# American Institute of Mathematical Sciences

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

## A rapidly converging phase field model

 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 $O(\varepsilon^2)$.
Citation: Xinfu Chen, G. Caginalp, Christof Eck. A rapidly converging phase field model. Discrete & Continuous Dynamical Systems - A, 2006, 15 (4) : 1017-1034. doi: 10.3934/dcds.2006.15.1017
 [1] Helmut Abels, Johannes Kampmann. Existence of weak solutions for a sharp interface model for phase separation on biological membranes. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 331-351. doi: 10.3934/dcdss.2020325 [2] Liupeng Wang, Yunqing Huang. Error estimates for second-order SAV finite element method to phase field crystal model. Electronic Research Archive, 2021, 29 (1) : 1735-1752. doi: 10.3934/era.2020089 [3] Weisong Dong, Chang Li. Second order estimates for complex Hessian equations on Hermitian manifolds. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020377 [4] Shasha Hu, Yihong Xu, Yuhan Zhang. Second-Order characterizations for set-valued equilibrium problems with variable ordering structures. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020164 [5] Shenglan Xie, Maoan Han, Peng Zhu. A posteriori error estimate of weak Galerkin fem for second order elliptic problem with mixed boundary condition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020340 [6] Soniya Singh, Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of second order impulsive systems with state-dependent delay in Banach spaces. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020103 [7] Elimhan N. Mahmudov. Infimal convolution and duality in convex optimal control problems with second order evolution differential inclusions. Evolution Equations & Control Theory, 2021, 10 (1) : 37-59. doi: 10.3934/eect.2020051 [8] Ying Lv, Yan-Fang Xue, Chun-Lei Tang. Ground state homoclinic orbits for a class of asymptotically periodic second-order Hamiltonian systems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1627-1652. doi: 10.3934/dcdsb.2020176 [9] Laura Aquilanti, Simone Cacace, Fabio Camilli, Raul De Maio. A Mean Field Games model for finite mixtures of Bernoulli and categorical distributions. Journal of Dynamics & Games, 2020  doi: 10.3934/jdg.2020033 [10] Yi-Long Luo, Yangjun Ma. Low Mach number limit for the compressible inertial Qian-Sheng model of liquid crystals: Convergence for classical solutions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 921-966. doi: 10.3934/dcds.2020304 [11] Björn Augner, Dieter Bothe. The fast-sorption and fast-surface-reaction limit of a heterogeneous catalysis model. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 533-574. doi: 10.3934/dcdss.2020406 [12] Guo-Niu Han, Huan Xiong. Skew doubled shifted plane partitions: Calculus and asymptotics. Electronic Research Archive, 2021, 29 (1) : 1841-1857. doi: 10.3934/era.2020094 [13] Serena Dipierro, Benedetta Pellacci, Enrico Valdinoci, Gianmaria Verzini. Time-fractional equations with reaction terms: Fundamental solutions and asymptotics. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 257-275. doi: 10.3934/dcds.2020137 [14] Joel Kübler, Tobias Weth. Spectral asymptotics of radial solutions and nonradial bifurcation for the Hénon equation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3629-3656. doi: 10.3934/dcds.2020032 [15] João Vitor da Silva, Hernán Vivas. Sharp regularity for degenerate obstacle type problems: A geometric approach. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1359-1385. doi: 10.3934/dcds.2020321 [16] Helmut Abels, Andreas Marquardt. On a linearized Mullins-Sekerka/Stokes system for two-phase flows. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020467 [17] Tian Ma, Shouhong Wang. Topological phase transition III: Solar surface eruptions and sunspots. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 501-514. doi: 10.3934/dcdsb.2020350 [18] Lin Shi, Dingshi Li, Kening Lu. Limiting behavior of unstable manifolds for spdes in varying phase spaces. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021020 [19] Tomáš Smejkal, Jiří Mikyška, Jaromír Kukal. Comparison of modern heuristics on solving the phase stability testing problem. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1161-1180. doi: 10.3934/dcdss.2020227 [20] Xinfu Chen, Huiqiang Jiang, Guoqing Liu. Boundary spike of the singular limit of an energy minimizing problem. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3253-3290. doi: 10.3934/dcds.2020124

2019 Impact Factor: 1.338