The modification of Smoothed Particle Hydrodynamics (SPH) method with Riemann Solver is called Godunov SPH. We further extend the Godunov SPH to the description of a medium with negative pressure. Under certain circumstances, the SPH method shows an unphysical instability that results in particle clustering. This instability is called the tensile instability. The tensile instability occurs in positive pressure regions in a regular fluid if a very large number of neighbor particles are used with certain shapes of kernel functions, and it is significant in negative pressure regions that emerge in stretched elastic bodies. We must suppress the tensile instability in SPH for calculations of elastic bodies. In this study, we develop a new technique to remove the tensile instability by extending the Godunov SPH method and conducting a linear stability analysis of the equation of motion for the extended method. We find that the tensile instability can be suppressed by choosing an appropriate order of interpolation in the equation of motion of the Godunov SPH method. We also derive an analytic solution for a Riemann solver for a simple equation of state of an elastic body, and construct a Godunov SPH method for the equation of state that allows negative pressure.
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