American Institute of Mathematical Sciences

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November  2006, 15(4): 1017-1034. doi: 10.3934/dcds.2006.15.1017

A rapidly converging phase field model

Xinfu Chen 1, , G. Caginalp 1, and  Christof Eck 2,

1. 

Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260
2. 

Institute of Applied Mathematics, University Erlangen-Nurenberg 91058 Erlangen

Received  December 2004 Published  May 2006

We propose a phase field model that approximates its limiting sharp interface model (free boundary problem) up to second order in interface thickness. A broad range of double-well potentials can be utilized so long as the dynamical coefficient in the phase equation is adjusted appropriately. This model thereby assures that computation with particular value of interface thickness $\varepsilon$, will differ at most by $O(\varepsilon^2$) from the limiting sharp interface problem. As an illustration, the speed of a traveling wave of the phase field model is asymptotically expanded to demonstrate that it differs from the speed of the traveling wave of the limit problem by $O(\varepsilon^2)$.
Keywords: Phase field model, second order asymptotics., sharp interface limit.
Mathematics Subject Classification: Primary: 80A22, 80M35; Secondary: 35K5.
Citation: Xinfu Chen, G. Caginalp, Christof Eck. A rapidly converging phase field model. Discrete and Continuous Dynamical Systems, 2006, 15 (4) : 1017-1034. doi: 10.3934/dcds.2006.15.1017
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