Discrete Model for an Ill-Posed Nonlinear Parabolic PDE

摘要

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 1/t-cubed. From this scaling law we show that the time-scale of the coarsening phase in the evolution of this 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.