The extension of time-marching computations to fluids with arbitrary e
quations of state is demonstrated by means of stability analyses, simp
lified problems, and practical applications. Most of the examples use
the properties of supercritical hydrogen for which the density varies
by more than an order of magnitude for small changes in pressure and t
emperature, but representative computations for incompressible fluids
and perfect gases are also given to demonstrate the generality of the
procedure. Because representative how velocities in typical supercriti
cal fluids applications are much lower than the speed of sound, conver
gence enhancement through eigenvalue control is often necessary. This
is accomplished through a generalization of earlier preconditioning me
thods that enables efficient computation of arbitrary equation of stat
e fluids, perfect gases, and incompressible fluids by a single procedu
re. The present approach thus provides a single method that is uniform
ly applicable to all equations of state.