  We present a new hydrodynamic scheme named Godunov Density-Independent
Smoothed Particle Hydrodynamics (GDISPH), that can accurately handle shock
waves and contact discontinuities without any manually tuned parameters. This
is in contrast to the standard formulation of smoothed particle hydrodynamics
(SSPH), which requires the parameters for an artificial viscosity term to
handle the shocks and struggles to accurately handle the contact
discontinuities due to unphysical repulsive forces, resulting in surface
tension that disrupts pressure equilibrium and suppresses fluid instabilities.
While Godunov SPH (GSPH) can handle the shocks without the parameters by using
solutions from a Riemann solver, it still cannot fully handle the contact
discontinuities. Density-Independent Smoothed Particle Hydrodynamics (DISPH),
one of several schemes proposed to handle contact discontinuities more
effectively than SSPH, demonstrates superior performance in our tests involving
strong shocks and contact discontinuities. However, DISPH still requires the
artificial viscosity term. We integrate the Riemann solver into DISPH in
several ways, yielding some patterns of GDISPH. The results of standard tests
such as the one-dimensional Riemann problem, pressure equilibrium,
Sedov-Taylor, and Kelvin-Helmholtz tests are favourable to GDISPH Case 1 and
GDISPH Case 2, as well as DISPH. We conclude that GDISPH Case 1 has an
advantage over GDISPH Case 2, effectively handling shocks and contact
discontinuities without the need for specific parameters or introducing any
additional numerical diffusion.
