A formulation of consistent particle hydrodynamics in strong form

In fluid dynamical simulations in astrophysics, large deformations are common and surface tracking is sometimes necessary. The smoothed particle hydrodynamics (SPH) method has been used in many 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 consistency, and thus the accuracy is of the 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 compressible fluid. Our method is based on a kernelweighted high-order fitting polynomial for intensive variables. With this approach, we can construct a scheme which solves all of the three problems 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 excellent results. 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).