One-dimensional gas dynamics equations of a polytropic gas in Lagrangian coordinates: Symmetry classification, conservation laws, difference schemes

Abstract A comprehensive analysis of the one-dimensional gas dynamics equations of a polytropic gas in Lagrangian coordinates is performed. One of the representations of these equations in Lagrangian coordinates is given by a single second-order partial differential equation. Symmetries of this equation are analyzed using the entropy for the group classification. Noether’s theorem is applied for constructing conservation laws. The obtained conservation laws are represented in the gas dynamics variables in Lagrangian coordinates and in Eulerian coordinates as well. Invariant and conservative difference schemes are discussed for the basic adiabatic case.

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

[2]  A. Aksenov,et al.  CRC Handbook of Lie Group Analysis of Differential Equations. Vol. 2. Applications in Engineering and Physical Sciences , 1995 .

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

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

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

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

[7]  A. Jamiołkowski.  Book reviewApplications of Lie groups to differential equations : Peter J. Olver (School of Mathematics, University of Minnesota, Minneapolis, U.S.A): Graduate Texts in Mathematics, Springer-Verlag, New York, Berlin, Heidelberg, Tokyo, 1986, XXVI+497pp. , 1989 .

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

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

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

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

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

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

[14]  O. V. Kaptsov,et al.  Applications of Group-Theoretical Methods in Hydrodynamics , 1998 .

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

[16]  B. Cantwell,et al.  Introduction to Symmetry Analysis , 2002 .

[17]  Sergey V. Meleshko,et al.  Symmetries of the shallow water equations in the Boussinesq approximation , 2019, Commun. Nonlinear Sci. Numer. Simul..

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

[19]  Sergey V. Meleshko,et al.  Methods for Constructing Exact Solutions of Partial Differential Equations: Mathematical and Analytical Techniques with Applications to Engineering , 2005 .

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

[21]  R. L. Seliger,et al.  Variational principles in continuum mechanics , 1968, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

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

[23]  S. Meleshko,et al.  Symmetries of the hyperbolic shallow water equations and the Green–Naghdi model in Lagrangian coordinates , 2016 .

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

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

[26]  Bruno Després,et al.  Lagrangian Gas Dynamics in Two Dimensions and Lagrangian systems , 2005 .

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

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

[29]  S. Griffis.  EDITOR , 1997, Journal of Navigation.

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

[31]  V. Dorodnitsyn,et al.  Symmetry-preserving difference schemes for some heat transfer equations , 1997, math/0402367.