Implementations and tests of Godunov-type particle hydrodynamics

Godunov-type particle hydrodynamics (GPH) is described. GPH inherits many good features from smoothed particle hydrodynamics (SPH), but it uses a Riemann solver to obtain the hydrodynamic acceleration and the rate of change of the internal energy of each particle. The grid-free nature of GPH converts a multidimensional problem into a locally one-dimensional problem, so that one only has to solve a one-dimensional Riemann problem, even in a globally three-dimensional situation. By virtue of the Riemann solver, it is unnecessary to introduce artificial viscosity in GPH. We have derived four different versions of GPH, and have performed a von Neumann stability analysis to understand the nature of GPH. GPH is stable for all wavelengths, while SPH is unstable for certain wavelengths. We have also performed eight tests in order to evaluate the performance of GPH. The results show that GPH can describe shock waves without artificial viscosity and prevents particle penetration. Furthermore, GPH shows better performance than SPH in a test involving velocity shear. GPH is easily implemented from SPH by simple replacement of the artificial viscosity with a Riemann solver, and appears to have some useful advantages over standard SPH.