Papers Published
Abstract:
We study a finite-difference discretization of an ill-posed
nonlinear parabolic partial differential equation. The PDE
is the one-dimensional version of a simplified
two-dimensional model for the formation of shear bands via
anti-plane shear of a granular medium. For the discretized
initial value problem, we derive analytically, and observed
numerically, a two-stage evolution leading to a
steady-state: (i) an initial growth of grid-scale
instabilities, and (ii) coarsening dynamics. Elaborating the
second phase, at any fixed time the solution has a piecewise
linear profile with a finite number of shear bands. In this
coarsening phase, one shear band after another collapses
until a steady-state with just one jump discontinuity is
achieved. The amplitude of this steady-state shear band is
derived analytically, but due to the ill-posedness of the
underlying problem, its position exhibits sensitive
dependence. Analyzing data from the simulations, we observe
that the number of shear bands at time $t$ decays like
$t^{-1/3}$. From this scaling law we show that the
time-scale of the coarsening phase in the evolution
of this model for granular media critically depends on the
discreteness of the model.
Our analysis also has implications to related ill-posed
nonlinear PDEs for the one-dimensional Perona-Malik equation
in image processing and to models for clustering
instabilities in granular materials.