  A second-order accurate and robust numerical scheme is developed for the
Kapila model to simulate compressible multiphase flows. The scheme is
formulated within the temporal-spatial coupling framework with the generalized
Riemann problem (GRP) solver applied as the cornerstone. The use of the GRP
solver enhances the capability of the resulting scheme to handle the stiffness
of the Kapila model in two ways. Firstly, in addition to Riemann solutions, the
time derivatives of flow variables at cell interfaces are obtained by the GRP
solver. The coupled values, i.e. Riemann solutions and time derivatives, lead
to a straightforward approximation to the velocity divergence at the next time
level, enabling a semi-implicit time discretization to the volume fraction
equation. Secondly, the use of time derivatives enables numerical fluxes to
comprehensively account for the effect of the source term, which includes
interactions between phases. The robustness of the resulting numerical scheme
is therefore further improved. Several challenging numerical experiments are
conducted to demonstrate the performance of the proposed finite volume scheme.
In particular, a test case with a nonlinear smooth solution is designed to
verify the numerical accuracy.
