A discrete model for an ill-posed nonlinear parabolic PDE

https://doi.org/10.1016/S0167-2789(01)00350-5Get rights and content

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.

Section snippets

Background

The degenerate parabolic PDE: ∂v∂t=divR∇v|∇v|,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

The continuous two-dimensional model

The dependent variable v=v(x,y,t) in Eq. (1.1) may be thought of as the velocity in the z-direction in a block of material undergoing anti-plane shearing. The equation for conservation of momentum equates acceleration, tv, at constant density (normalized to unity), with the divergence of stress, ∇·τ. Modeling stress by the vector τ=R∇v/|∇v|, in which R is the matrix representing a rotation counter-clockwise through a constant angle α: R=cosαsinαsinαcosα,with 0<α<π/2, we arrive at Eq. (1.1).

The discrete steady-state solutions

The trivial solution, wn≡0, is an equilibrium of Eq. (2.25). In this section we determine the nontrivial equilibrium solutions of Eq. (2.25). We show that the number of solutions depends on φ, corresponding to the three cases identified in Eq. (2.23).

Stability of the equilibria

The following result on the stability and long-time evolution of solutions of Eq. (2.25)for the cases defined in Eq. (3.7) will be proved in this section.

Proposition 3

On the stability of equilibrium solutions of Eq. (2.25):

Case 1′: Only strong shock solutions of Eq. (2.25) with a single jump discontinuity are stable.

Case 2′: Strong shock solutions with a single discontinuity and the trivial zero solution are the only stable equilibria.

Case 3′: The zero solution is stable and in fact globally attracting.

In

Representative dynamic simulations

In this section, we present the results of a representative set of numerical simulations of the dynamics of system (2.25). These simulations illustrate some of the differences in behavior in the three cases in Proposition 3. The simulations also guide the analysis of the nonlinear dynamics in the following sections.

We begin with a brief discussion of the numerical methods used for the simulations. As was discussed above, while the continuum PDE (2.10) is ill-posed, for any finite N the discrete

Scaling laws and considerations of the continuum limit

In this section, we study in detail the coarsening dynamics of solutions of Eq. (2.25) in case 1′ with initial conditions w(x,0)=w(x)=Asin(πx),with 0<|A|≪Amax,where Amax=|smax|/π. For this range of A, the condition xw(x)>smax,is satisfied everywhere and coarsening dynamics is observed throughout the entire interval, 0≤x≤1. Recall from Section 2.2 that in case 1, the trivial solution w≡0 is linearly ill-posed and small-amplitude initial data satisfying Eq. (6.2)will be unstable everywhere in 0≤x

Intermediate dynamics: coarsening

The dynamic simulations of the previous sections suggest that while we have thoroughly studied the steady-states and asymptotic stability, a complete understanding of the behavior of Eq. (2.25)requires an examination of the complicated intermediate dynamics of the system as well. For ill-posed initial data, i.e. for data with max(wx)>smax, the formation of a large number of jumps in the solution creates very unstable intermediate states. From Proposition 4, we know that there are no stable

Acknowledgments

DGS was supported by NSF Grant DMS 9803305. MS was supported by ARO Grant DAAG55-98-1-0128 and NSF Grants DMS 9818900 and DMS 0073841. TPW was supported by an Alfred P. Sloan foundation fellowship. We wish to thank D. Lohse for sharing a preprint of his group’s work with us [35], [36].

References (36)

  • J.W. Cahn

    Spinodal decomposition

    Trans. Metal. Soc. AIME

    (1968)
  • F. Catté et al.

    Image selective smoothing and edge detection by nonlinear diffusion

    SIAM J. Numer. Anal.

    (1992)
  • P.G. de Gennes

    Granular matter: a tentative view

    Rev. Mod. Phys.

    (1999)
  • J. Eggers

    Sand as Maxwell’s demon

    Phys. Rev. Let.

    (1999)
  • L.C. Evans et al.

    Motion of level sets by mean curvature. I

    J. Differential Geom.

    (1991)
  • F.X. Garaizar

    Numerical computations for anti-plane shear in a granular flow model

    Quart. Appl. Math.

    (1994)
  • K. Höllig, J.A. Nohel, A diffusion equation with a nonmonotone constitutive function, in: Systems of Nonlinear Partial...
  • J.M. Hyman, B. Nicolaenko, S. Zaleski, Order and complexity in the Kuramoto–Sivashinsky model of weakly turbulent...
  • Cited by (25)

    • Volume scavenging of networked droplets

      2019, Physica D: Nonlinear Phenomena
    • Solution of the scalar wave equation over very long distances using nonlinear solitary waves: Relation to finite difference methods

      2012, Journal of Computational Physics
      Citation Excerpt :

      With this equilibrium state, error accumulation does not occur – high order becomes unnecessary and NSW’s can stably propagate indefinitely without spreading or dispersing, even with moderate discretization error. (ii) In order to preserve a pulse indefinitely under perturbations from discretization that can cause the pulse to spread, it is necessary that the outer regions of that pulse should be contracting, or anti-diffusive [14]. On the other hand, the inner region of a pulse should be diffusive in order to prevent divergence.

    • Transient and self-similar dynamics in thin film coarsening

      2009, Physica D: Nonlinear Phenomena
      Citation Excerpt :

      The energetics favor successive redistribution of the fluid mass and coalescence of drops in a process called coarsening. Coarsening dynamics occur in many settings: fluid dynamics [8], granular materials [9–11], materials science [12,11,13–16], and image processing [17–19]. In each application strong instabilities rapidly drive the system into a very complicated early state.

    • Convergence for long-times of a semidiscrete peronamalik equation in one dimension

      2011, Mathematical Models and Methods in Applied Sciences
    View all citing articles on Scopus
    View full text