arXiv subjects

V. A. Dorodnitsyn

Publications and source records attributed to V. A. Dorodnitsyn.

12 recordsLinked to original sources

arXiv2025

Symmetry-Preserving Finite-Difference Schemes and Auto-Bäcklund Transformations for the Schwarz Equation

It is demonstrated that one of the equations from the Lie classification list of second-order ODEs is a first integral of the Schwarz equation. As symmetry-preserving finite-difference schemes have been previously constructed for both equations, the preservation of a similar connection between these schemes is studied. It is shown that the schemes for the Schwarz equation and the second-order ODE (with an arbitrary constant $C$) can be related through a Bäcklund-type difference transformation. In addition, previously unexamined aspects of the difference scheme for the second-order ODE are discussed, including its singular solution and the complete set of difference first integrals for the case $C^2=4$.

math.NA↗

arXiv2024

Invariant conservative finite-difference schemes for the one-dimensional shallow water magnetohydrodynamics equations in Lagrangian coordinates

Invariant finite-difference schemes for the one-dimensional shallow water equations in the presence of a magnetic field for various bottom topographies are constructed. Based on the results of the group classification recently carried out by the authors, finite-difference analogues of the conservation laws of the original differential model are obtained. Some typical problems are considered numerically, for which a comparison is made between the cases of a magnetic field presence and when it is absent (the standard shallow water model). The invariance of difference schemes in Lagrangian coordinates and the energy preservation on the obtained numerical solutions are also discussed.

math.NA↗

arXiv2023

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.

physics.flu-dyn↗

arXiv2023

Symmetries and first integrals for variational ODEs with delay

A Lagrangian formalism for variational second-order delay ordinary differential equations (DODEs) is developed. The Noether operator identity for a DODE is established, which relates the invariance of a Lagrangian function with the appropriate variational equations and the conserved quantities. The identity is used to formulate Noether-type theorems that give the first integrals for DODE with symmetries. Relations between the invariance of the variational second-order DODEs and the invariance of the Lagrangian functions are also analyzed. Several examples illustrate the theoretical results.

math-ph↗

arXiv2022

Symmetries and conservation laws of the one-dimensional shallow water magnetohydrodynamics equations in Lagrangian coordinates

Symmetries of the one-dimensional shallow water magnetohydrodynamics equations (SMHD) in Gilman's approximation are studied. The SMHD equations are considered in case of a plane and uneven bottom topography in Lagrangian and Eulerian coordinates. Symmetry classification separates out all bottom topographies which yields substantially different admitted symmetries. The SMHD equations in Lagrangian coordinates were reduced to a single second order PDE. The Lagrangian formalism and Noether's theorem are used to construct conservation laws of the SMHD equations. Some new conservation laws for various bottom topographies are obtained. The results are also represented in Eulerian coordinates. Invariant and partially invariant solutions are constructed.

physics.flu-dyn↗

arXiv2022

Invariant finite-difference schemes with conservation laws preservation for one-dimensional MHD equations

Invariant finite-difference schemes are considered for one-dimensional magnetohydrodynamics (MHD) equations in mass Lagrangian coordinates for the cases of finite and infinite conductivity. For construction these schemes previously obtained results of the group classification of MHD equations are used. On the basis of the classical Samarskiy-Popov scheme new schemes are constructed for the case of finite conductivity. These schemes admit all symmetries of the original differential model and have difference analogues of all of its local differential conservation laws. Among the conservation laws there are previously unknown ones. In the case of infinite conductivity, conservative invariant schemes constructed as well. For isentropic flows of a polytropic gas proposed schemes possess the conservation law of energy and preserve entropy on two time layers. This is achieved by means of specially selected approximations for the equation of state of a polytropic gas. Also, invariant difference schemes with additional conservation laws are proposed. A new scheme for the case of finite conductivity is tested numerically for various boundary conditions which shows accurate preservation of difference conservation laws.

math.NA↗

arXiv2021

Conservative invariant finite-difference schemes for the modified shallow water equations in Lagrangian coordinates

The one-dimensional modified shallow water equations in Lagrangian coordinates are considered. It is shown the relationship between symmetries and conservation laws in Lagrangian coordinates, in mass Lagrangian variables, and Eulerian coordinates. For equations in Lagrangian coordinates an invariant finite-difference scheme is constructed for all cases for which conservation laws exist in the differential model. Such schemes possess the difference analogues of the conservation laws of mass, momentum, energy, the law of center of mass motion for horizontal, inclined and parabolic bottom topographies. Invariant conservative difference scheme is tested numerically in comparison with naive approximation invariant scheme.

math.NA↗

arXiv2020

Symmetries, conservation laws and difference schemes of the (1+2)-dimensional shallow water equations in Lagrangian coordinates

The two-dimensional shallow water equations in Eulerian and Lagrangain coordinates are considered. Lagrangian and Hamiltonian formalism of the equations is given. The transformations mapping the two-dimensional shallow water equations with a circular or plane bottom into the gas dynamics equations of a polytropic gas with polytropic exponent $γ=2$ is represented. Group properties of the equations are considered, and the group classification for the case of the elliptic paraboloid bottom topography is performed. The properties of the two-dimensional shallow water equations in Lagrangian coordinates are discussed from the discretization point of view. New invariant conservative finite-difference schemes for the equations and their one-dimensional reductions are constructed. The schemes are derived either by extending the known one-dimensional schemes or by direct algebraic construction based on some assumptions on the form of the energy conservation law. Among the proposed schemes there are schemes possessing conservation laws of mass and energy.

physics.flu-dyn↗

arXiv2020

Invariant conservative difference schemes for shallow water equations in Eulerian and Lagrangian coordinates

The one-dimensional shallow water equations in Eulerian coordinates are considered. Relations between symmetries and conservation laws for the potential form of the equations, and symmetries and conservation laws in Eulerian coordinates are shown. An invariant difference scheme for equations in Eulerian coordinates with arbitrary bottom topography is constructed. It possesses all the finite-difference analogues of the conservation laws. Some bottom topographies require moving meshes in Eulerian coordinates, which are stationary meshes in mass Lagrangian coordinates. The developed invariant conservative difference schemes are verified numerically using examples of flow with various bottom topographies.

math.NA↗

arXiv2020

Symmetries, conservation laws, invariant solutions and difference schemes of the one-dimensional Green-Naghdi equations

The paper is devoted to the Lie group properties of the one-dimensional Green-Naghdi equations describing the behavior of fluid flow over uneven bottom topography. The bottom topography is incorporated into the Green-Naghdi equations in two ways: in the classical Green-Naghdi form and in the approximated form of the same order. The study is performed in Lagrangian coordinates which allows one to find Lagrangians for the analyzed equations. Complete group classification of both cases of the Green-Naghdi equations with respect to the bottom topography is presented. Applying Noether's theorem, the obtained Lagrangians and the group classification, conservation laws of the one-dimensional Green-Naghdi equations with uneven bottom topography are obtained. Difference schemes which preserve the symmetries of the original equations and the conservation laws are constructed. Analysis of the developed schemes is given. The schemes are tested numerically on the example of an exact traveling-wave solution.

physics.flu-dyn↗

arXiv2020

Invariant Conservation Law-Preserving Discretizations of Linear and Nonlinear Wave Equations

Symmetry- and conservation law-preserving finite difference discretizations are obtained for linear and nonlinear one-dimensional wave equations on five- and nine-point stencils, using the theory of Lie point symmetries of difference equations, and the discrete direct multiplier method of conservation law construction. In particular, for the linear wave equation, an explicit five-point scheme is presented that preserves the discrete analogs of its basic geometric point symmetries, and six of the corresponding conservation laws. For a class of nonlinear wave equations arising in hyperelasticity, a nine-point implicit scheme is constructed, preserving four point symmetries and three local conservation laws. Other discretization of the nonlinear wave equations preserving different subsets of conservation laws are discussed.

math-ph↗

arXiv2019

Shallow water equations in Lagrangian coordinates: symmetries, conservation laws and its preservation in difference models

The one-dimensional shallow water equations in Eulerian and Lagrangian coordinates are considered. It is shown the relationship between symmetries and conservation laws in Lagrangian (potential) coordinates and symmetries and conservation laws in mass Lagrangian variables. For equations in Lagrangian coordinates with a flat bottom an invariant difference scheme is constructed which possesses all the difference analogues of the conservation laws: mass, momentum, energy, the law of center of mass motion. Some exact invariant solutions are constructed for the invariant scheme, while the scheme admits reduction on subgroups as well as the original system of equations. For an arbitrary shape of bottom it is possible to construct an invariant scheme with conservation of mass and momentum or energy. Invariant conservative difference scheme for the case of a flat bottom tested numerically in comparison with other known schemes.

math-ph↗