A discrete model for an ill-posed nonlinear parabolic PDE

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. © 2001 Elsevier Science B.V. All rights reserved.Keywords: Nonlinear PDE; Ill-posed equations; Nonlinear diffusion; Granular medium; Shear bands BackgroundThe degenerate parabolic PDE:in which R is a rotation matrix, arises from a simplified model of the velocity field of a sheared granular material [33]. This equation is ill-posed, a property typical of continuum models of granular media [25,31,32,34]. To study the dynamics of this model, in this paper, we analyze one particular finite-difference approximation of the PDE (1.1).More precisely, we discretize in space, and study the resulting system of ODEs, which we refer to as the discrete model. This model is a form of regularization of Eq. (1.1); the discrete model is well-posed, mollifying instabilities at the highest frequencies.

Upper bounds on the coarsening rate of discrete, ill‐posed nonlinear diffusion equations

“…Denoting population density by v, (1) arises for external fields proportional to the number of entities present. The dynamics of (8) share many qualitative features with an ill-posed nonlinear PDE that arises in image processing [25,26] and granular flow [29,28]:…”

“…Intending to smooth noisy regions of the image without blurring edges (object boundaries), they required that f , the primitive of R, satisfy exactly the conditions discussed above. Equation (13) has also been studied as a simplified one-dimensional model for the formation of shear bands in a granular medium [29]. In this case, f is defined by…”

“…Although the equations are well-posed, they resemble finite difference approximations of ill-posed nonlinear diffusion equations, leading us to call them discrete ill-posed nonlinear diffusion equations in the title. The equations arise in the study of population dynamics as a class of reinforced random walks on a lattice [14,24], and in granular flow as a simplified one-dimensional model for the formation of shear bands by anti-plane shear in a granular medium [29]. They also have strong connections to an algorithm introduced by Perona and Malik for image denoising [10,26].…”

“…In granular flow, the coarsening represents a merging of shear bands, and in population dynamics it represents collective movement to population centers. Our contribution in this paper is to show how recently developed methods for analyzing coarsening rates can be applied to these models to answer questions raised previously [29].…”

“…Indeed the coarsening rates depend on the uniform grid cell width, h, for the finite differences involved. Numerical experiments show that coarsening slows to a halt as h → 0 [29], so any useful bound on the coarsening rates must account for this dependence. Although our results can be stated in arbitrary space dimensions, for clarity we present them for one and two space dimensions only, and limit our discussion in the first few sections to just one dimension.…”