  This paper investigates the application of mesh adaptation techniques in the
Non-Ideal Compressible Fluid Dynamic (NICFD) regime, a region near the
vapor-liquid saturation curve where the flow behavior significantly departs
from the ideal gas model, as indicated by a value of the fundamental derivative
of gasdynamics less than one. A recent interpolation-free finite-volume
adaptive scheme is exploited to modify the grid connectivity in a conservative
way, and the governing equations for compressible inviscid flows are solved
within the Arbitrary Lagrangian Eulerian framework by including special
fictitious fluxes representing volume modifications due to mesh adaptation.The
absence of interpolation of the solution to the new grid prevents spurious
oscillations that may make the solution of the flow field in the NICFD regime
more difficult and less robust.Non-ideal gas effects are taken into account by
adopting the polytropic Peng-Robinson thermodynamic model. The numerical
results focus on the problem of a piston moving in a tube filled with siloxane
$\mathrm{MD_4M}$, a simple configuration which can be the core of experimental
research activities aiming at investigating the thermodynamic behavior of NICFD
flows. Several numerical tests involving different piston movements and initial
states in 2D and 3D assess the capability of the proposed adaption technique to
correctly capture compression and expansion waves, as well as the generation
and propagation of shock waves, in the NICFD and in the non-classical regime.
