Second-order energy stable schemes for the new model of the Cahn-Hilliard-MHD equations (original) (raw)

References

  1. Abdou, M.A., et al.: On the exploration of innovative concepts for fusion chamber technology fusion. Fusion Eng. Des. 54, 181–247 (2001)
    Google Scholar
  2. Cahn, J.W., Hilliard, J.E.: Free energy of a nonuniform system. I. Interfacial free energy. J. Chem. Phys. 28, 258–267 (1958)
    MATH Google Scholar
  3. Chen, R., Ji, G., Yang, X., Zhang, H.: Decoupled energy stable schemes for phase field vesicle membrane model. J. Comput. Phys. 302, 509–523 (2015)
    MathSciNet MATH Google Scholar
  4. Chen, R., Yang, X., Zhang, H.: Second order, linear, and unconditionally energy stable schemes for a hydrodynamic model of smectic-A liquid crystals. SIAM J. Sci. Comput. 39, A2808–A2833 (2017)
    MathSciNet MATH Google Scholar
  5. Chen, R., Yang, X., Zhang, H.: Decoupled, energy stable scheme for hydrodynamic Allen-Cahn phase field moving contact line model. J. Comput. Math. 36, 661–681 (2018)
    MathSciNet MATH Google Scholar
  6. Chen, W., Feng, W., Liu, Y., Wang, C., Wise, S.M.: A second order energy stable scheme for the Cahn-Hilliard-Hele-Shaw equations. Discrete Cont. Dyn. Sys. B 24. https://doi.org/10.3934/dcdsb.2018090 (2016)
  7. Cyr, E.C., Shadid, J.N., Tuminaro, R.S., Pawlowski, R.P., Chacón, L.: A new approximate block fractorization preconditioner for two-dimensional incompressible (reduced) resistive MHD. SIAM J. Sci. Comput. 35, B701–B730 (2013)
    MATH Google Scholar
  8. Feng, X.: Fully discrete finite element approximations of the Navier-Stokes-Cahn-Hilliard diffuse interface model for two-phase fluid flows. SIAM J. Numer. Anal. 44, 1049–1072 (2006)
    MathSciNet MATH Google Scholar
  9. Grün, G.: On convergent schemes for diffuse interface models for two-phase flow of incompressible fluids with general mass densities. SIAM J. Numer. Anal. 51, 3036–3061 (2013)
    MathSciNet MATH Google Scholar
  10. Guo, Z., Lin, P., Lowengrub, J.S.: A numerical method for the quasi-incompressible Cahn-Hilliard-Navier-Stokes equations for variable density flows with a discrete energy law. J. Comput. Phys. 276, 486–507 (2014)
    MathSciNet MATH Google Scholar
  11. Gerbeau, J.F., Le Bris, C., Leliévre, T.: Mathematical Methods for the Magnetohydrodynamics of Liquid Metals. Oxford University Press, Oxford (2006)
    MATH Google Scholar
  12. Gunzburger, M.D., Meir, A.J., Peterson, J.S.: On the existence, uniqueness, and finite element approximation of solutions of the equations of stationary incompressible magnetohydrodynamics. Math. Comp. 56, 523–563 (1991)
    MathSciNet MATH Google Scholar
  13. Han, D., Wang, X.: A second order in time, decoupled, unconditionally stable numerical scheme for the Cahn-Hilliard-Darcy system. J. Sci. Comput. 77, 1210–1233 (2018)
    MathSciNet MATH Google Scholar
  14. Hiptmair, R., Li, L., Mao, S., Zheng, W.: A fully divergence-free finite element method for magneto-hydrodynamic equations. Math. Mod. Meth. Appl. Sci. 24, 659–695 (2018)
    MATH Google Scholar
  15. Ingram, R.: A new linearly extrapolated Crank-Nicolson time-stepping scheme for the Navier-Stokes equations. Math. Comp. 82, 1953–1973 (2013)
    MathSciNet MATH Google Scholar
  16. van Kan, J.: A second-order accurate pressure-correction scheme for viscous incompressible flow. SIAM J. Sci. Statist. Comput. 7, 870–891 (1986)
    MathSciNet MATH Google Scholar
  17. Lee, H., Lowengrub, J.S., Goodman, J.: Modeling pinchoff and reconnection in a Hele-Shaw cell. II. Analysis and simulation in the nonlinear regime. Phys. Fluids. 14, 514–545 (2002)
    MathSciNet MATH Google Scholar
  18. Li, L., Zheng, W.: A robust solver for the finite element approximation of stationary incompressible MHD equations in 3D. J. Comput. Phys. 351, 254–270 (2017)
    MathSciNet MATH Google Scholar
  19. Li, X., Qiao, Z.H., Zhang, H.: An unconditionally energy stable finite difference scheme for a stochastic Cahn-Hilliard equation. Sci. China Math. 59, 1815–1834 (2016)
    MathSciNet MATH Google Scholar
  20. Li, X., Qiao, Z.H., Zhang, H.: A second-order convex-splitting scheme for the Cahn-Hilliard equation with variable interfacial parameters. J. Comput. Math. 35, 693–710 (2017)
    MathSciNet MATH Google Scholar
  21. Liu, C., Shen, J., Yang, X.: Decoupled energy stable schemes for a phase-field model of two-phase incompressible flows with variable density. J. Sci. Comput. 62, 601–622 (2014)
    MathSciNet MATH Google Scholar
  22. Ma, Y., Hu, K., Hu, X., Xu, J.: Robust preconditioners for incompressible MHD models. J. Comput. Phys. 316, 721–746 (2016)
    MathSciNet MATH Google Scholar
  23. Moreau, R.: Magnetohydrodynamics. Kluwer Academic Publishers, Dordrecht, Boston (1990)
    MATH Google Scholar
  24. Ni, M.-J., Munipalli, R., Huang, P., Morley, N.B., Abdou, M.A.: A current density conservative scheme for incompressible MHD flows at a low magnetic Reynolds number. Part I. On a rectangular collocated grid system. J. Comp. Phys. 227, 174–204 (2007)
    MATH Google Scholar
  25. Ni, M.-J., Munipalli, R., Huang, P., Morley, N.B., Abdou, M.A.: A current density conservative scheme for incompressible MHD flows at a low magnetic Reynolds number. Part II: on an arbitrary collocated mesh. J. Comp. Phys. 227, 205–228 (2007)
    MATH Google Scholar
  26. Phillips, E.G., Elman, H.C., Cyr, E.C., Shadid, J.N., Pawlowski, R.P.: A block preconditioner for an exact penalty formulation for stationary MHD. SIAM J. Sci Comput. 36, B930–B951 (2014)
    MathSciNet MATH Google Scholar
  27. Phillips, E.G., Shadid, J.N., Cyr, E.C., Elman, H.C., Pawlowski, R.P.: Block preconditioners for stable mixed nodal and edge finite element representations of incompressible resistive MHD. SIAM J. Sci. Comput. 38, B1009–B1031 (2016)
    MathSciNet MATH Google Scholar
  28. Planas, R., Badia, S., Codina, R.: Approximation of the inductionless MHD problem using a stabilized finite element method. J. Comput. Phys. 230, 2977–2996 (2011)
    MathSciNet MATH Google Scholar
  29. Prohl, A.: Convergent finite element discretizations of the nonstationary incompressible magnetohydrodynamics system. ESAIM Math. Model Num. Anal. 42, 1065–1087 (2008)
    MathSciNet MATH Google Scholar
  30. Qiao, Z.H., Tang, T., Xie, H.: Error analysis of a mixed finite element method for molecular beam epitaxy model. SIAM J. Numer. Anal. 53, 184–205 (2015)
    MathSciNet MATH Google Scholar
  31. Schötzau, D.: Mixed finite element methods for stationary incompressible magneto Chydrodynamics. Numer. Math. 96, 771–800 (2004)
    MathSciNet MATH Google Scholar
  32. Shen, J.: On error estimates of the projection methods for the Navier-Stokes equations: second-order schemes. Math. Comp. 65, 1039–1065 (1996)
    MathSciNet MATH Google Scholar
  33. Shen, J., Yang, X.: Decoupled energy stable schemes for phase-field models of two-phase complex fluids. SIAM J. Sci. Comput. 36, 122–145 (2014)
    MathSciNet MATH Google Scholar
  34. Shen, J., Yang, X.: Decoupled energy stable schemes for phase-field models of two-phase incompressible flows. SIAM J. Numer. Anal. 53, 279–296 (2015)
    MathSciNet MATH Google Scholar
  35. Shen, J., Yang, X., Yu, H.: Efficient energy stable numerical schemes for a phase field moving contact line model. J. Comput. Phys. 284, 617–630 (2015)
    MathSciNet MATH Google Scholar
  36. Shen, J., Yang, X.F.: Numerical approximations of Allen-Cahn and Cahn-Hilliard equations. Discete Cont. Dyn. Sys.-A 28, 1669–1691 (2010)
    MathSciNet MATH Google Scholar
  37. Shen, J., Xu, J., Yang, J.: The scalar auxiliary variable (SAV) approach for gradient flows. J. Comput. Phys. 353, 407–416 (2018)
    MathSciNet MATH Google Scholar
  38. Shen, J., Xu, J.: Convergence and error analysis for the scalar auxiliary variable (SAV) scheme to gradient flows. SIAM J. Num. Anal. 56, 2895–2912 (2018)
    MathSciNet MATH Google Scholar
  39. Xu, Z., Zhang, H.: Stabilized semi-implicit numerical scheme for the Cahn-Hilliard with variable interfacial parameters. J. Comput. Appl. Math. 346, 307–322 (2019)
    MathSciNet MATH Google Scholar
  40. Xu, Z., Yang, X.F., Zhang, H., Xie, Z.Q.: Efficient and Linear Schemes for Anisotropic Cahn-Hilliard Equations Using the Stabilized Invariant Energy Quadratization (S-IEQ) Approach, Comm. Comput. Phys. Online Publishing. https://doi.org/10.1016/j.cpc.2018.12.019 (2019)
  41. Yang, X.: Linear, first and second-order, unconditionally energy stable numerical schemes for the phase field model of homopolymer blends. J. Comput. Phys. 327, 294–316 (2016)
    MathSciNet MATH Google Scholar
  42. Yang, X., Zhao, J., Wang, Q., Shen, J.: Numerical approximations for a three components Cahn-Hilliard phase-field model based on the invariant energy quadratization method. Math. Mod. Meth. Appl. Sci. 27, 1993–2030 (2017)
    MathSciNet MATH Google Scholar
  43. Zhang, J., Han, T.Y., Yang, J.C., Ni, M.J.: On the spreading of impacting drops under the influence of a vertical magnetic field. J. Fluid Mech. 809. https://doi.org/10.1017/jfm.2016.725 (2016)
  44. Zhang, J., Ni, M.J.: What happens to the vortex structures when the rising bubble transits from zigzag to spiral. J. Fluid Mech. 828, 353–373 (2017)
    MathSciNet MATH Google Scholar
  45. Zhang, J., Ni, M.J.: Direct numerical simulations of incompressible multiphase magnetohydrodynamics with phase change. J. Comput. Phys. 375, 717–746 (2018)
    MathSciNet MATH Google Scholar

Download references