  Godunov Smoothed Particle Hydrodynamics (Godunov SPH) method is a
computational fluid dynamics method that utilizes a Riemann solver and achieves
the second-order accuracy in space. In this paper, we extend the Godunov SPH
method to elastic dynamics by incorporating deviatoric stress tensor that
represents the stress for shear deformation or anisotropic compression.
Analogously to the formulation of the original Godunov SPH method, we formulate
the equation of motion, the equation of energy, and the time evolution equation
of deviatoric stress tensor so that the resulting discretized system achieves
the second-order accuracy in space. The standard SPH method tends to suffer
from the tensile instability that results in unphysical clustering of particles
especially in tension-dominated region. We find that the tensile instability
can be suppressed by selecting appropriate interpolation for density
distribution in the equation of motion for the Godunov SPH method even in the
case of elastic dynamics. Several test calculations for elastic dynamics are
performed, and the accuracy and versatility of the present method are shown.
