subroutine  update_q(qnp1, qn, x_i, FF_iph, delta_t, Nx, Ng)

c------------------------------------------------------------
c        This routine implements the time-stepping procedure
c        for the Runge-Kutta method.  Note that the numerical 
c        flux vector FF_iph(i) is the flux evaluated at 
c        x_(i+1/2).
c------------------------------------------------------------

         implicit      none

         character*8   cdnm
         parameter   ( cdnm = 'update_q' ) 

         logical       ltrace
         parameter   ( ltrace = .false. )

         integer       Nx
         integer       Ng

         real*8        dx
         real*8        delta_t

         real*8        qnp1     ( Nx )  
         real*8        qn       ( Nx )  
         real*8        x_i      ( Nx )
         real*8        FF_iph   ( Nx )

         integer       i

c============================================================
c============================================================
c------------------------------------------------------------
c Note that x_i is the coordinates of the cell centres.
c We only want the difference (x_(i+1/2) - x_(i-1/2)) which is 
c constant and equal to (x_(i+1) - x_(i)).
c------------------------------------------------------------
         dx = x_i(2) - x_i(1)
         if ( ltrace ) then
            write(0,*) cdnm,':       Nx = ', Nx
            write(0,*) cdnm,':       Ng = ', Ng
            write(0,*) cdnm,':       dx = ', dx
            write(0,*) cdnm,':  delta_t = ', delta_t
         endif

c------------------------------------------------------------
c We only use the equation to update the real cells
c not the ghost cells.
c------------------------------------------------------------
          
         do i = Ng+1, Nx-Ng
            qnp1(i) = qn(i) - delta_t / dx *
     &             ( FF_iph(i) - FF_iph(i-1) )
         enddo
         return
         end