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Publications [#244214] of Thomas P. Witelski
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 Witelski, TP; Schaeffer, DG; Shearer, M, A discrete model for an illposed nonlinear parabolic PDE,
Physica D: Nonlinear Phenomena, vol. 160 no. 34
(2001),
pp. 189221, ISSN 01672789 [S01672789(01)003505], [doi]
(last updated on 2017/12/15)
Abstract: We study a finitedifference discretization of an illposed nonlinear parabolic partial differential equation. The PDE is the onedimensional version of a simplified twodimensional model for the formation of shear bands via antiplane shear of a granular medium. For the discretized initial value problem, we derive analytically, and observed numerically, a twostage evolution leading to a steadystate: (i) an initial growth of gridscale 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 steadystate with just one jump discontinuity is achieved. The amplitude of this steadystate shear band is derived analytically, but due to the illposedness 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 t1/3. From this scaling law, we show that the timescale 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 illposed nonlinear PDEs for the onedimensional PeronaMalik equation in image processing and to models for clustering instabilities in granular materials. © 2001 Elsevier Science B.V. All rights reserved.


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