Skip to main content
×
×
Home

A Convex-Splitting Scheme for a Diffuse Interface Model with Peng-Robinson Equation of State

  • Qiujin Peng (a1)
Abstract
Abstract

We present a convex-splitting scheme for the fourth order parabolic equation derived from a diffuse interface model with Peng-Robinson equation of state for pure substance. The semi-implicit scheme is proven to be uniquely solvable, mass conservative, unconditionally energy stable and L convergent with the order of . The numerical results verify the effectiveness of the proposed algorithm and also show good agreement of the numerical solution with laboratory experimental results.

Copyright
Corresponding author
*Corresponding author. Email: pengqiujin@ruc.edu.cn (Q. J. Peng)
References
Hide All
[1] Anderson D., McFadden G. B. and Wheeler A., Diffuse-interface methods in fluid mechanics, Ann. Rev. Fluid Mech., 30 (1998), pp. 139165.
[2] Baskaran A., Lowengrub J. S., Wang C. and Wise S. M., Convergence analysis of a second order convex splitting scheme for the modified phase field crystal equation, SIAM J. Numer. Anal., 51 (2013), pp. 28512873.
[3] Cahn J. W. and Hilliard J. E., Free energy of a nonuniform system I. Interfacial free energy, J. Chem. Phys., 28 (1958), pp. 258267.
[4] Cahn J. W., Free energy of a nonuniform system II. Thermodynamic basis, J. Chem. Phys., 30 (1959), pp. 11211124.
[5] Cahn J. W., On spinodal decomposition, Acta Metallurgica, 9 (1961), pp. 795801.
[6] Chen W., Conde S., Wang C., Wang X. and Wise S., A linear energy stable scheme for a thin film model without slope selection, J. Sci. Comput., 52 (2012), pp. 546562.
[7] Eyre D. J., Unconditionally gradient stable time marching the Cahn–Hilliard equation, in Bullard J. W., Kalia R., Stoneham M. and Chen L. Q., (eds.) Computational and Mathematical Models of Microstructural Evolution, 53 (1998), pp. 16861712.
[8] Firoozabadi A., Thermodynamics of Hydrocarbon Reservoirs, McGraw-Hill, New York, 1999.
[9] Gu S., Zhang H. and Zhang Z., An energy-stable finite-difference scheme for the binary fluid-surfactant system, J. Comput. Phys., 270 (2014), pp. 416431.
[10] Guo J., Wang C., Wise S. M. and Yue X., An H2 convergence of a second-order convex-splitting, finite difference scheme for the three-dimensional Cahn-Hilliard equation, Commun. Math. Sci., 14(2) (2016), pp. 489515.
[11] Hu Z., Wise S. M., Wang C. and Lowengrub J. S., Stable and efficient finite-difference nonlinear-multigrid schemes for the phase field crystal equation, J. Comput. Phys., 228 (2009), pp. 53235339.
[12] Kou J. and Sun S., An adaptive finite element method for simulating surface tension with the gradient theory of fluid interfaces, J. Comput. Appl. Math., 255 (2014), pp. 593604.
[13] Kou J. and Sun S., Unconditionally stable methods for simulating multi-component two-phase interface models with Peng-Robinson equation of state and various boundary conditions, J. Comput. Appl. Math., (2015).
[14] Kou J., Sun S. and Wang X., Efficient numerical methods for simulating surface tension of multi-component mixtures with the gradient theory of fluid interfaces, Comput. Methods Appl. Mech. Eng., 292 (2015), pp. 92106.
[15] Li J., Sun Z. Z. and Zhao X., A three level linearized compact difference scheme for the Cahn-Hilliard equation, Sci. China Math., 55(4) (2012), pp. 805826.
[16] Li D. and Qiao Z., On second order semi-implicit Fourier spectral methods for 2D Cahn-Hilliard equations, J. Sci. Comput., (2016), DOI: 10.1007/s10915-016-0251-4.
[17] Li D., Qiao Z. and Tang T., Characterizing the stabilization size for semi-implicit Fourier-Spetral method to phase field equations, SIAM J. Numer. Anal., 54 (2016), pp. 16531681.
[18] Lin H. and Duan Y., Surface tension measurements of propane (r–290) and isobutane (r–600a) from (253 to 333)K, J. Chem. Eng. Data, 48 (2003), pp. 13601363.
[19] Liao H. and Sun Z., Maximum norm error bounds of ADI and compact ADI methods for solving parabolic equations, Numer. Methods Partial Differential Equations, 26 (2010), pp. 3760.
[20] Peng D. Y. and Robinson D. B., A new two-constant equation of state, Industrial and Engineering Chemistry Fundamentals, 15(1) (1976), pp. 5964.
[21] Peng Q., Qiao Z. and Sun S., Stability and convergence analysis of second-order schemes for a diffuse interface model with Peng-Robinson equation of state, J. Comput. Math., doi:10.4208/jcm.1611-m2016-0623.
[22] Qiao Z., Sun Z. and Zhang Z., The stability and convergence of two linearized finite difference schemes for the nonlinear epitaxial growth model, Numer. Methods Partial Differential Equations, 28(6) (2012), pp. 18931915.
[23] Qiao Z. and Sun S., Two-phase fluid simulation using a diffuse interface model with Peng–Robinson equation of state, SIAM J. Sci. Comput., 36(4) (2014), pp. B708B728.
[24] Qiao Z., Sun Z. and Zhang Z., Stability and convergence of second-order schemes for the nonlinear epitaxial growth model without slope selection. Math. Comput., 84 (2015), pp. 653674.
[25] Robinson D. B., Peng D. and Chung S. Y., The development of the Peng-Robinson equation and its application to phase equilibrium in a system containing methanol, Fluid Phase Equilibria, 24 (1985), pp. 2541.
[26] Shen J., Wang C., Wang X. and Wise S. M., Second–order convex splitting schemes for gradient flows with Ehrlich-Schwoebel type energy: application to thin film epitaxy, SIAM J. Numer. Anal., 50 (2012), pp. 105125.
[27] Shen J. and Yang X., Numerical approximations of Allen–Cahn and Cahn–Hilliard equations, Discrete Contin. Dyn. Syst. Ser. A, 28 (2010), pp. 16691691.
[28] Sun Z., A second–order accurate linearized difference scheme for the two–dimensional Cahn–Hilliard equation, Math. Comput., 64(212) (1995), pp. 14631471.
[29] Van Der Waals J. D., “verhandel. konink. akad. weten. amsterdam” vol. 1 no. 8 (dutch) 1893, translation of van der Waals J. D. (The thermodynamic theory of capillarity under the hypothesis of a continuous density variation), J. Stat. Phys., 1979.
[30] Vignal P., Dalcin L., Brown D. L., Collier N. and Calo V. M., An energy-stable convex splitting for the phase-field crystal equation, Comput. Struct., 158 (2015), pp. 355368.
[31] Wang C., Wang X. and Wise S. M., Unconditionally stable schemes for equations of thin film epitaxy, Discrete Contin. Dyn. Syst. Ser. A, 28 (2010), pp. 405423.
[32] Wang C. and Wise S. M, An energy stable and convergent finite-difference scheme for the modified phase field crystal equation, SIAM J. Numer. Anal., 49 (2011), pp. 945969.
[33] Wise S. M., Wang C. and Lowengrub J. S., An energy-stable and convergent finite-difference scheme for the phase field crystal equation, SIAM J. Numer. Anal., 47 (2009), pp. 22692288.
[34] Xu C. and Tang T., Stability analysis of large time-stepping methods for epitaxial growth models, SIAM J. Numer. Anal., 44 (2006), pp. 17591779.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Advances in Applied Mathematics and Mechanics
  • ISSN: 2070-0733
  • EISSN: 2075-1354
  • URL: /core/journals/advances-in-applied-mathematics-and-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Keywords:

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 48 *
Loading metrics...

Abstract views

Total abstract views: 147 *
Loading metrics...

* Views captured on Cambridge Core between 11th July 2017 - 22nd January 2018. This data will be updated every 24 hours.