# Numerical investigation of the discrete solution of phase-field equation

## Main Article Content

## Abstract

In this article, we deal with the numerical solution of the phase-field equation. The numerical solution is based on the lattice Boltzmann method compared with the finite difference method. First, a short introduction to the mathematical and numerical model is presented and the implementation of two different methods is briefly investigated. Then, the results of both methods are compared and the experimental order of convergence is determined. One of the significant drawbacks of the finite difference method is that a sufficiently fine computational mesh is crucial for accurate results. The advantage of the lattice Boltzmann method is that it produces accurate results on courser meshes.

## Article Details

How to Cite

Eichler, P., Malík, M., Oberhuber, T., & Fučík, R.
(2020).
Numerical investigation of the discrete solution of phase-field equation.

*Proceedings Of The Conference Algoritmy,*, 111 - 120. Retrieved from http://www.iam.fmph.uniba.sk/amuc/ojs/index.php/algoritmy/article/view/1565/824
Section

Articles

## References

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[14] M. S. Allen and J. W. Cahn, Mechanisms of phase transformations within the miscibility gap of Fe-rich Fe-Al alloys, Acta Metallurgica, 1976, 24.5: 425-437.

[15] Z. Guo and C. Shu, Lattice Boltzmann method and its applications in engineering, World Scientific, 2013.

[16] T. Krüger, H. Kusumaatmaja, A. Kuzmin, O. Shardt, G. Orest and E. Viggen, The lattice Boltzmann method, Springer International Publishing, 2017, 10: 978-3.

[17] D. d'Humi'ere, Generalized lattice-Boltzmann equations, Rarefied gas dynamics, 1992.

[18] M. Geier, A. Greiner and J. G. Korvink, Cascaded digital lattice Boltzmann automata for high Reynolds number flow, Physical Review E, 2006, 73.6: 066705.

[19] M. Geier, M. Schönherr, A. Pasquili and M. Krafczyk, The cummulant lattice Boltzmann equation in three dimensions: Theory and validation, Computers & Mathematics with Applications, 2015, 70.4: 507-547.

[20] S. Chikatamarla, S. Ansumali and I. Karlin, Entropic lattice Boltzmann models for hydro-dynamics in three dimensions, Physical review letters, 2006, 97.1: 010201.

[21] J. G. Zhou, Lattice Boltzmann methods for shallow water flows,Berlin: Springer, 2004.

[22] M. Geier, A. Fakhari, and T. Lee, Conservative phase-field lattice Boltzmann model for interface tracking equation, Physical Review E, 2015, 91.6: 063309.

[23] A. Fakhari, M. Geier, and D. Bolster, A simple phase-field model for interface tracking in three dimensions, Computers & Mathematics with Applications, 2019, 78.4: 1154-1165.

[24] J. C. Butcher, Numerical methods for ordinary differential equations, John Wiley & Sons, 2016.

[25] Template Numerical Library, https://tnl-project.org/.

[2] Y. Sun and C. Beckermann, Sharp interface tracking using the phase-field equation, Journal of Computational Physics, 2007, 220.2: 626-653.

[3] J. W, Boettinger, J. A. Beckermann, and A. Karma, Phase-field simulation of solidification, Annual review of materials research, 2002, 32.1: 163-194.

[4] D. Jacqmin, Calculation of two-phase Navier–Stokes flows using phase-field modeling, Journal of Computational Physics, 1999, 155.1: 96-127.

[5] S. Luckhaus, Solutions for the two-phase stefan problem with the Gibbs—Thomson law for the melting temperature. In: Fundamental contributions to the continuum theory of evolving phase interfaces in solids, Springer, Berlin, Heidelberg, 1999. p. 317-327.

[6] H. Greberg, G. V. Paolini, J. Satherley, J. Penfold, and S. Nordholm, Generalized van

der Waals theory of interfaces in simple fluid mixtures, Journal of colloid and interface science, 2001, 235.2: 334-343.

[7] M. E. Gurtin, On the Two-Phase Stefan Problem with Interfacial Energy and Entropy, Technical Summary Report, 1985.

[8] J. A. Sethian, Level set methods: Evolving interfaces in geometry, fluid mechanics, computer vision, and materials science, Cambridge: Cambridge University Press, 1996.

[9] P.-H. Chiu and Y.-T. Lin, A conservative phase field method for solving incompressible two-phase flows, Journal of Computational Physics, 2011, 230.1: 185-204.

[10] A. K. Gunstensen, D. H. Rothman, S. Zaleski and G. Zanetti, Lattice Boltzmann model of immis- cible fluids, Physical Review A, 1991, 43.8: 4320.

[11] X. W. Shan and H. D. Chen, Lattice Boltzmann model for simulating flows with multiple phases and components, Physical review E, 1993, 47.3: 1815.

[12] M. R. Swift, E. Orlandi, W. R. Osborn, and J. M. Yeomans, Lattice Boltzmann simulations of liquid-gas and binary fluid systems, Physical Review E, 1996, 54.5: 5041.

[13] M. R. Swift, W. R. Osborn, and J. M. Yeomans, Lattice Boltzmann simulation of nonideal fluids, Physical review letters, 1995, 75.5: 830.

[14] M. S. Allen and J. W. Cahn, Mechanisms of phase transformations within the miscibility gap of Fe-rich Fe-Al alloys, Acta Metallurgica, 1976, 24.5: 425-437.

[15] Z. Guo and C. Shu, Lattice Boltzmann method and its applications in engineering, World Scientific, 2013.

[16] T. Krüger, H. Kusumaatmaja, A. Kuzmin, O. Shardt, G. Orest and E. Viggen, The lattice Boltzmann method, Springer International Publishing, 2017, 10: 978-3.

[17] D. d'Humi'ere, Generalized lattice-Boltzmann equations, Rarefied gas dynamics, 1992.

[18] M. Geier, A. Greiner and J. G. Korvink, Cascaded digital lattice Boltzmann automata for high Reynolds number flow, Physical Review E, 2006, 73.6: 066705.

[19] M. Geier, M. Schönherr, A. Pasquili and M. Krafczyk, The cummulant lattice Boltzmann equation in three dimensions: Theory and validation, Computers & Mathematics with Applications, 2015, 70.4: 507-547.

[20] S. Chikatamarla, S. Ansumali and I. Karlin, Entropic lattice Boltzmann models for hydro-dynamics in three dimensions, Physical review letters, 2006, 97.1: 010201.

[21] J. G. Zhou, Lattice Boltzmann methods for shallow water flows,Berlin: Springer, 2004.

[22] M. Geier, A. Fakhari, and T. Lee, Conservative phase-field lattice Boltzmann model for interface tracking equation, Physical Review E, 2015, 91.6: 063309.

[23] A. Fakhari, M. Geier, and D. Bolster, A simple phase-field model for interface tracking in three dimensions, Computers & Mathematics with Applications, 2019, 78.4: 1154-1165.

[24] J. C. Butcher, Numerical methods for ordinary differential equations, John Wiley & Sons, 2016.

[25] Template Numerical Library, https://tnl-project.org/.