A new conservative finite difference scheme for 1D Cahn–Hilliard equation coupled with elasticity

References

[1] E. Bonetti, P. Colli, W. Dreyer, G. Gilardi, G. Schimperna and J. Sprekels, On a model for phase separation in binary alloys driven by mechanical effects, Phys. D 165 (2002), no. 1–2, 48–65. 10.1016/S0167-2789(02)00373-1Search in Google Scholar

[2] D. Furihata and T. Matsuo, Discrete Variational Derivative Method, Chapman & Hall/CRC Numer. Anal. Sci. Comput., CRC Press, Boca Raton, 2010. 10.1201/b10387Search in Google Scholar

[3] H. Garcke, On Cahn–Hilliard systems with elasticity, Proc. Roy. Soc. Edinburgh Sect. A 133 (2003), no. 2, 307–331. 10.1017/S0308210500002419Search in Google Scholar

[4] H. Garcke, On a Cahn–Hilliard model for phase separation with elastic misfit, Ann. Inst. H. Poincaré Anal. Non Linéaire 22 (2005), no. 2, 165–185. 10.1016/j.anihpc.2004.07.001Search in Google Scholar

[5] H. Garcke and U. Weikard, Numerical approximation of the Cahn–Larché equation, Numer. Math. 100 (2005), no. 4, 639–662. 10.1007/s00211-004-0578-xSearch in Google Scholar

[6] M. E. Gurtin, Generalized Ginzburg–Landau and Cahn–Hilliard equations based on a microforce balance, Phys. D 92 (1996), no. 3–4, 178–192. 10.1016/0167-2789(95)00173-5Search in Google Scholar

[7] E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations, Springer Ser. Comput. Math. 31, Springer, Berlin, 2006. Search in Google Scholar

[8] Z. Kosowski and I. Pawł ow, Unique global solvability of the Fried–Gurtin model for phase transitions in solids, Topol. Methods Nonlinear Anal. 24 (2004), no. 2, 209–237. 10.12775/TMNA.2004.026Search in Google Scholar

[9] A. Miranville, Long-time behavior of some models of Cahn–Hilliard equations in deformable continua, Nonlinear Anal. Real World Appl. 2 (2001), no. 3, 273–304. 10.1016/S0362-546X(00)00104-8Search in Google Scholar

[10] A. Miranville, The Cahn–Hilliard equation and some of its variants, AIMS Math. 2 (2017), 479–544. 10.3934/Math.2017.2.479Search in Google Scholar

[11] M. Okumura, A stable and structure-preserving scheme for a non-local Allen–Cahn equation, Jpn. J. Ind. Appl. Math. 35 (2018), no. 3, 1245–1281. 10.1007/s13160-018-0326-8Search in Google Scholar

[12] M. Okumura and D. Furihata, A structure-preserving scheme for the Allen–Cahn equation with a dynamic boundary condition, Discrete Contin. Dyn. Syst. 40 (2020), no. 8, 4927–4960. 10.3934/dcds.2020206Search in Google Scholar

[13] I. Pawł ow and W. M. Zaja̧czkowski, Classical solvability of 1-D Cahn–Hilliard equation coupled with elasticity, Math. Methods Appl. Sci. 29 (2006), no. 7, 853–876. 10.1002/mma.715Search in Google Scholar

[14] I. Pawłow and W. M. Zaja̧czkowski, Global existence and uniqueness of weak solutions to Cahn–Hilliard–Gurtin system in elastic solids, Parabolic and Navier-Stokes Equations, Banach Center Publ. 81, Polish Academy of Sciences, Warsaw (2008), 337–368. 10.4064/bc81-0-22Search in Google Scholar

[15] I. Pawł ow and W. M. Zaja̧czkowski, Strong solvability of 3-D Cahn–Hilliard system in elastic solids, Math. Methods Appl. Sci. 31 (2008), no. 8, 879–914. 10.1002/mma.946Search in Google Scholar

[16] I. Pawł ow and W. M. Zaja̧czkowski, Global regular solutions to Cahn–Hilliard system coupled with viscoelasticity, Math. Methods Appl. Sci. 32 (2009), no. 17, 2197–2242. 10.1002/mma.1131Search in Google Scholar

[17] I. Pawł ow and W. M. Zaja̧aczkowski, Long time behaviour of a Cahn–Hilliard system coupled with viscoelasticity, Ann. Polon. Math. 98 (2010), no. 1, 1–21. 10.4064/ap98-1-1Search in Google Scholar

[18] J. M. Sanz-Serna and M. P. Calvo, Numerical Hamiltonian Problems, Appl. Math. Math. Comp. 7, Chapman & Hall, London, 1994. 10.1007/978-1-4899-3093-4Search in Google Scholar

[19] K. Shimura and S. Yoshikawa, Error estimate for structure-preserving finite difference schemes of the one-dimensional Cahn–Hilliard system coupled with viscoelasticity, Regularity and Asymptotic Analysis for Critical Cases of Partial Differential Equations, RIMS Kôkyûroku Bessatsu B82, Research Institute for Mathematical Sciences (RIMS), Kyoto (2020), 159–175. Search in Google Scholar

[20] D. Wegner, Existence for coupled pseudomonotone-strongly monotone systems and application to a Cahn–Hilliard model with elasticity, Nonlinear Anal. 113 (2015), 385–400. 10.1016/j.na.2014.10.018Search in Google Scholar

[21] S. Yoshikawa, Energy method for structure-preserving finite difference schemes and some properties of difference quotient, J. Comput. Appl. Math. 311 (2017), 394–413. 10.1016/j.cam.2016.08.008Search in Google Scholar

[22] S. Yoshikawa, Remarks on energy methods for structure-preserving finite difference schemes—small data global existence and unconditional error estimate, Appl. Math. Comput. 341 (2019), 80–92. 10.1016/j.amc.2018.08.030Search in Google Scholar