We describe a constructive algorithm for obtaining smooth
solutions of a nonlinear, nonhyperbolic pair of balance laws
modeling incompressible two-phase flow in one space dimension and
time. Solutions are found as stationary solutions of a related
hyperbolic system, based on the introduction of an artificial time
variable.
As may be expected for such nonhyperbolic systems, in general the
solutions obtained do not satisfy both components of the given
initial data. This deficiency may be overcome, however, by
introducing an alternative “solution" satisfying both components
of the initial data and an approximate form of a corresponding
linearized system.