(u_{n+1}+u_{n-1})+ R(u_n)]/(u_{n+1}-u_{n-1}),$$ where $P$, $Q$ and $R$ are arbitrary analytic functions is shown to have the dimension 1 \le \mbox{dim}L \le 5. When $P$, $Q$ and $R$ are specific second order polynomials in $u_n$ (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 $u_{n+N}=u_n$ and thus to reduce the differential--difference equation to a system of $N$ coupled ordinary three points difference equations

One-dimensional MHD flows with cylindrical symmetry: Lie symmetries and conservation laws

A recent paper considered symmetries and conservation laws of the plane one-dimensional flows for magnetohydrodynamics in the mass Lagrangian coordinates. This paper analyses the one-dimensional magnetohydrodynamics flows with cylindrical symmetry in the mass Lagrangian coordinates. The medium is assumed inviscid and thermally non-conducting. It is modeled by a polytropic gas. Symmetries and conservation laws are found. The cases of finite and infinite electric conductivity need to be analyzed separately. For finite electric conductivity $\sigma (\rho,p)$ we perform Lie group classification, which identifies $\sigma (\rho,p)$ cases with additional symmetries. The conservation laws are found by direct computation. For cases with infinite electric conductivity variational formulations of the equations are considered. Lie group classifications are obtained with the entropy treated as an arbitrary element. A variational formulation allows to use the Noether theorem for computation of conservation laws. The conservation laws obtained for the variational equations are also presented in the original (physical) variables

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

Grape selection for resistance to biotic and abiotic environmental factors

Most of the viticultural regions of the USSR are located under conditions of limiting biotic and abiotic factors, with frosts, drought, fungal diseases, phylloxera, mites, grape berry moths and some others being of primary importance. The main breeding organizations have been creating for more than 40 years new table and wine cultivars with complex resistance according to long-term programs. These cultivars are own-rooted and capable of wintering in outdoor culture with a limited amount of spray treatments, if any. In crossing, Amur grape and its hybrids, cultivars Seibel and Seyve Villard and some others are used as donors of resistance. Using biophysical and cytoembryological methods, gametes are treated with physical and chemical mutagenic factors in order to increase the variability range of F(1) seedlings, aiming at higher efficiency of selection. The process of selection is accelerated if seedlings are grown hydroponically. Analysis of the F(1) hybrid population determines the nature of the inheritance of valuable agricultural characters and the selection of pairs. The in vitro method is used when seedlings are grown from non-vital seeds, callus embryoids and in accelerated propagation of valuable genotypes providing virus and bacteria elimination. More than 50 cultivars with complex resistance have been bred during 35 years. More than 10 of them have been recommended for culture (Moldova, Lyana, Vostorg, Sukholimanski biely, Pervenets Magaracha, and others), while the remainder are being tested in different viticultural regions of the Soviet Union