• Previous Article
    Quasi-$m$-accretivity of Schrödinger operators with singular first-order coefficients
  • DCDS Home
  • This Issue
  • Next Article
    Asymptotic behavior of a Cahn-Hilliard equation with Wentzell boundary conditions and mass conservation
December  2008, 22(4): 1065-1080. doi: 10.3934/dcds.2008.22.1065

Stable discretizations of the Cahn-Hilliard-Gurtin equations

1. 

Université de Poitiers, Laboratoire de Mathématiques et Applications, Téléport 2 - BP 30179, SP2MI, Boulevard Marie et Pierre Curie, 86962 Futuroscope Chasseneuil, France

2. 

Laboratoire de Mathématiques et Applications UMR CNRS 6086, Université de Poitiers, Téléport 2 - BP 30179, Boulevard Marie et Pierre Curie, 86962 Futuroscope Chasseneuil, France

Received  March 2007 Revised  June 2007 Published  September 2008

We study space and time discretizations of the Cahn-Hilliard-Gurtin equations with a polynomial nonlinearity. We first consider a space semi-discrete version of the equations, and we prove in particular that any solution converges to a steady state (as in the continuous case). Then, we study the numerical stability of the fully discrete scheme obtained by applying the Euler backward scheme to the space semi-discrete problem. In particular, we show that this fully discrete problem is unconditionally stable. Numerical simulations in one space dimension conclude the paper.
Citation: Sami Injrou, Morgan Pierre. Stable discretizations of the Cahn-Hilliard-Gurtin equations. Discrete & Continuous Dynamical Systems - A, 2008, 22 (4) : 1065-1080. doi: 10.3934/dcds.2008.22.1065
[1]

Benoît Merlet, Morgan Pierre. Convergence to equilibrium for the backward Euler scheme and applications. Communications on Pure & Applied Analysis, 2010, 9 (3) : 685-702. doi: 10.3934/cpaa.2010.9.685

[2]

Alain Haraux. Some applications of the Łojasiewicz gradient inequality. Communications on Pure & Applied Analysis, 2012, 11 (6) : 2417-2427. doi: 10.3934/cpaa.2012.11.2417

[3]

Zhuchun Li, Yi Liu, Xiaoping Xue. Convergence and stability of generalized gradient systems by Łojasiewicz inequality with application in continuum Kuramoto model. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 345-367. doi: 10.3934/dcds.2019014

[4]

Maurizio Grasselli, Morgan Pierre. Convergence to equilibrium of solutions of the backward Euler scheme for asymptotically autonomous second-order gradient-like systems. Communications on Pure & Applied Analysis, 2012, 11 (6) : 2393-2416. doi: 10.3934/cpaa.2012.11.2393

[5]

Monika Eisenmann, Etienne Emmrich, Volker Mehrmann. Convergence of the backward Euler scheme for the operator-valued Riccati differential equation with semi-definite data. Evolution Equations & Control Theory, 2019, 8 (2) : 315-342. doi: 10.3934/eect.2019017

[6]

Janosch Rieger. The Euler scheme for state constrained ordinary differential inclusions. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2729-2744. doi: 10.3934/dcdsb.2016070

[7]

Yeping Li, Jie Liao. Stability and $ L^{p}$ convergence rates of planar diffusion waves for three-dimensional bipolar Euler-Poisson systems. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1281-1302. doi: 10.3934/cpaa.2019062

[8]

Yanqing Wang. A semidiscrete Galerkin scheme for backward stochastic parabolic differential equations. Mathematical Control & Related Fields, 2016, 6 (3) : 489-515. doi: 10.3934/mcrf.2016013

[9]

Weidong Zhao, Jinlei Wang, Shige Peng. Error estimates of the $\theta$-scheme for backward stochastic differential equations. Discrete & Continuous Dynamical Systems - B, 2009, 12 (4) : 905-924. doi: 10.3934/dcdsb.2009.12.905

[10]

Weidong Zhao, Yang Li, Guannan Zhang. A generalized $\theta$-scheme for solving backward stochastic differential equations. Discrete & Continuous Dynamical Systems - B, 2012, 17 (5) : 1585-1603. doi: 10.3934/dcdsb.2012.17.1585

[11]

Jyoti Mishra. Analysis of the Fitzhugh Nagumo model with a new numerical scheme. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 781-795. doi: 10.3934/dcdss.2020044

[12]

Francesco C. De Vecchi, Andrea Romano, Stefania Ugolini. A symmetry-adapted numerical scheme for SDEs. Journal of Geometric Mechanics, 2019, 11 (3) : 325-359. doi: 10.3934/jgm.2019018

[13]

Carlo Morosi, Livio Pizzocchero. On the constants in a Kato inequality for the Euler and Navier-Stokes equations. Communications on Pure & Applied Analysis, 2012, 11 (2) : 557-586. doi: 10.3934/cpaa.2012.11.557

[14]

Wolf-Jüergen Beyn, Janosch Rieger. The implicit Euler scheme for one-sided Lipschitz differential inclusions. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 409-428. doi: 10.3934/dcdsb.2010.14.409

[15]

Bahareh Akhtari, Esmail Babolian, Andreas Neuenkirch. An Euler scheme for stochastic delay differential equations on unbounded domains: Pathwise convergence. Discrete & Continuous Dynamical Systems - B, 2015, 20 (1) : 23-38. doi: 10.3934/dcdsb.2015.20.23

[16]

Michele Coti Zelati. Remarks on the approximation of the Navier-Stokes equations via the implicit Euler scheme. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2829-2838. doi: 10.3934/cpaa.2013.12.2829

[17]

G. Bellettini, Giorgio Fusco, Nicola Guglielmi. A concept of solution and numerical experiments for forward-backward diffusion equations. Discrete & Continuous Dynamical Systems - A, 2006, 16 (4) : 783-842. doi: 10.3934/dcds.2006.16.783

[18]

S. J. Li, Z. M. Fang. On the stability of a dual weak vector variational inequality problem. Journal of Industrial & Management Optimization, 2008, 4 (1) : 155-165. doi: 10.3934/jimo.2008.4.155

[19]

M. D. Todorov, C. I. Christov. Conservative numerical scheme in complex arithmetic for coupled nonlinear Schrödinger equations. Conference Publications, 2007, 2007 (Special) : 982-992. doi: 10.3934/proc.2007.2007.982

[20]

Yohan Penel. An explicit stable numerical scheme for the $1D$ transport equation. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 641-656. doi: 10.3934/dcdss.2012.5.641

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (6)
  • HTML views (0)
  • Cited by (8)

Other articles
by authors

[Back to Top]