Orateur
Description
This talk deals with the stability analysis of discrete shock profiles
for systems of conservation laws. These profiles correspond to
approximations of shocks of systems of conservation laws by
conservative finite difference schemes. Discontinuous solutions
appear naturally in the study of systems of conservation laws,
which can model many physical situations, such as gas dynamics.
Existence and stability of discrete shock profiles for each stable
shock of the approximated system of conservation laws is seen as an
improved consistency condition and implies that the finite difference
scheme should be able to approach discontinuities fairly precisely.
The aim of the talk is to review some stability results regarding
discrete shock profiles and to present a recent effort to extend them.
More precisely, most results known up until recently are focused on the
stability of discrete shock profiles associated with shocks of small
amplitude. The talk will focus on a nonlinear orbital stability result
for discrete shock profiles in quite a general setting, where the
smallness assumption on the shock's amplitude is replaced by a spectral
stability assumption on the linear operator obtained by linearizing the
numerical scheme about the discrete shock profile. This nonlinear
orbital stability result relies on a precise description of the Green's
function of the linearization about discrete profiles.
