Numerische Analyse äquivarianter
Evolutionsgleichungen
11-016 Jens Rottmann-Matthes.
Stability and freezing of nonlinear waves in
first order hyperbolic PDEs
It is a well-known problem to derive nonlinear stability of a
traveling wave from the spectral stability of a linearization.
In this paper we prove such a result for a large class of
hyperbolic systems.
To cope with the unknown asymptotic phase, the problem is
reformulated as a partial differential algebraic equation for
which asymptotic stability becomes the usual Lyapunov
stability. The stability proof is then based on linear
estimates from a previous paper and a careful analysis of the
nonlinear terms. Moreover, we show that the freezing method
is well-suited for the long time simulation and numerical
approximation of the asymptotic behavior.
The theory is illustrated by two numerical examples, including a
hyperbolic version of the Hodgkin-Huxley equations.
