One-dimensional flows of a polytropic gas: Lie group classification, conservation laws, invariant and conservative difference schemes

The paper considers one-dimensional flows of a polytropic gas in the Lagrangian coordinates in three cases: plain one-dimensional flows, radially symmetric flows and spherically symmetric flows. The one-dimensional flow of a polytropic gas is described by one second-order partial differential equation in the Lagrangian variables. Lie group classification of this PDE is performed. Its variational structure allows to construct conservation laws with the help of Noether's theorem. These conservation laws are also recalculated for the gas dynamics variables in the Lagrangian and Eulerian coordinates. Additionally, invariant and conservative difference schemes are provided.

[1]  V. A. Dorodnitsyn,et al.  Shallow water equations in Lagrangian coordinates: Symmetries, conservation laws and its preservation in difference models , 2020, Commun. Nonlinear Sci. Numer. Simul..

[2]  A. A. Samarskii,et al.  Completely conservative difference schemes , 1969 .

[3]  V. Dorodnitsyn,et al.  A Heat Transfer with a Source: the Complete Set of Invariant Difference Schemes , 2003, math/0309139.

[4]  N. N. Yanenko,et al.  Systems of Quasilinear Equations and Their Applications to Gas Dynamics , 1983 .

[5]  N. Ibragimov.  Transformation groups applied to mathematical physics , 1984 .

[6]  E. Toro.  Riemann Solvers and Numerical Methods for Fluid Dynamics , 1997 .

[7]  G. Webb.  Magnetohydrodynamics and Fluid Dynamics: Action Principles and Conservation Laws , 2018 .

[8]  J. Marsden,et al.  Introduction to mechanics and symmetry , 1994 .

[9]  D. Levi,et al.  Continuous symmetries of difference equations , 2005, nlin/0502004.

[10]  I. Akhatov,et al.  Nonlocal symmetries. Heuristic approach , 1991 .

[11]  J. Meinhardt,et al.  Symmetries and differential equations , 1981 .

[12]  J. Marsden,et al.  A mathematical introduction to fluid mechanics , 1979 .

[13]  Continuous symmetries of Lagrangians and exact solutions of discrete equations , 2003, nlin/0307042.

[14]  P. Olver.  Applications of lie groups to differential equations , 1986 .

[15]  G. Zank,et al.  Scaling symmetries, conservation laws and action principles in one-dimensional gas dynamics , 2009 .

[16]  Yvonne Koch,et al.  Variational Principles Of Continuum Mechanics , 2016 .

[17]  G. Bluman,et al.  Applications of Symmetry Methods to Partial Differential Equations , 2009 .

[18]  A. Cheviakov,et al.  Invariant conservation law-preserving discretizations of linear and nonlinear wave equations , 2020, Journal of Mathematical Physics.

[19]  I. G. BONNER CLAPPISON.  Editor , 1960, The Electric Power Engineering Handbook - Five Volume Set.

[20]  FINITE-DIFFERENCE ANALOG OF THE NOETHER THEOREM , 1993 .

[21]  V. Dorodnitsyn,et al.  Symmetries and Integrability of Difference Equations: Lagrangian and Hamiltonian Formalism for Discrete Equations: Symmetries and First Integrals , 2011 .

[22]  A. Samarskii.  The Theory of Difference Schemes , 2001 .

[23]  Sergey V. Meleshko,et al.  One-dimensional gas dynamics equations of a polytropic gas in Lagrangian coordinates: Symmetry classification, conservation laws, difference schemes , 2018, Commun. Nonlinear Sci. Numer. Simul..

[24]  V. Dorodnitsyn.  Applications of Lie Groups to Difference Equations , 2010 .

[25]  W. Miller,et al.  Group analysis of differential equations , 1982 .

[26]  T. N. Stevenson,et al.  Fluid Mechanics , 2021, Nature.

[27]  V. Dorodnitsyn,et al.  Symmetries, Conservation Laws, Invariant Solutions and Difference Schemes of the One-dimensional Green-Naghdi Equations , 2020, Journal of Nonlinear Mathematical Physics.

[28]  S. Meleshko,et al.  Conservation laws of the two-dimensional gas dynamics equations , 2018, International Journal of Non-Linear Mechanics.

[29]  N. Ibragimov,et al.  Elementary Lie Group Analysis and Ordinary Differential Equations , 1999 .

[30]  Roman Kozlov,et al.  Conservative difference schemes for one-dimensional flows of polytropic gas , 2019, Commun. Nonlinear Sci. Numer. Simul..

[31]  N. Ibragimov.  A new conservation theorem , 2007 .

[32]  J. Pain,et al.  Gas Dynamics , 2018, Nature.

[33]  A. Sjöberg,et al.  Non-local symmetries and conservation laws for one-dimensional gas dynamics equations , 2004, Appl. Math. Comput..

[34]  James E. Goehring,et al.  THE ENGLISH TRANSLATION , 1986 .

[35]  Roman Kozlov,et al.  Lie group classification of second-order ordinary difference equations , 2000 .