The Navier-Stokes-Korteweg (NSK) equations are a classica
l diffuse-interface
model for compressible two-phase flows. As direct numerical
simulations based
on the NSK system are quite expensive and in some cases even im
possible, we
consider a relaxation of the NSK system, for which robust num
erical methods
can be designed. However, time steps for explicit numerical
schemes depend on
the relaxation parameter and therefore numerical simulati
ons in the relaxation
limit are very inefficient. To overcome this restriction, we p
ropose an implicit-
explicit asymptotic-preserving finite volume method. We pr
ove that the new
scheme provides a consistent discretization of the NSK syst
em in the relaxation
limit and demonstrate that it is capable of accurately and effi
ciently computing
numerical solutions of problems with realistic density rat
ios and small interfacial
widths.