November  2016, 36(11): 6379-6411. doi: 10.3934/dcds.2016076

Convergence of space-time discrete threshold dynamics to anisotropic motion by mean curvature

1. 

Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, United States

2. 

Department of Mathematics, Purdue University, 150 N. University Street, West Lafayette, IN 47907

Received  January 2015 Revised  July 2016 Published  August 2016

We analyze the continuum limit of a thresholding algorithm for motion by mean curvature of one dimensional interfaces in various space-time discrete regimes. The algorithm can be viewed as a time-splitting scheme for the Allen-Cahn equation which is a typical model for the motion of materials phase boundaries. Our results extend the existing statements which are applicable mostly in semi-discrete (continuous in space and discrete in time) settings. The motivations of this work are twofolds: to investigate the interaction between multiple small parameters in nonlinear singularly perturbed problems, and to understand the anisotropy in curvature for interfaces in spatially discrete environments. In the current work, the small parameters are the spatial and temporal discretization step sizes: $\triangle x = h$ and $\triangle t = \tau$. We have identified the limiting description of the interfacial velocity in the (i) sub-critical ($h \ll \tau$), (ii) critical ($h = O(\tau)$), and (iii) super-critical ($h \gg \tau$) regimes. The first case gives the classical isotropic motion by mean curvature, while the second produces intricate pinning and de-pinning phenomena, and anisotropy in the velocity function of the interface. The last case produces no motion (complete pinning).
Citation: Oleksandr Misiats, Nung Kwan Yip. Convergence of space-time discrete threshold dynamics to anisotropic motion by mean curvature. Discrete and Continuous Dynamical Systems, 2016, 36 (11) : 6379-6411. doi: 10.3934/dcds.2016076
References:
[1]

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, National Bureau of Standards, Applied Mathematics Series, 1964.

[2]

G. Barles and C. Georgelin, A simple proof of convergence for an approximation scheme for computing motions by mean curvature, SIAM J. Numer. Anal., 32 (1995), 484-500. doi: 10.1137/0732020.

[3]

G. Barles, H. Soner and P. Souganidis, Front propagation and phase field theory, SIAM J. Control Optim., 31 (1993), 439-469. doi: 10.1137/0331021.

[4]

B. Bence, J. Merriman and S. Osher, Motion of multiple functions: A level set approach, J. Comput. Phys., 112 (1994), 334-363. doi: 10.1006/jcph.1994.1105.

[5]

L. Bronsard and R. Kohn, Motion by mean curvature as the singular limit of Ginzburg-Landau dynamics, J. Differential Equations, 90 (1991), 211-237. doi: 10.1016/0022-0396(91)90147-2.

[6]

X. Chen, Generation and propagation of interfaces for reaction-diffusion equations, J. Differential Equations, 96 (1992), 116-141. doi: 10.1016/0022-0396(92)90146-E.

[7]

X. Chen, C. Elliott, A. Gardiner and J. Zhao, Convergence of numerical solutions to the Allen-Cahn equation, Appl. Anal., 69 (1998), 47-56 .

[8]

L. Evans, Convergence of an algorithm for mean curvature motion, Indiana Univ. Math. J., 42 (1993), 533-557. doi: 10.1512/iumj.1993.42.42024.

[9]

L. Evans, H. Soner and P. Souganidis, Phase transitions and generalized motion by mean curvature, Comm. Pure Appl. Math., 45 (1992), 1097-1123. doi: 10.1002/cpa.3160450903.

[10]

T. Ilmanen, Convergence of the Allen-Cahn equation to Brakke's motion by mean curvature, J. Differential Geom., 38 (1993), 417-461.

[11]

H. Ishii, G. Pires and P. Souganidis, Threshold dynamics type approximation schemes for propagating fronts, J. Math. Soc. Japan, 51 (1999), 267-308. doi: 10.2969/jmsj/05120267.

[12]

P. de Mottoni and M. Schatzman, Geometrical evolution of developed interfaces, Trans. Amer. Math. Soc., 347 (1995), 1533-1589. doi: 10.1090/S0002-9947-1995-1672406-7.

[13]

P. Souganidis and G. Barles, Convergence of approximation scheme for fully nonlinear second order equations, Asymptotic Analysis, 4 (1991), 271-283.

[14]

P. Souganidis and G. Barles, A new approach to front propagation problems: theory and applications, Arch. Rational Mech. Anal., 141 (1998), 237-296. doi: 10.1007/s002050050077.

[15]

G. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, 1922.

show all references

References:
[1]

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, National Bureau of Standards, Applied Mathematics Series, 1964.

[2]

G. Barles and C. Georgelin, A simple proof of convergence for an approximation scheme for computing motions by mean curvature, SIAM J. Numer. Anal., 32 (1995), 484-500. doi: 10.1137/0732020.

[3]

G. Barles, H. Soner and P. Souganidis, Front propagation and phase field theory, SIAM J. Control Optim., 31 (1993), 439-469. doi: 10.1137/0331021.

[4]

B. Bence, J. Merriman and S. Osher, Motion of multiple functions: A level set approach, J. Comput. Phys., 112 (1994), 334-363. doi: 10.1006/jcph.1994.1105.

[5]

L. Bronsard and R. Kohn, Motion by mean curvature as the singular limit of Ginzburg-Landau dynamics, J. Differential Equations, 90 (1991), 211-237. doi: 10.1016/0022-0396(91)90147-2.

[6]

X. Chen, Generation and propagation of interfaces for reaction-diffusion equations, J. Differential Equations, 96 (1992), 116-141. doi: 10.1016/0022-0396(92)90146-E.

[7]

X. Chen, C. Elliott, A. Gardiner and J. Zhao, Convergence of numerical solutions to the Allen-Cahn equation, Appl. Anal., 69 (1998), 47-56 .

[8]

L. Evans, Convergence of an algorithm for mean curvature motion, Indiana Univ. Math. J., 42 (1993), 533-557. doi: 10.1512/iumj.1993.42.42024.

[9]

L. Evans, H. Soner and P. Souganidis, Phase transitions and generalized motion by mean curvature, Comm. Pure Appl. Math., 45 (1992), 1097-1123. doi: 10.1002/cpa.3160450903.

[10]

T. Ilmanen, Convergence of the Allen-Cahn equation to Brakke's motion by mean curvature, J. Differential Geom., 38 (1993), 417-461.

[11]

H. Ishii, G. Pires and P. Souganidis, Threshold dynamics type approximation schemes for propagating fronts, J. Math. Soc. Japan, 51 (1999), 267-308. doi: 10.2969/jmsj/05120267.

[12]

P. de Mottoni and M. Schatzman, Geometrical evolution of developed interfaces, Trans. Amer. Math. Soc., 347 (1995), 1533-1589. doi: 10.1090/S0002-9947-1995-1672406-7.

[13]

P. Souganidis and G. Barles, Convergence of approximation scheme for fully nonlinear second order equations, Asymptotic Analysis, 4 (1991), 271-283.

[14]

P. Souganidis and G. Barles, A new approach to front propagation problems: theory and applications, Arch. Rational Mech. Anal., 141 (1998), 237-296. doi: 10.1007/s002050050077.

[15]

G. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, 1922.

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