A discrete model for an ill-posed nonlinear parabolic PDE

Published

Journal Article

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. © 2001 Elsevier Science B.V. All rights reserved.

Full Text

Duke Authors

Cited Authors

  • Witelski, TP; Schaeffer, DG; Shearer, M

Published Date

  • December 15, 2001

Published In

Volume / Issue

  • 160 / 3-4

Start / End Page

  • 189 - 221

International Standard Serial Number (ISSN)

  • 0167-2789

Digital Object Identifier (DOI)

  • 10.1016/S0167-2789(01)00350-5

Citation Source

  • Scopus