  The Navier-Stokes-Korteweg (NSK) equations are a classical diffuse-interface
model for compressible two-phase flow. As direct numerical simulations based on
the NSK system are quite expensive and in some cases even impossible, we
consider a relaxation of the NSK system, for which robust numerical methods can
be designed. However, time steps for explicit numerical schemes depend on the
relaxation parameter and therefore numerical simulations in the relaxation
limit are very inefficient. To overcome this restriction, we propose an
implicit-explicit asymptotic-preserving finite volume method. We prove that the
new scheme provides a consistent discretization of the NSK system in the
relaxation limit and demonstrate that it is capable of accurately and
efficiently computing numerical solutions of problems with realistic density
ratios and small interfacial widths.
