  The HLLC Riemann solver, which resolves both the shock waves and contact
discontinuities, is popular to the computational fluid dynamics community
studying compressible flow problems with mesh methods. Although it was reported
to be used in meshless methods, the crucial information and procedure to
realise this scheme within the framework of meshless methods were not clarified
fully. Moreover, the capability of the meshless HLLC solver to deal with
compressible liquid flows is not completely clear yet as very few related
studies have been reported. Therefore, a comprehensive investigation of a
dimensional non-split HLLC Riemann solver for the least-square meshless method
is carried out in this study. The stiffened gas equation of state is adopted to
capacitate the proposed method to deal with single-phase gases and/or liquids
effectively, whilst direct applying the perfect gas equation of state for
compressible liquid flows might encounter great difficulties in correlating the
state variables. The spatial derivatives of the Euler equations are computed by
a least-square approximation and the flux terms are calculated by the HLLC
scheme in a dimensional non-split pattern. Simulations of gas and liquid shock
tubes, moving shock passing a cylinder, internal supersonic flows in channels
and external transonic flows over aerofoils are accomplished. The current
approach is verified by extensive comparisons of the produced numerical
outcomes with various available data such as the exact solutions, finite volume
mesh method results, experimental measurements or other reference results.
