**Papers Published**

- Thomas P. Witelski, D. G. Schaeffer, and M. Shearer,
*A Discrete Model for an Ill-posed Nonlinear Parabolic PDE*, Physica D, 160 (2001) pp. 189--221. [S0167-2789(01)00350-5] .

(last updated on 2002/10/10)**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.