  We present a framework for constructing a first-order hyperbolic system whose
solution approximates that of a desired higher-order evolution equation.
Constructions of this kind have received increasing interest in recent years,
and are potentially useful as either analytical or computational tools for
understanding the corresponding higher-order equation. We perform a systematic
analysis of a family of linear model equations and show that for each member of
this family there is a stable hyperbolic approximation whose solution converges
to that of the model equation in a certain limit. We then show through several
examples that this approach can be applied successfully to a very wide range of
nonlinear PDEs of practical interest.
