Preprint E30/2012
EFFICIENT NUMERICAL CALCULATION OF LARGE SHOCKS ARISING FROM SMALL DATA

Dan Marchesin | Matos, Vítor | Hime, Gustavo

**Keywords: **
Conservation laws | large viscous solutions | parallel computation.

In a recent paper, stable solutions were proved to arise
from small amplitude data for an example of conservation laws with a parabolic term possessing the identity as viscosity matrix. The initial data analyzed therein lies outside the elliptic region, and the viscous solutions are related to a bifurcation of planar vector fields with nilpotent singularities studied by Dumortier, Roussarie and Sotomaior (DRS). For non–identity viscosity matrices, the DRS bifurcation is located at the border of the Majda–Pego instability region. The stability proof does not easily generalize for other viscosity matrices: in this work, this issue is studied using numerical simulation. We use a custom high–performance parallel Newton solver for the non–linear system arising from the Crank–Nicolson finite difference discretization. We present numerical evidence that the stable shocks do not arise from arbitrary small Riemann data with non–identity viscous matrices. However, some of the stable shocks arise from small data if one initial state lies in the elliptic region. In some of these cases, the
solution seems to be unaffected by small perturbations.