  In fluid dynamical simulations in astrophysics, large deformations are common
and surface tracking is sometimes necessary. Smoothed Particle Hydrodynamics
(SPH) method has been used in many of such simulations. Recently, however, it
has been shown that SPH cannot handle contact discontinuities or free surfaces
accurately. There are several reasons for this problem. The first one is that
SPH requires that the density is continuous and differentiable. The second one
is that SPH does not have the consistency, and thus the accuracy is zeroth
order in space. In addition, we cannot express accurate boundary conditions
with SPH. In this paper, we propose a novel, high-order scheme for
particle-based hydrodynamics of compress- ible fluid. Our method is based on
kernel-weighted high-order fitting polynomial for intensive variables. With
this approach, we can construct a scheme which solves all of the three prob-
lems described above. For shock capturing, we use a tensor form of
von-Neumann-Richtmyer artificial viscosity. We have applied our method to many
test problems and obtained excel- lent result. Our method is not conservative,
since particles do not have mass or energy, but only their densities. However,
because of the Lagrangian nature of our scheme, the violation of the
conservation laws turned out to be small. We name this method Consistent
Particle Hydrodynamics in Strong Form (CPHSF).
