# Preprint 2009-034

# The linear appearance theorem for a class of non linear non homogeneous hyperbolic systems involving a transport equation

## Alain-Yves LeRoux

**Abstract:**
The linear appearance theorem states that,
for a wide class of non linear hyperbolic systems,
when a source term occurs,
some solutions are also solutions to a linear homogeneous system,
which means that the corresponding profiles
are simply translated with a constant velocity.
This allows to solve some problems
by combining a sequence of such profiles
separated by shock waves.
Several examples are reported,
such as the roll waves in hydraulics,
acoustics waves as solution of gas dynamics systems in a duct,
or a rarefaction wave in fluids,
seen as limit of kinds of saw waves towards a Riemann invariant,
as for the nonlinear homogeneous case.
Some new numerical schemes adapted to the source term are presented,
and tested on examples.