  A numerical procedure was developed for solving equations for compressible
granular multiphase flows in which the particle volume fraction can range
dynamically from very dilute to very dense. The procedure uses a
low-dissipation and high-order numerical method that can describe shocks and
incorporates a particulate model based on kinetic theory. The algorithm
separates edges of a computational cell into gas and solid sections where gas-
and granular-phase Riemann problems are solved independently. Solutions from
these individual Riemann problems are combined to assemble the fully coupled
convective fluxes and nonconservative terms for both phases. The technique
converges under grid refinement even with very high volume fraction granular
interfaces. The method can advect sharp granular material interfaces that
coincide with multi-species gaseous contact surfaces without violating the
pressure nondisturbing condition. The procedure also reproduces known features
from multiphase shock tube problems, granular shocks, transmission angles of
compaction waves, and shock wave and dust-layer interactions. This approach is
relatively straightforward to use in an existing code based on Godunov's method
and can be constructed from standard compressible solvers for the gas-phase and
a modified AUSM$^+$-up scheme for the particle phase.
