Abstract
We introduce a variant of a recently proposed method of rotated lattices for numerical treatment of moving boundary problems. The usual lattice introduced for numerical computation of phase-field models gives rise to unphysical metastable states and anisotropy. In the present case we rotate and shift the lattice by random angles and fractions of a lattice constant. We show that a twelve point interpolation formula is adequate to keep numerical interpolation errors sufficiently localized. This removes the unphysical metastabilities and makes the model fully isotropic. This is demonstrated by a few example-calculations for dendritic pattern formation.
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References
L.D. Landau, E.M. Lifshitz, Statistical Physics, Pergamon Press Oxford, 1969
J.S. Langer,Lectures in the Theory of Pattern Formation in Chance and Matter, Les Houches 1986, edited by J. Souletie et al. (North Holland) 1987; J.S. Langer, Rev. Mod. Phys. 52, 1 (1980)
D. Temkin in: Crystallization Processes, p. 15 (Consultants Bureau, New York 1966)
H. Müller-Krumbhaar, Phys. Rev. B 10, 1308 (1974)
G. Gilmer, H.J. Leamy, K. A. Jackson, H. Reiss, J. Crystal Growth 24, 495 (1974)
J. Weeks, G. H. Gilmer, J. Chem. Phys. 63, 3136 (1975)
S.T. Chui, J.D. Weeks, Phys. Rev. B 14, 4978 (1976)
H. Müller-Krumbhaar, Z. Physik B 25, 287 (1976)
Y. Saito, H. Müller-Krumbhaar, J. Chem. Phys. 74, 721 (1981)
M. Kardar, G. Parisi, Y. Zhang, Phys. Rev. Lett. 56, 889 (1986)
J.P.v.d. Eerden and H. Müller-Krumbhaar, Phys. Rev. Lett. 57, 2431 (1986)
R. Jullien, J. Kertesz, P. Meakin and D.E. Wolf, “Surface Disordering: Growth, Roughening and Phase Transitions”, Nova Science Publ., Commack, 1993.
B. Halperin, P. Hohenberg, S.K., Ma, Phys. Rev. B 10, 139 (1974); P. Hohenberg, B. Halperin Rev. Mod. Phys. 49, 435 (1977).
R. Bausch and G. Rose, Physica A 210, 352 (1994).
G. Caginalp, P. Fife, Phys. Rev. B 33, 7792 (1986); G. Caginalp, Arch. Rat. Mech. Anal. 92, 205 (1986).
H. Collins and H. Levine, Phys. Rev. B 31, 6119 (1985).
A.A. Wheeler, B.T. Murray, R.J. Schaefer Physica D 66, 243 (1993).
A.A. Wheeler, W. Boettinger, G. McFadden, Phys. Rev. E 47, 1893 (1993).
R. Kupferman, O. Shochet and E. Ben-Jacob, Phys. Rev. B 46, 16045 (1992); R. Kupferman, O. Shochet and E. Ben-Jacob, Phys, Rev. E (to appear).
A. Karma, Phys. Rev. E 49, 2245 (1994).
S. Smale, Bull. of AMS 73, 747 (1963).
B. Chirikov, Phys. Rep. 52, 263 (1979).
S. Aubry, in Solitons and Condensed Matter Physics, Springer Series in Solid-State Science 8, ed. A. Bishop, T. Schneider (Springer, N.Y. 1978); S. Aubry and P. Le Daeron, Physica 8D, 381 (1983).
M. Berry, “Regular and irregular motion”, in S. Journa ed., Amer. Inst. Phys. Conf. Proceedings 46, 16 (1978).
W. Mullins, R. Sekerka, J. Appl. Phys. 34, 323 (1963).
H. Müller-Krumbhaar and W. Kurz, “Solidification” in: Materials Science, Vol. 5, ed. P. Haasen, VCH-Verlag, Weinheim 1991
E.A. Brener and V.I. Mel'nikov, Adv. Phys., 40, 53 (1991).
E. Brener, H. Müller-Krumbhaar, D. Temkin, Europhys. Letters 17, 535 (1992)
T. Ihle, H. Müller-Krumbhaar, Phys. Rev. Letters 70, 3083 (1993); Phys. Rev. E 49, 2972 (1994).
G. Dziuk, Num. Math. 58, 603 (1991); A. Schmidt, Ph.D. thesis, Math. Dept., Univ. Freiburg, (to be published).
H. Müller-Krumbhaar, T. Ihle and A. Boesch, in Proceedings of a NATO-ASI workshop on “Spatio-temporal Patterns in Nonequilibrium Systems”, eds. P.E. Cladis, P. Palffy-Muhoray, Santa Fe Institute, April 13–17, 1993 (Addison Wesley, Reading, 1995).
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Bösch, A., Müller-Krumbhaar, H. & Shochet, O. Phase-field models for moving boundary problems: Controlling metastability and anisotropy. Z. Physik B - Condensed Matter 97, 367–377 (1995). https://doi.org/10.1007/BF01307490
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DOI: https://doi.org/10.1007/BF01307490