A new model for interface problems with capillary effects in compressible fluids is presented together with a specific numerical method to treat capillary flows and pressure waves propagation. This new multiphase hyperbolic model is in agreement with physical principles of conservation and respects the second law of thermodynamics. A new numerical method is also proposed where the global system of equations is split into several submodels. Each submodel is hyperbolic and can be solved with an adequate numerical method. This model is tested on experiments on droplet breakup induced by a shock wave. (This is a joint work with K. Schmidmayer, F. Petitpas, E. Daniel, and N. Favrie.)