A discrete model for an ill-posed nonlinear parabolic PDE
Section snippets
Background
The 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
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, . Modeling stress by the vector , in which R is the matrix representing a rotation counter-clockwise through a constant angle α: 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 with where Amax=|smax|/π. For this range of A, the condition 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].
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