title: Symmetries, conservation laws, invariant solutions and difference schemes of the one-dimensional Green-Naghdi equations
authors: V. A. Dorodnitsyn E. I. Kaptsov S. V. Meleshko
text: 
  
  [
      R. Nevgi^1, K. R. Priolkar^1, L. Righi^2, M. Solzi^3, F. Cugini^3, E. Dias^4 and A. K. Nigam^4
      
  ==================================================================================================
  
  empty
  
  
  
  
  
  ABSTRACT
  
  
  
  
  
  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.
  
  
  
  
  
  
  
  
  
  
  § INTRODUCTION
  
  
  
  An ideal fluid flows under the force of gravity can be modeled
   by means of the Euler equations. However, the full Euler equations
  are too complicated for describing waves on the surfaces of ideal
  fluid, in particular, because of free surface being a
  part of the solution. This difficulty motivated scientists for
  deriving simpler equations. One class of such equations is a class
  of shallow water equations. The classical approach of deriving the shallow
  water equations consists of approximation of the Euler equations for
  the irrotational flows. The hierarchy of the shallow water
  approximations is considered with respect to the shallowness
  parameter δ=h_0/L, where h_0 is the mean depth of the
  fluid, L is the typical length scale of the wave
  <cit.>[See also the references therein.].
  In particular, the Green-Naghdi equations, derived for
  describing the two-dimensional  fluid flow over an uneven bottom,
  are accurate to the dispersive terms of order δ^2. The
  Green-Naghdi system of equations is the generalization of the
  equations derived first by Serre <cit.> and later by
  Su and Garden <cit.> to describe the
  one-dimensional propagation of fully nonlinear and weakly dispersive
  surface gravity waves over flat bottom. In the present paper we
  study the one-dimensional case of the Green-Naghdi equations.
  
  
  Due to the significance of the Green-Naghdi model, there has been
  increasing interest in the numerical solving of the Green-Naghdi equations.
  The following numerical approaches have been developed for the Green-Naghdi
  system, based on either the finite difference, hybrid finite-difference
  finite-volume, pseudospectral and Galerkin/finite-element methods.
  Review of these methods can be found in <cit.>.
  A numerical scheme developed in <cit.> is based
  on a different approach: the idea is to replace the dispersive Green-Naghdi
  equations by approximate hyperbolic equations. This approach was also applied
  in <cit.>.
  
  In the present paper we use an approach based on the group
  properties of the Green-Naghdi equations. The group analysis method
  <cit.> yields  exact solutions of
  differential equations, conservation laws via Noether's theorem and
  a basis for invariant finite-difference schemes construction.
  Applications of group analysis to the Green-Naghdi equations with a horizontal
  bottom topography in
  Eulerian and Lagrangian coordinates were studied in
  <cit.>.
  In particular, the authors of <cit.>
  applied Nother's theorem for finding conservation laws of the
  Green-Naghdi equations. Notice that in order to apply the Noether
  theorem for finding conservation laws one needs to know an admitted
  Lie group. Another requirement is an existence of a Lagrangian
  providing that the underline equations are in the Euler-Lagrange
  form. This is one of the main advantages of the Lagrangian coordinates
  used in the present paper.
  
  One of the main objectives of the present paper among the group classification
  and conservation laws is to
  construct a numerical scheme which inherits the group properties of
  the original equations.
  Earlier, in <cit.>, this method was applied to the hyperbolic shallow water equations  with an arbitrary bottom topography considered in Eulerian coordinates.
  An invariant difference scheme possessing all difference analogues
  of the conservation laws was constructed there.
  
  
  
  
  The one-dimensional Green-Naghdi equations describing
  surface gravity waves over uneven bottom have the form <cit.>
  
      [                        ∂/∂ th+∂/∂ x(hu)=0,; d/dtu+g∂/∂ x(h+H_b)=ϵ/h(∂/∂ xA+B∂/∂ xH_b), ]
  
  where h is the depth of the layer of fluid, u is the velocity,
  t is time, x is the Eulerian coordinate, H_b(x) is the
  function describing the bottom topography, ϵ=δ^2,
  d/dt=∂/∂ t+u∂/∂ x
  is the material derivative, and
  
      A=h^2d/dt(h/3u_x-1/2uH_bx),   B=hd/dt(h/2u_x-uH_bx),
  
  
  Assuming that
  
      H_b(x)=H_b0+ϵ D(x),   (H_b0=const),
  
  or
  
      H_b(x)=D(ϵ x),
  
  and cutting terms of order O(ϵ^2), one obtains equations (<ref>) with the mild slope approximation, where
  
      A=h^2d/dt(h/3u_x),   B=0.
  
  
  
  It should be noted here that for ε=0 equations (<ref>) become
  the hyperbolic shallow water equations. Symmetries, conservation laws and numerical
  schemes based on their group properties of the one-dimensional hyperbolic shallow
  water equations with different type of bottom topographies have been  analyzed in
  Eulerian and Lagrangian coordinates in  <cit.>.
  
  In the present paper the one-dimensional equations (<ref>)
  in mass Lagrangian coordinates (s,t) with the functions A and
  B of the form either (<ref>) or (<ref>)
  are studied.
  In <cit.>, it is shown that the
  Green-Naghdi equations with horizontal plane bottom are equivalent
  to the Euler-Lagrange equation written in the mass Lagrangian coordinates.
  For the one-dimensional
  Green-Naghdi equations with uneven bottom there is the problem of finding a Lagrangian. If the Lagrangian
  is found, then one can apply Noether's theorem for finding conservation
  laws.
  
  Another objective of the present paper is to construct conservative
  numerical schemes which preserve both a symmetry of original differential
  equations and difference analogs of conservation laws.
  
  
  
  The paper is organized as follows.  Section 2 is devoted to the
  study of equations (<ref>), (<ref>) in
  Lagrangian coordinates:  corresponding Lagrangians are found, and
  the group classification is carried out. Applying Noether's theorem,
  conservation laws in Lagrangian and Eulerian coordinates are
  derived. Then, the similar study of equations (<ref>),
  (<ref>) is given in Section 3, where it is also
  shown that equations (<ref>), (<ref>)
  with a flat bottom topography H_b=qx+β are locally
  equivalent to the Green-Naghdi equations with a horizontal bottom
  topography H_b=const. Preliminary information essential
  for further construction of conservative finite-difference schemes
  is given in Section 4. A new three-layer invariant conservative
  finite-difference scheme for the Green-Naghdi equations with a
  horizontal flat bottom topography is constructed in Section 5. At
  the end of Section 5, the possibilities of extending the obtained
  difference scheme to the case of an arbitrary bottom profile are
  discussed. Application of the scheme for analysis of traveling
  wave type solutions of the Green-Naghdi equations are considered in
  Section 6. The results are summarized in Conclusion.
  
  
  
  § EQUATIONS (<REF>), (<REF>) IN LAGRANGIAN
  COORDINATES
  
  
  
  
   §.§ Eulerian and Lagrangian coordinates
  
  
  Relations between Lagrangian coordinates (t,ξ) and Eulerian coordinates
  (t,x) for the one-dimensional case are defined by the condition
  x=φ(t,ξ), where the function φ(t,ξ) is the solution
  of the Cauchy problem
  
      φ_t(t,ξ)=u(t,φ(t,ξ)),   φ(t_0,ξ)=ξ.
  
  In Lagrangian coordinates, the general solution of the mass conservation
  law equation is
  
      h(t,φ(t,ξ))=h_0(ξ)/φ_ξ(t,ξ),
  
  where h_0(ξ) is an arbitrary function of the integration such
  that h(t_0,ξ)=h_0(ξ). Introducing the mass Lagrangian coordinate
  s <cit.> by the equation
  
      ξ=α(s),
  
  where α^ '(s)=h_0(α(s)), one obtains that in
  the mass Lagrangian coordinates (t,s)
  
      h̃(t,s)≡ h(t,φ(t,α(s)))=1/φ̃_s(t,s).
  
  Here the functions φ̃(t,s) and φ(t,ξ) are
  related by the formula
  
      φ̃(t,s)=φ(t,α(s)).
  
  Hence, the mass Lagrangian coordinates are defined by the equations
  
      φ̃_t(s,t)=ũ(s,t),  φ̃_s(s,t)=h̃^-1(s,t).
  
  where ũ(s,t)=u(φ(α(s),t),t). The sign tilde
   is further omitted.  In order to derive representations
  of the Green-Naghdi equations in the mass Lagrangian coordinates one
  can use the following relations:
  
      [                             u_x=φ_tsh,   u_t=φ_tt-uu_x,   h_x=-φ_ssh^3,;                     u_xx=φ_tss+u_xh_xh^-1,   u_tx=φ_ttsh-(uu_xx+u_x^2),; u_tt=φ_ttt-(u^2u_xx+2uu_tx+uu_x^2+u_xu_t),   h_xx=-φ_sssh^4+3h_x^2h^-1,;            u_txx=φ_ttssh^2-(uu_xxx+3u_xu_xx-h^-1(uu_xx+u_x^2+u_tx)h_x). ]
  
  The corresponding equation becomes
  
      [ ϵ(-3H_b^'''φ_t^2φ_s^5-6H_b^''H_b^'φ_t^2φ_s^6+6H_b^''φ_t^2φ_s^3φ_ss-6H_b^''φ_tφ_tsφ_s^4-3H_b^''φ_ttφ_s^5-6H_b^'^2φ_ttφ_s^6;                             +6H_b^'φ_ttφ_s^3φ_ss-6H_b^'φ_ts^2φ_s^3-8φ_ttsφ_sφ_ss+2φ_ttssφ_s^2+20φ_ts^2φ_ss-8φ_tsφ_tssφ_s);                                                                                  +6φ_s^3(-H_b^'φ_s^3g-φ_ttφ_s^3+φ_ssg)=0. ]
  
  
  
  
  
   §.§ Search for a Lagrangian
  
  
  For finding a Lagrangian for which equation (<ref>) is
  the Euler-Lagrange equation one has to solve the following problem.
  Let L(t,s,φ_t,φ_s,φ_tt,φ_ts,φ_ss)
  be a corresponding Lagrangian. Then,
   substituting L into
  the equation[Notations related with the variational analysis follow to the formulations given in <cit.>.]
   δ
  L/δφ=0, excluding the derivative φ_ttss
  found from equation (<ref>), and splitting it with
  respect to the parametric derivatives
  
      φ_ttt,   φ_tts,   φ_tss,   φ_sss,   φ_tttt,   φ_ttts,   φ_tsss,   φ_ssss,
  
  one obtains an overdetermined system of equations for the function
  L. The general solution of these equations such that 2 L_φ_ttφ_ss+ L_φ_tsφ_ts≠0
  gives the sought Lagrangian. Calculations, performed in symbolic manipulation system Reduce <cit.>, give that the general solution
  contains several arbitrary constants and unknown functions. These
  functions satisfy a compatible system of partial differential equations.
  As the general form of the Lagrangian is cumbersome, we only present here
  a particular case of the Lagrangian
  
      L=1/2φ_t^2(1+ϵ(H_b^'^2-H_b^'φ_s^-3φ_ss+1/2H_b^''φ_s^-1))+1/6φ_s^-4(ϵφ_ts^2-3gφ_s^2(2Q_bφ_ss+φ_s)).
  
  where Q_b^'(φ)=H_b(φ).
  
  
  
   §.§ Group analysis of equation (<ref>)
  
  
  To find equivalence transformations we used the infinitesimal
  criterion <cit.>. For this purpose the
  determining equations for the components of generators of
  one-parameter groups of equivalence transformations were derived.
  The solution of these determining equations gives the general form
  of elements of the equivalence algebra of the class
  (<ref>). The basis elements of the equivalence algebra
  of the class (<ref>) are
  
      X_1^e=φ∂_φ+4s∂_s+t∂_t+2H_b∂_H_b,   X_2^e=∂_H_b,
  
  
      X_3^e=φ∂_φ+s∂_s+t∂_t+2ϵ∂_ϵ,   X_4^e=t∂_t-2g∂_g,
  
  
      X_5^e=∂_φ,   X_6^e=∂_s,   X_7^e=∂_t.
  
  
  In the group classification we use the transformations corresponding
  to the generators X_1^e and X_2^e which are
  
      φ̃=e^aφ,   t̃=e^at,   s̃=e^4as,   H̃_b=e^2aH_b
  
  and
  
      H̃_b=H_b+a,
  
  where a is a group parameter and only changeable variables are presented.
  The generators X_3^e and X_4^e allow reducing the constants
  g and ϵ to the simple case g=1 and ϵ=1.
  
  It should be noted that the Galilean transformation corresponding
  to the generator X_9^e=t∂_φ is absent among
  the equivalence transformations. This is related with the property
  that the bottom does not depend on time t.
  
  There are also the obvious involutions
  
      E_1:    t̃=-t,
  
  
      E_2:    φ̃=-φ,   s̃=-s.
  
  
  Calculations show that the kernel of admitted Lie algebras is defined
  by the generators
  
  
      ∂_t,   ∂_s.
  
  An extension of the kernel only occurs for a linear bottom
  
      H_b=qx+β,
  
  where without loss of generality one can assume that β=0. The
  extension is defined by the generators
  
      ∂_φ,   t∂_φ,   t∂_t+4s∂_s+2φ∂_φ.
  
  
   The same Lie group is admitted by the equations with
  the horizontal bottom (q=0) studied in <cit.>.
   The coincidence of the admitted
  Lie groups proposes to assume that the Green-Naghdi equations with
  a flat bottom topography (<ref>) are equivalent
  to the Green-Naghdi equations with a horizontal bottom q=0. We
  have checked that if such a transformation exists, then it is not
  among the point transformations of the form
  
      h̅=f_1(t,x)u+f_2(t,x)h+f_0(t,x),   u̅=g_1(t,x)u+g_2(t,x)h+g_0(t,x),
  
  
      x̅=γ_1(t,x),   t̅=γ_2(t,x).
  
  We have also checked that it is not among the point transformations
  of the form
  
      φ̅=f_1(t,s)φ+f_0(t,s),   s̅=γ_1(t,s),   t̅=γ_2(t,s).
  
  
  
   Equation (<ref>) with flat bottom topography
  (<ref>) have the same representations of invariant
  solutions as the Green-Naghdi equations with a horizontal bottom q=0
  given in <cit.>. 
  
  
  
   §.§ Conservation laws 
  
  
  A conservation law of equations either (<ref>),
   (<ref>) or (<ref>),
   (<ref>) in Lagrangian coordinates is
  considered in the following local form
  
      D_tT^t+D_sT^s=0,
  
  where the densities T^t and T^s depend on
  
      (t,s,φ,φ_t,φ_s,φ_tt,φ_ts,φ_ss,φ_ttt,φ_tts,φ_tss,φ_sss).
  
  Its counterpart in Eulerian coordinates has the form
  
      D_t^eT^t+D_xT^x=0,
  
  where
  
      ^eT^t=hT^t,  T^x=huT^t+T^s.
  
  
  
  Notice that if a generator
  X=ξ^t∂_t+ξ^s∂_s+ζ∂_φ
  is either variational or divergently variational, then there exist
  such functions B^t and B^s that <cit.>
  
      X L+ L(D_tξ^t+D_sξ^s) = D_tB^t+D_sB^s.
  
  Hence, for either variational or divergently variational generator X, one has that
  
      δ/δφ(X L+ L(D_tξ^t+D_sξ^s))=0.
  
  
  
  
    §.§.§ Conservation laws corresponding to the kernel of admitted Lie algebras
  
  
  Conservation laws are obtained by applying Noether's theorem.
  
  The generator ∂_s provides the conservation law with the
  densities:
  
      T_1^t=ϵφ_s^-3φ_tss-3φ_t(2φ_s(1+ϵ H_b^'^2)+ϵ H_b^'')+ϵφ_s^-4φ_ss(6H_b^'φ_tφ_s^2-5φ_ts),
  
  
      T_1^s=ϵφ_s^-3φ_tts+3φ_t^2(1+ϵ H_b^'^2)-6φ_s^-2(ϵ H_b^'φ_tφ_ts+gφ_s)-6gH_b.
  
  In Eulerian coordinates the counterpart conservation law has the
  densities
  
      ^eT_1^t=ϵ h^2v_xx+4ϵ hh_xv_x-6v-3ϵ v(H_b^''h+2H_b^'^2+2H_b^'h_x),
  
  
      T_1^x=ϵ h^2(v_tx+v_x^2)+2ϵ h^2vv_xx+2ϵ hvv_x(2h_x-3H_b^')-6g(H_b+h)-3v^2(1+H_b^''ϵ h+H_b^'^2ϵ+2H_b^'ϵ h_x).
  
  
  The generator ∂_t gives the conservation law of energy with the
  densities:
  
      T_2^t=-6φ_t^2(1+ϵ H_b^'^2)+φ_s^-5(2ϵφ_tφ_tssφ_s-8ϵφ_tφ_tsφ_ss-6φ_s^4g-3H_b^''ϵφ_t^2φ_s^4+6H_b^'ϵφ_t^2φ_s^2φ_ss+12Q_bφ_s^3φ_ss),
  
  
      T_2^s=φ_s^-4(2ϵφ_tφ_tts-2ϵφ_ttφ_ts-6φ_tφ_s^2g-3H_b^''ϵφ_t^3φ_s^2-6H_b^'ϵφ_t^2φ_tsφ_s-12Q_bφ_tsφ_s^2-12φ_tφ_s^3gH_b).
  
  In Eulerian coordinates the densities of the corresponding conservation
  law are
  
      ^eT_2^t=1/2(-3H_b^''ϵ h^2v^2-6H_b^'^2ϵ hv^2-6H_b^'ϵ hh_xv^2-12Q_bh_x+2ϵ h^3vv_xx+6ϵ h^2h_xvv_x-6gh^2-6hv^2),
  
  
      [ T_2^x=-3H_b^''ϵ h^2v^3-3H_b^'^2ϵ hv^3-3H_b^'ϵ h^2v^2v_x-3H_b^'ϵ hh_xv^3-6Q_bhv_x-6Q_bh_xv+2ϵ h^3v^2v_xx+ϵ h^3vv_tx;                                                                 -ϵ h^3v_tv_x+3ϵ h^2h_xv^2v_x-6gH_bhv-6gh^2v-3hv^3. ]
  
  
  
  
  
    §.§.§ Flat bottom topography 
  
  
  In this case one has that H_b=qx.
  
  Using the general solution for a Lagrangian, it can be shown that
  the generator X=t∂_t+4s∂_s+2φ∂_φ
  does not satisfy the condition (<ref>).
  
  The generator t∂_φ corresponding to the Galilean
  transformation provides the conservation law  with the densities:
  
      T_1^t=3gqt^2+6(1+ϵ q^2)(tφ_t-φ)-3ϵφ_s^-1q+tφ_s^-5(2ϵφ_ss(φ_ts-3qφ_tφ_s^2)-ϵφ_tssφ_s),
  
  
      T_1^s=φ_s^-4(t(6ϵφ_tφ_tsφ_sq-ϵφ_tts+3φ_s^2g)+ϵφ_ts-3ϵφ_tφ_s^2q).
  
  In Eulerian coordinates this conservation law gives a the center-of-mass law:
  
      ^eT_1^t=h(6(ϵ q ^2+1)(tv-x)-3ϵ q h+6ϵ q h_xtv-ϵ h^2tv_xx-3ϵ hh_xtv_x+3gq t^2),
  
  
      [ T_1^x=h(6ϵ q ^2tv^2-6ϵ q ^2vx+6ϵ q htvv_x-6ϵ q hv+6ϵ q h_xtv^2-2ϵ h^2tvv_xx-ϵ h^2tv_tx-ϵ h^2tv_x^2+ϵ h^2v_x;                                                                     -3ϵ hh_xtvv_x+3gq t^2v+3ght+6tv^2-6vx). ]
  
  
  The generator ∂_φ provides the conservation law
  with the densities:
  
      T_2^t=6φ_t(1+ϵ q^2)+6gqt+φ_s^-5(2ϵφ_ss(2φ_ts-3qφ_tφ_s^2)-ϵφ_tssφ_s),
  
  
      T_2^s=φ_s^-4(6ϵφ_tφ_tsφ_sq-ϵφ_tts+3φ_s^2g),
  
  which in Eulerian coordinates corresponds the conservation law of momentum with the densities:
  
      ^eT_2^t=h(6ϵ q^2v+6ϵ qh_xv-ϵ h^2v_xx-3ϵ hh_xv_x+6gqt+6v),
  
  
      T_2^x=h(6ϵ q^2v^2+6ϵ qhvv_x+6ϵ qh_xv^2-2ϵ h^2vv_xx-ϵ h^2v_tx-ϵ h^2v_x^2-3ϵ hh_xvv_x+6gqtv+3gh+6v^2).
  
  
  
  
  § EQUATIONS (<REF>), (<REF>) IN LAGRANGIAN
  COORDINATES
  
  
  
  
   §.§ Lagrangian coordinates
  
  
  Similar analysis of equations (<ref>), (<ref>)
  gives that in mass Lagrangian coordinates equations (<ref>),
  (<ref>) are equivalent to the equation
  
      ϵ/3(φ_ttssφ_s^2-4φ_s(φ_ssφ_tts
      +φ_tsφ_tss)+10φ_ts^2φ_ss)
      -φ_s^3(φ_ttφ_s^3-gφ_ss)
      =gϵφ_s^6H_b^'.
  
  
  Following the same method of finding a Lagrangian for equation (<ref>) as in the previous section, we derived the general form of the Lagrangian. A particular form of the Lagrangian providing equation (<ref>) as the Euler-Lagrange
  equation δ L/δφ=0
  is
  
      L=1/2(φ_t^2-1/2gφ_s^-1)+1/6ϵφ_ts^2φ_s^-4-ϵ gH_b.
  
  
  
  
  
   §.§ Equivalence transformations
  
  
  
  
    §.§.§ Equivalence group
  
  
  Calculations give that equivalence group is defined by the generators
  
      Y_1^e=t∂_t+4s∂_s+2φ∂_φ+2H_b∂_H_b,   X_2^e=∂_H_b,
  
  
      Y_3^e=2t∂_t+5s∂_s+3φ∂_φ+2ϵ∂_ϵ,   X_4^e=t∂_t-2g∂_g,
  
  
      X_5^e=∂_φ,   X_6^e=∂_s,   X_7^e=∂_t.
  
  As in the previous case there are also the obvious involutions
  
      E_1:    t̃=-t,
  
  
      E_2:    φ̃=-φ,   s̃=-s.
  
  
  
  
  
    §.§.§ Flat bottom
  
  
  Consider the topography
  
      H_b=qx+β.
  
  Because of the equivalence transformation corresponding to the generator
  Y_1^e, one can assume that q^2=1. Direct calculations
  show that the change either
  
      t̃=t+1/2gϵ,   x̃=qx+1/2gϵ t^2+k_1t+k_2,
  
  
      ũ=qu+gϵ t+k_1,   h̃=h,
  
  or
  
      t̃=k(t+k/2gϵ),   x̃=k^2(qx+1/2gϵ t^2)+k_2,
  
  
      ũ=k(qu+gϵ t),   h̃=k^2h,
  
  reduces equations (<ref>), (<ref>) to
  the horizontal flat bottom H̃_b=0. Here k≠0, k_1
  and k_2 are arbitrary constants. As the case with the horizontal
  flat bottom is studied in <cit.>,
  it is excluded from our further consideration.
  
  
  The property of the reduction of the shallow water equations with a flat bottom topography to the equations with the horizontal flat bottom is known for the one-dimensional hyperbolic shallow water equations <cit.>. For the two-dimensional hyperbolic shallow water equations it is proven in <cit.>.
  
  
  
  
   §.§ Group classification
  
  
  The kernel of admitted Lie algebras is defined by the generators
  
      ∂_t,   ∂_s.
  
  
  Extensions of the kernel occur for
  
      H_b(x)=kx^2+α x+β,   (k≠0),
  
  where without loss of generality one can assume that α=0 and
  β=0. The extensions depend on the sign of k:
  
  if k=q^2/2ϵ g>0, then the extension is given by
  the generators
  
      sin(qt)∂_φ,    cos(qt)∂_φ.
  
  
  if k=-q^2/2ϵ g<0, then the extension is given
  by the generators
  
      e^qt∂_φ,    e^-qt∂_φ.
  
  
  
  Similar extensions of a kernel of admitted Lie groups occur for the one-dimensional hyperbolic shallow water equations with a parabolic bottom topography
  <cit.>. In the case of two-dimensional hyperbolic shallow equations with a constant Coriolis parameter such type of extensions also occur for a  circular paraboloid  <cit.>. The author of <cit.> noted that for a circular parabolic bottom the admitted Lie algebra found in <cit.> is isomorphic to the Lie algebra admitted by the classical shallow water equations with a horizontal bottom H_b = const and zero Coriolis parameter. In  <cit.> it is proven that two-dimensional hyperbolic shallow equations with a constant Coriolis parameter with a circular parabolic bottom topography are locally equivalent to the classical gas dynamics equations.
  
  
  
  
  
   §.§ Conservation laws 
  
  
  
  
    §.§.§ Kernel of admitted Lie algebras
  
  
  The kernel of admitted Lie algebras ∂_s and ∂_t
  gives the conservation laws with the densities, respectively:
  
  
      T_1^t=ϵφ_s^-4(φ_tssφ_s-5φ_tsφ_ss)-6φ_tφ_s,
  
  
      T_1^s=φ_s^-3(ϵφ_tts-6φ_s^2g)+3φ_t^2-6ϵ gH_b.
  
  and
  
      T_2^t=ϵφ_s^-5φ_t(φ_tssφ_s-4φ_tsφ_ss)-3(2ϵ gH_b+φ_t^2+φ_s^-1g),
  
  
      T_2^s=φ_s^-4(ϵφ_tφ_tts-ϵφ_ttφ_ts-3gφ_tφ_s^2).
  
  Counterparts of these conservation laws in Eulerian coordinates have
  the densities:
  
      ^eT_1^t=ϵ h^2v_xx+4ϵ hh_xv_x-6v,
  
  
      T_1^x=ϵ h^2(v_tx+v_x^2)+2ϵ h^2vv_xx+4ϵ hh_xvv_x-6gh-3v^2-6ϵ gH_b,
  
  and
  
      ^eT_2^t=h(ϵ h^2vv_xx+3ϵ hh_xvv_x-3v^2-3gh-6ϵ gH_b),
  
  
      T_2^x=h(ϵ h^2v(2vv_xx+v_tx)-ϵ h^2v_tv_x+3ϵ hh_xv^2v_x-6ghv-3v^3-6ϵ gH_bv).
  
  
  
  
  
    §.§.§ Parabolic bottom topography
  
  
  Using the general solution for a Lagrangian, it can be shown that
  the generators e^qt∂_φ and e^-qt∂_φ
  corresponding to the case k=-q^2/(2ϵ g) do not satisfy
  the condition (<ref>). The other two generators sin(qt)∂_φ
  and cos(qt)∂_φ corresponding to the case k=q^2/(2ϵ g)
  provide the conservation laws, respectively:
  
  
      T_3^t=6cos(qt)φ_sqs+φ_s^-5sin(qt)(4ϵφ_tsφ_ss-ϵφ_tssφ_s+6φ_tφ_s^5),
  
  
      T_3^s=6qs(sin(qt)φ q-cos(qt)φ_t)+φ_s^-4(ϵcos(qt)φ_tsq-ϵsin(qt)φ_tts+3sin(qt)φ_s^2g),
  
  and
  
      T_4^t=6(cos(qt)φ_t-sin(qt)φ_sqs)+ϵcos(qt)φ_s^-5(4φ_tsφ_ss-φ_tssφ_s),
  
  
      T_4^s=6qs(cos(qt)φ q+sin(qt)φ_t)-φ_s^-4(cos(qt)ϵφ_tts-3cos(qt)φ_s^2g+sin(qt)ϵφ_tsq).
  
  
  In Eulerian coordinates the counterparts of these conservation laws
  contain the variable s, which satisfy the equations
  
      s_t=-hv,  s_x=h.
  
  These conservation laws have the densities:
  
      ^eT_3^t=6cos(qt)sq+sin(qt)h(6v-ϵ h^2v_xx-3ϵ hh_xv_x),
  
  
      T_3^x=cos(qt)ϵ qh^3v_x+sin(qt)(6sq^2x+6hv^2-2ϵ h^3vv_xx-ϵ h^3v_tx-ϵ h^3v_x^2-3ϵ h^2h_xvv_x+3gh^2),
  
  and
  
      ^eT_4^t=cos(qt)h(ϵ h^2v_xx+3ϵ hh_xv_x-6v)+6sin(qt)sq,
  
  
      T_4^x=cos(qt)(2ϵ h^3vv_xx+ϵ h^3v_tx+ϵ h^3v_x^2+3ϵ h^2h_xvv_x-3gh^2-6sq^2x-6hv^2)+sin(qt)ϵ qh^3v_x.
  
  
  
  
  
  
  
  § PRELIMINARY ANALYSIS OF THE GREEN-NAGHDI EQUATIONS FOR CONSTRUCTING FINITE-DIFFERENCE SCHEMES
  
  
  
  
   §.§ Conservative form of the equations in Lagrangian variables
  
  
  Before construction of finite-difference schemes
  we rewrite the Green-Naghdi equations for a horizontal bottom topography
  
  
      2γ(x_s ^2x_ttss-4x_sx_ssx_tts-4x_s x_tsx_tss+10x_ts^2x_ss)
      +x_s^3(2
      x_ss-x_s^3x_tt)=0,
  
  where the following notations are used: φ = x and γ∝ε.
  In a conservative form the latter equation becomes
  
      D_t(x_t) + D_s(
              1/x_s^2)
          - 2γ D_s(
              x_s x_tts - 2 x_ts^2/x_s^5) = 0,
  
  where it is assumed that x_s ≠ 0.
  
  
  The local conservation laws of momentum, energy, and center-of-mass law
  can be rewritten as follows
  
      ∂_x:      D_t(x_t) + D_s(
              1/x_s^2
              - 2γ (x_s x_tts - 2 x_ts^2)/x_s^5) = 0,
  
  
      t∂_x:      D_t(t x_t - x) + D_s(
              t(1/x_s^2
              - 2γ (x_s x_tts - 2 x_ts^2)/x_s^5)) = 0,
  
  
      ∂_t:     
        D_t(
              x_t^2/2 + 1/x_s  + γx_ts^2/x_s^4)
        + D_s(
              x_t (1/x_s^2
              - 2γ (x_s x_tts - 2 x_ts^2)/x_s^5)) = 0.
  
  
  The conservation law of mass automatically follows from the symmetry of second derivatives x_ts=x_st.
  
  
  
   §.§ The Green-Naghdi equations in hydrodynamic variables
  
  
  Here we represent the Green-Naghdi equations in hydrodynamic variables u(t,s), ρ(t,s),
  where ρ[We use ρ instead of h, as h is usually accepted in numerical analysis for  the finite-difference step.] is the depth of the fluid over the bottom and s, as before, is the mass Lagrangian coordinate.
  
  It turns out to be especially convenient to use hydrodynamic
  variables in finite-difference space <cit.>.
  This allows transition from a three-layer finite-difference schemes to a two-layer ones.
  
  The shallow water equations in hydrodynamic variables are obtained by the change
  
  
      x_s = 1/ρ,
            
        x_t = u.
  
  The Green-Naghdi equations become
  
      (1/ρ)_t = u_s,
  
  
      u_t + 2ρρ_s = 2γ(ρ^4 (u_ts - 2 ρ u_s^2))_s,
  
  where the first equation follows from the symmetry of second-order derivatives of the variable x.
  
  The variable p(t,s) which was introduced in <cit.> by the relation
  
      p = ρ^2,
  
  allows one to rewrite (<ref>) as follows
  
      u_t + p_s = 2γ(p^2 (u_ts - 2 √(p) u_s^2))_s.
  
  
  The local conservation laws of momentum (<ref>), energy (<ref>), and
  center-of-mass law (<ref>)  become
  
      D_t(u) + D_s(
              ρ^2
              - 2γ p^2(u_ts - 2√(p) u_s^2)
          ) = 0,
  
  
      D_t(t u - x) + D_s(
              t[ ρ^2
              - 2γ p^2(u_ts - 2√(p) u_s^2)
          ]) = 0,
  
  
      D_t(
              u^2/2 + p/ρ + γ (pρ u_s)^2
        )
        + D_s(
              u [ ρ^2
              - 2γ p^2(u_ts - 2√(p) u_s^2)
              ]
          ) = 0.
  
  
  
  
  
  
  
  
  § INVARIANT DIFFERENCE SCHEMES
  
  
  
  
  
   §.§ Invariant conservative scheme
  
  
  For constructing difference scheme one needs
  9-nods stencil (see Figure <ref>),
  which has three-layers in time
  
      (𝐭,𝐬,𝐱) =
        (
              t_n, t_n-1, t_n+1;
              s_m, s_m-1, s_m+1;
              x^n_m, x^n_m+1, x^n+1_m,
              x^n+1_m, x^n+1_m+1, x^n+1_m-1,
              x^n-1_m, x^n-1_m+1, x^n-1_m-1
          ),
  
  
      x^n+p_m+q = x(t_n+p, s_m+q),
              
          p, q ∈ℤ,
  
  
  
  
  Here n- and m-indices correspond to the time and space layers appropriately, and
  the point (n,m) is fixed. For the sake of brevity we also use the following notation <cit.>
  
      (𝐭,𝐬,𝐱) =
        (
              t, ť, t̂;
              s, s_-, s_+;
              x, x_+, x_-,
              x̂, x̂_+, x̂_-,
              x̌, x̌_+, x̌_-,
          )
  
  Here and further the simplest orthogonal regular mesh is considered
  
      1.5[                   τ_- = τ_+ = τ = t_n+p+1 - t_n+p = const,; h_- = h_+ = h = s_m+q+1 - s_m+q = const,
            
        p, q ∈ℤ, ]
  
  which is invariant with respect to the operators admitted by the
  original differential equation (see also finite-difference
  invariants (<ref>)).
  
  Indices of finite-difference variables are shifted by the finite-difference shift operators
  ±τS and ± sS that defined as follows
  
      1.5[      +τS(f(t_n, s_m)) = f(t_n + τ, s_m) = f(t̂, s) = f̂,
              -τS(f(t_n, s_m)) = f(t_n - τ, s_m) = f(ť, s) = f̌,; +sS(f(t_n, s_m)) = f(t_n, s_m + h) = f(t, s_+) = f_+,
              -sS(f(t_n, s_m)) = f(t_n, s_m - h) = f(t, s_-) = f_-. ]
  
  The finite-difference total differentiation operators are defined through the shifts as
  
      +τD = +τS - 1/t_n+1 - t_n,
            -τD = 1 - -τS/t_n - t_n-1,
            +sD = +sS - 1/s_m+1 - s_m,
            -sD = 1 - -sS/s_m - s_m-1,
  
  and the following notation is used for difference derivatives
  
      x_t = +τD(x),
              
          x_ť = -τD(x),
              
          x_tť = +τD-τD(x),
              
          x_tťt = +τD-τD+τD(x),
  
  
      x_s = +sD(x),
              
          x_s̅ = -sD(x),
              
          x_ss̅ = +sD-sD(x),
              
          x_ss̅s = +sD-sD+sD(x),
              etc.
  
  Notice that the operators
   ±τS, ± sS,
  ±τD and ± sD
  commute in any combination on uniform orthogonal
  meshes so that the following relations are valid
  
      x̌_t = x_ť,
              x̌_tt = x_ťt = x_tť,
              
          x^-_s = x_s̅,
              
          x^-_ss = x_s̅s = x_ss̅,
              etc.
  
  
  On the 9-nods stencil there was constructed invariant
  scheme for shallow water equations in <cit.>:
  
      x_ťt
              + D[1/x̂_s x̌_s] = 0,
              τ̂ = τ̌ = τ = const,
              
          h_+  = h_- = h = const,
  
  which possesses the whole number of appropriate conservation laws.
  
  
  
  
  
  Analysis of the scheme along with consideration
  of difference invariants (<ref>) suggests
  the following invariant extension for equation (<ref>)
  
  
      x_ťt
              + D[1/x̂_s x̌_s]
        - 2 γD[
              1/x̂_s^2 x̌_s^2(
                  x_tťs - 2 x_ts x_ťs/x_s)
        ]
          = 0,
  
  
      τ_- = τ_+ = τ = const,
              
          h_- = h_+ = h = const.
  
  The scheme can also be obtained with the help of the finite-difference analog
  of so-called the `direct method' (see <cit.>
  and <cit.> for discussion in detail).
  
  
  Scheme (<ref>) approximates equation (<ref>) to the order O(τ^2 + h^2).
  It possesses the following local difference conservation laws:
  
    
    * Conservation law of mass:
    
      D(x̂_s) - D(x_t^+) = 0.
  
    It automatically follows from the commutativity
    of the finite-difference differentiation on a uniform orthogonal mesh;
  
    
    * Conservation law of momentum:
    
      D(
              x_t
          )
          + D[
              1/x̂_s x̌_s
              - 2 γ/x̂_s^2 x̌_s^2(
                  x_tťs - 2 x_ts x_ťs/x_s)
          ] = 0;
  
  
    
    * Center-of-mass law:
    
      D(
              t x_t - x
          )
          + D[
              t 1/x̂_s x̌_s
              - t 2 γ/x̂_s^2 x̌_s^2(
                  x_tťs - 2 x_ts x_ťs/x_s)
          ] = 0;
  
  
    
    * Conservation law of energy:
    
      D[
              x_t^2/2
                  + 1/2(
                      1/x_s + 1/x̂_s)
              +γ{D(1/x_s)}^2
          ]
          + D[
              x_t^+ + x̌_t^+/2{1/x̂_s x̌_s
                  - 2 γ/x̂_s^2 x̌_s^2(
                      x_tťs - 2 x_ts x_ťs/x_s)
              }] = 0.
  
  
  
  
  
   §.§ Invariant representation of scheme (<ref>)
  
  
  Scheme (<ref>) admits the same Lie algebra as its differential counterpart, i.e.,
  the generators
  
      X_1=∂_t,  
      X_2=∂_s ,  
      X_3=∂_x,  
      X_4=t∂_x,  
      X_5=t∂_t+4s ∂_s +2x∂_x.
  
  In 15-dimensional space of the difference stencil (<ref>), there are 15-5=10 difference
  invariants:
  
      1.75[                   I_1 = τ_+/τ_-,
              
          I_2 = h_+/h_-,
              
          I_3 = h_-/τ_-^4,;     I_4 = x_+ - x/τ_-^2,
            
          I_5 = x̌_+ - x̌/τ_-^2,
            
          I_6 = x - x_-/τ_-^2,; I_7 = x̂ - x̂_-/τ_-^2,
            
          I_8 = x̂_+ - x̂/τ_-^2,
            
          I_9 = x̌ - x̌_-/τ_-^2,;                                       I_10 = (x̂ - x̌) t + (x - x̂) ť + (x̌ - x) t̂/τ_-^3. ]
  
  Using the difference  invariants (<ref>), scheme (<ref>) can be represented  as follows
  
      I_10/ I_1
      + I_3 (
              I_2^2/I_5 I_8 - 1/I_7 I_9)
      + 2γ I_3^2/I_1{I_2^3/I_8 I_5^2(
                      1 - 2  I_5/I_4
                      + (  I_4- I_5) I_1- I_4+2  I_5/ I_8)
               
      
              +  2  I_9 -  I_6/ I_6  I_7 I_9^2
               - (  I_6- I_9)  I_1 + 2  I_9 - I_6/ I_7^2 I_9^2} = 0,
  
  
      I_1 = 1,
                  
          I_2 = 1.
  
  
  Hence, the invariant scheme and mesh are constructed. They possess the
  difference analogs of all conservation laws.
  
  
  All the generators (<ref>) preserve the mesh uniformness and
  orthogonality (see criterion in <cit.>). This
  requirement is necessary for constructing invariant difference
  schemes on uniform orthogonal meshes, and it is often satisfied for
  the hydrodynamics-type schemes in Lagrangian coordinates.
  
  
  
  
   §.§ Scheme (<ref>) in hydrodynamic variables
  
  
  Here we introduce finite-difference hydrodynamic variables through the relations constructed
  in <cit.> for the shallow water difference scheme, namely
  
      x_t = u,
          
      x_s = 1/√(p),
  
  
      1/√(p̌) + 1/√(p) = 2/ρ̌.
  
  The latter relation is an implicit approximation of equation (<ref>),
  which allows one to preserve difference conservation laws.
  
  Relations (<ref>) and (<ref>) allow one to write the scheme (<ref>)
  in hydrodynamic variables on two time layers, i.e., to obtain a two-level scheme.
  But at the same time the additional equations must be added to the scheme.
  Hence, scheme (<ref>) becomes
  
      1.75[                              D( 1/ρ) - D(
              u^+ + ǔ^+/2) = 0,;                                      D(u) + D(
              W
          ) = 0,; x_t = u,
              
          x_s = 1/√(p),
              1/√(p̌) + 1/√(p)= 2/ρ̌,;                                           h_+ = h_-,     τ_+ = τ_-, ]
  
  where
  
      W =
            [
                  4/ρρ̌
                  - 2/√(p)( 1/ρ + 1/ρ̌)
                  + 1/p]^-1
            - 2 γ p p̌ρ^2/(2√(p) - ρ)^2(
                      ǔ_ts - 2 u_sǔ_s √(p)).
  
  The scheme is defined on 6-point stencil (see Figure <ref>).
  One can check that equations of the system
  approximate system (<ref>), (<ref>) to the order O(τ + h).
  Notice that the first equation of system (<ref>) follows from
  the equality
  
      D(x_s + x̂_s) = D(x_t^+ + x̌_t^+)
  
  that is correct on the uniform orthogonal mesh.
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  The conservation laws of  scheme (<ref>) are the
  following:
  
      
    * The conservation of mass
      
      D( 1/ρ) - D(
                  u^+ + ǔ^+/2) = 0;
  
  
    
    * The conservation law of momentum:
      
      D( u ) + D(W) = 0;
  
  
    
    * The center-of-mass law:
      
      D(t u - x) + D(t W) = 0;
  
  
    
    * The conservation law of energy:
      
      D(
                  u^2/2 + p/2 √(p) - ρ
                  +  γ[pρ u_s/2√(p) - ρ]^2
              )
               + D(
                  u^+ + ǔ^+/2  W
              ) = 0.
  
  
  
  
  
  
  
   §.§ On extensions of the scheme to the case of arbitrary bottom
  
  
  For model (<ref>), (<ref>)
  invariant scheme (<ref>) can be extended to the case of an arbitrary bottom H(x)
  in the same manner as it was done in <cit.>.
  Indeed,  adding
  
      DH(x) + DH(x)/x_t + x̌_t
  
  to the left-hand side of equation (<ref>), one gets the scheme that
  preserves the conservation laws of mass (<ref>) and energy:
  
      D[
              x_t^2/2
                  + 1/2(
                      1/x_s + 1/x̂_s)
              +γ{D(1/x_s)}^2
              - H(x) - H(x̂)
          ]
          + D[
              x_t^+ + x̌_t^+/2{1/x̂_s x̌_s
                  - 2 γ/x̂_s^2 x̌_s^2(
                      x_tťs - 2 x_ts x_ťs/x_s)
              }] = 0.
  
  The construction of an invariant scheme that conserves both energy and momentum meets considerable difficulties.
  A separate example of invariant scheme that conserves momentum was proposed in <cit.>
  for the shallow water equations with an arbitrary bottom.
  It can be also generalized to the case of the Green-Naghdi equations.
  
  
  
  § TRAVELLING-WAVE TYPE SOLUTION FOR THE FINITE-DIFFERENCE SCHEME
  
  
  
  
   §.§ Travelling-wave type solution in Lagrangian coordinates
  
  
  To analyze some numerical properties of the constructed scheme, we
  consider Serre's travelling-wave type solution <cit.>
  of the Green-Naghdi equations. In Eulerian coordinates, this type of
  solutions is invariant with respect to the generator ∂_t
  and has the form[Usually the independent variable in traveling waves has the form x-ct. Because of the Galilean
  transformation the constant c can be set to zero.]
  
      u(x)=-k/ρ(x),
          ρ(x)=R_0+A^2(μ x),
  
  where
  
      k^2=2R_0^2(A+R_0),
          μ^2=A/8γ R_0^2(A+R_0).
  
  In Lagrangian coordinates, the solution corresponds
  to the set of solutions invariant with respect
  to the generator ∂_t + ∂_s <cit.>.
  It has the following representation
  
      x(t,s) = ψ(λ),     λ = s - t.
  
  The reduced equation (<ref>) is
  
      2γ( ψ^(4)(ψ^')^2
          - 8 ψ^'''ψ^''ψ^'
          + 10 (ψ^'')^3
          )
          + ψ^'' (ψ^')^3
          (
              2 - (ψ^')^3
          ) = 0,
  
  According to conservative form (<ref>) of the equations,
  it can be rewritten as
  
      ψ^''
          + D_λ((ψ^')^-2)
          - 2 γ D_λ(
              (ψ^')^-4(
                      ψ^'''
                      - 2 (ψ^'')^2/ψ^')
          ) = 0.
  
  
  
  
  
  
  
  
  
  
  Using the change
  
      ψ^'' = Θ(ζ),
          ψ^' = ζ,
          
      D_λ = Θ D_ζ,
  
  equation (<ref>) becomes
  
      {
          1 + D_ζ(1/ζ^2)
          - 2γ D_ζ[
              Θ/ζ^4(
                      Θ^' - 2Θ/ζ)
          ]
      }Θ = 0.
  
  If Θ = 0, then one gets the trivial solution
  
      x = λ.
  
  In case Θ≠ 0 equation (<ref>) has the first integral
  
      ζ + 1/ζ^2
          - 2γΘ/ζ^4(
                      Θ^' - 2Θ/ζ) = B = const,
  
  and a particular solution corresponding to Serre's solution (<ref>) is
  
      Θ^2(ζ) = ζ^3/2γ(ζ - 1/R_0)^2(ζ - 1/R_0 + A).
  
  In Eulerian coordinates it has the form
  
      ρ̃_x^2 = ρ̃^3/2γ(
                      1/ρ̃ - 1/R_0)^2
                  (
                      1/ρ̃ - 1/R_0 + A),
                      ρ̃ = ρ̃(t,x),
  
  and in mass Lagrangian coordinates it is
  
      ρ_s^2 = ρ/2γ(
                      1/ρ - 1/R_0)^2
                  (
                      1/ρ - 1/R_0 + A),
                      ρ = ρ(t,s).
  
  Notice that Serre's solution satisfy the higher-order equation (<ref>)
  only if R_0 and A are related as follows
  
      A = 1 - 2R_0^3/2R_0^2.
  
  
  
  
  
  
  
  In order to perform computations in mass Lagrangian coordinates, one must find the initial distribution
  which correspond to solution (<ref>).
  Hence, one must solve the Cauchy problem (<ref>).
  For Serre's solution it has the form
  
      1/(x(0,s))_s = R_0 + A ^2(μ x(0, s)),
          
      x(0, s_0) = x_0,
  
  where
  
      s_0 = ∫_0^x_0 (R_0 + A ^2(μ z))dz
  
  is mass of the fluid at point x=x_0.
  Then, the initial conditions must satisfy the following relations
  
      x(0, s) = z_0(s)/μ,
          where  
        e^2z_0(μ (s_0 - s) + A + z_0 R_0)
          + μ (s_0 - s) - A + z_0 R_0 = 0.
  
  The solution for R_0 = 0.75, γ=1 is given in Figure <ref> (left). The center of the soliton solution is shifted to the point x_c = 50.
  The right side of Figure <ref> demonstrates the numerical error of the transformations.
  The numerical solutions in Eulerian (<ref>) and mass Lagrangian (<ref>)
  coordinates for x ∈ [0,180] are given in Figure <ref>.
  The original analytical solution is not periodic and it is stable for small linear perturbations <cit.>.
  However, numerical calculations implement a periodic like solution (it was mentioned in <cit.>).
  Notice that this does not contradict the linear analysis since numerical calculations introduce finite perturbations. The quasiperiodicity of the numerical solution is confirmed by calculations
  with very small steps. Variation of steps from large to small does not affect the final solution sufficiently.
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
   §.§ Reduction of the scheme
  
  
  Reduction of the invariant scheme  on the subgroup ∂_t + α∂_s
  is the same as  the reduction for the shallow water
  equations scheme <cit.>.
  The reduced scheme is
  
      α^2 ψ̌_λλ
          + -λD(
              1/ψ̂_λψ̌_λ)
          - 2 γα^2 -λD[
              1/(ψ̂_λψ̌_λ)^2(
                  ψ̌_λλλ
                  - 2 ψ̌_λλψ_λλ/ψ_λ)
          ] = 0,
  
  
      Δλ = λ̂ - λ = λ - λ̌
          = const,
  
  where ψ = ψ(λ) = ψ(s - α t).
  
  Equation (<ref>) is a finite-difference analog of equation (<ref>).
  To guarantee that new mesh spacing Δλ matches the original
  mesh nodes on the plane <cit.> (see Figure <ref>), one should assume that
  
      Δλ = h = ατ
  
  
  Below h and τ are considered equal, which corresponds to α=1.
  
  
  
  
  It is worth mentioning that the difference shift operators in mass Lagrangian coordinates
  are related to the shift operators in λ–coordinates as follows
  
      ±τS = ∓λS,
              ± hS = ±λS.
  
  
  
  Changing the variables
  
      ψ_λλ = Θ(ζ),
            ψ̌_λλ = Θ(ζ̌) = Θ̌,
            ζ = ψ_λ,
            -λD = Θ̌-ζD,
  
  one gets a difference analog of ODE (<ref>):
  
      {
              1
              + -ζD(1/ζ̂ζ̌)
              - 2γ-ζD[
                  Θ̌/(ζ̂ζ̌)^2(
                      Θ̌_ζ
                      - 2 Θ/ζ)
              ]
          }Θ̌ = 0,
  
  
      Δζ = ζ̂ - ζ = ζ - ζ̌ = const.
  
  If Θ̌ = 0, then the appropriate solution is x = λ
  as it is in the differential case considered above.
  If Θ̌≠ 0, then one can derive the following difference first integral
  
      ζ̂
          + 1/ζ̂ζ̌
          - 2γΘ̌/(ζ̂ζ̌)^2(
                  Θ̌_ζ
                  - 2 Θ/ζ) = B_1 = const,
  
  which is the finite-difference analogue of integral (<ref>).
  The control of the values of the first integral (<ref>) for  Serre's solution
  near the center of the soliton for the chosen parameters is given in Figure <ref>.
  It shows high accuracy of the first integral conservation.
  
  
  The solution of the reduced scheme for the initial parameters,
  obtained by using Runge-Kutta method for equation (<ref>),
  is presented in Figure <ref>.
  Notice that, as in the previous section, the numerical solution is periodic like.
  
  
  
  
  
  
  
   §.§ Evolution of the Serre's solution for a perturbed scheme
  
  
  Consider the second equation of scheme (<ref>)
  
      D(u) + D(
                  [
                  4/ρρ̌
                  - 2/√(p)( 1/ρ + 1/ρ̌)
                  + 1/p]^-1)
            = 2γD( Q [
                      ǔ_ts - 2 u_sǔ_s √(p)] ),
  
  where Q = (p p̌ρ^2)/(2√(p) - ρ)^2.
  
  The term D(Q ǔ_ts) ∼ǔ_tss̅ is proportional to (τ h^2)^-1
  which takes on  large values. One can avoid it by putting γ∼τ h^2.
  But in this case the equations become the shallow water equations which are out of our interest here.
  Because of the large coefficients even for very small gradients of u
  the scheme cannot be numerically solved by standard methods.
  There are at least two known approaches to correct this situation. The
  first approach <cit.> is based on the splitting the scheme into a system of difference equations
  and
  calculating the flow Q ǔ_ts separately. It is used, for example, in magnetohydrodynamics,
  where the coefficient of thermal conductivity often tends to infinity.
  Another approach <cit.> goes beyond finite difference methods.
  The hybrid scheme is constructed by splitting the Green-Naghdi equation
  into two parts. The finite-difference part approximates the shallow water equation,
  and the finite-volume part of the scheme approximates the γ–terms of the equation.
  
  In order to avoid these difficulties
  we consider the following perturbed version of equation (<ref>)
  
      D(u) + D(
              [
              4/ρρ̌
              - 2/√(p)( 1/ρ + 1/ρ̌)
              + 1/p]^-1)
        = 2γD( Q[
                  αǔ_ts - 2 β u_sǔ_s √(p)] ),
  
  where 0 < α⩽ 1 and 0 < β⩽ 1 are some constant values.
  Tending the coefficient α to zero, one can neglect
  effects of the term proportional to ǔ_tss̅.
  This particular problem is also of independent interest.
  To clarify the physical meaning of this case, we refer to the paper <cit.>,
  where it was considered a similar to (<ref>) perturbed form
  of the Green-Naghdi equations in Eulerian coordinates.
  The perturbation coefficient α is responsible for the stability of the soliton solution.
  If its value tends to zero, the solution show dispersive qualities in time. In contrast to α,
  the value of the perturbation coefficient β does not essentially affect the profile of the free surface.
  
  In particular, the evolution of the Serre's solution for α=0.001, β=1, R_0=0.75 and γ=1 is presented in Figure <ref>.
  For the represented solution we explore a viscous version of the scheme, where the following change was used
  
      ρ_m^n ↦ρ_m^n - ντD(ρ_m^n),
           
       u_m^n ↦ u_m^n - ντD(ρ_m^n u_m^n),
           
       m > 2.
  
  Here ν is a small viscosity coefficient.
  The dispersive wave forms are very similar to the results presented in <cit.> for small values of α.
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  § CONCLUSION
  
  
  Group analysis of the one-dimensional Green-Naghdi equations
  describing the behavior of fluid flow over uneven bottom
  topography is given in the present paper. The Green-Naghdi equations
  are considered in two forms: in the classical form (<ref>),
  (<ref>) and in the mild slope approximation form
  (<ref>), (<ref>) which is of the same
  order as the original Green-Naghdi equations. Analysis of the
  studied equations is performed in Lagrangian coordinates. Working in
  the Lagrangian coordinates allowed us to find Lagrangians turning
  the analyzed equations into the Euler-Lagrange equations. It is
  shown that equations (<ref>), (<ref>)
  with a flat bottom topography H_b=qx+β are locally
  equivalent to the Green-Naghdi equations with a horizontal bottom
  topography H_b=const. Complete group classification of
  both cases of the Green-Naghdi equations with respect to the
  function H_b describing the uneven bottom topography is
  presented. Applying the Noether theorem, the developed
  Lagrangians and performed group classification, conservation laws
  of the one-dimensional Green-Naghdi equations with uneven bottom
  topography are obtained.
  
  
  An invariant conservative finite-difference scheme is constructed
  for the Green-Naghdi equations for the case of a flat bottom
  topography. The scheme possesses the conservation laws of mass,
  momentum, energy and the center-of-mass law. This scheme is also
  represented in hydrodynamic variables. The representation in
  hydrodynamic variables simplifies its numerical implementation. The
  reduction of the invariant scheme on a subgroup is carried out
  similarly to the reduction of the corresponding differential
  equations. As a result of the reduction an ordinary
  finite-difference equation is obtained. This equation possesses a
  first integral, which is well preserved on  Serre's exact solution.
  Using the example of Serre's solution  further numerical analysis of
  the scheme is performed. The stationary solution, obtained by the
  proposed scheme, has the same qualitative properties as the
  solutions calculated by the Runge-Kutta methods. The time evolution
  of Serre's solution is considered by the example of a specific
  perturbed version of the scheme which allows one to avoid working
  with large velocity gradients terms. The latter invariant scheme was
  generalized for an arbitrary bottom topography. This scheme also
  possesses the conservation laws of mass and energy.
  
  
  
  § ACKNOWLEDGEMENTS
  
  The research  was supported by Russian Science Foundation Grant No
  18-11-00238 "Hydrodynamics-type equations: symmetries, conservation
  laws, invariant difference schemes".
  E.I.K. acknowledges Suranaree University of Technology for
  Full-time Master Researcher Fellowship.
  
  
  
  
  
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