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January 2019, 24(1): 149-182. doi: 10.3934/dcdsb.2018090

A second order energy stable scheme for the Cahn-Hilliard-Hele-Shaw equations

1. 

School of Mathematical Sciences, Fudan University, Shanghai 200433, China

2. 

Mathematics Department, University of Tennessee, Knoxville, TN 37996, USA

3. 

Mathematics Department, University of Massachusetts North Dartmouth, MA 02747, USA

* Corresponding author

Received  November 2016 Revised  September 2017 Published  March 2018

We present a second-order-in-time finite difference scheme for the Cahn-Hilliard-Hele-Shaw equations. This numerical method is uniquely solvable and unconditionally energy stable. At each time step, this scheme leads to a system of nonlinear equations that can be efficiently solved by a nonlinear multigrid solver. Owing to the energy stability, we derive an $\ell^2 (0, T; H_h^3)$ stability of the numerical scheme. To overcome the difficulty associated with the convection term $\nabla · (\phi \mathit{\boldsymbol{u}})$, we perform an $\ell^∞ (0, T; H_h^1)$ error estimate instead of the classical $\ell^∞ (0, T; \ell^2)$ one to obtain the optimal rate convergence analysis. In addition, various numerical simulations are carried out, which demonstrate the accuracy and efficiency of the proposed numerical scheme.

Citation: Wenbin Chen, Wenqiang Feng, Yuan Liu, Cheng Wang, Steven M. Wise. A second order energy stable scheme for the Cahn-Hilliard-Hele-Shaw equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (1) : 149-182. doi: 10.3934/dcdsb.2018090
References:
[1]

A. BaskaranZ. Guan and J. S. Lowengrub, Energy stable multigrid method for local and non-local hydrodynamic models for freezing, Comput. Methods in Appl. Mech. Eng., 299 (2016), 22-56. doi: 10.1016/j.cma.2015.10.011.

[2]

A. BaskaranZ. HuJ. LowengrubC. WangS. M. Wise and P. Zhou, Energy stable and efficient finite-difference nonlinear multigrid schemes for the modified phase field crystal equation, J. Comput. Phys., 250 (2013), 270-292. doi: 10.1016/j.jcp.2013.04.024.

[3]

A. BaskaranJ. LowengrubC. Wang and S. M. Wise, Convergence analysis of a second order convex splitting scheme for the modified phase field crystal equation, SIAM J. Numer. Anal., 51 (2013), 2851-2873. doi: 10.1137/120880677.

[4]

J.W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. Ⅰ. Interfacial free energy, J. Chem. Phys., 28 (1958), 258-267.

[5]

W. ChenY. LiuC. Wang and S. M. Wise, Convergence analysis of a fully discrete finite difference scheme for Cahn-Hilliard-Hele-Shaw equation, Math. Comp., 85 (2016), 2231-2257. doi: 10.1090/mcom3052.

[6]

C. CollinsJ. Shen and S. M. Wise, An efficient, energy stable scheme for the Cahn-Hilliard-Brinkman system, Commun. Comput. Phys., 13 (2013), 929-957. doi: 10.4208/cicp.171211.130412a.

[7]

A. DiegelX. Feng and S. M. Wise, Convergence analysis of an unconditionally stable method for a Cahn-Hilliard-Stokes system of equations, SIAM J. Numer. Anal., 53 (2015), 127-152. doi: 10.1137/130950628.

[8]

A. DiegelC. WangX. Wang and S. M. Wise, Convergence analysis and error estimates for a second order accurate finite element method for the Cahn-Hilliard-Navier-Stokes system, Numer. Math., 137 (2017), 495-534. doi: 10.1007/s00211-017-0887-5.

[9]

A. DiegelC. Wang and S. M. Wise, Stability and convergence of a second order mixed finite element method for the Cahn-Hilliard equation, IMA J. Numer. Anal., 36 (2016), 1867-1897. doi: 10.1093/imanum/drv065.

[10]

W. E and J.-G. Liu, Projection method Ⅲ: Spatial discretization on the staggered grid, Math. Comp., 71 (2002), 27-47.

[11]

W. FengZ. GuoJ. Lowengrub and S. M. Wise, A mass-conservative adaptive FAS multigrid solver for cell-centered finite difference methods on block-structured, locally-cartesian grids, J. Comput. Phys., 352 (2018), 463-497. doi: 10.1016/j.jcp.2017.09.065.

[12]

W. FengA.J. SalgadoC. Wang and S. M. Wise, Preconditioned steepest descent methods for some nonlinear elliptic equations involving p-Laplacian terms, J. Comput. Phys., 334 (2017), 45-67. doi: 10.1016/j.jcp.2016.12.046.

[13]

X. Feng and S. M. Wise, Analysis of a Darcy-Cahn-Hilliard diffuse interface model for the Hele-Shaw flow and its fully discrete finite element approximation, SIAM J. Numer. Anal., 50 (2012), 1320-1343. doi: 10.1137/110827119.

[14]

H.B. FrieboesF. JinY. ChuangS. M. WiseJ. S. Lowengrub and V. Cristini, Three-dimensional multispecies nonlinear tumor growth-Ⅱ: Tumor invasion and angiogenesis, J. Theor. Biol., 264 (2010), 1254-1278. doi: 10.1016/j.jtbi.2010.02.036.

[15]

Z. GuanJ. S. Lowengrub and C. Wang, Convergence analysis for second order accurate schemes for the periodic nonlocal Allen-Cahn and Cahn-Hilliard equations, Math. Methods Appl. Sci., 40 (2017), 6836-6863. doi: 10.1002/mma.4497.

[16]

Z. GuanJ.S. LowengrubC. Wang and S. M. Wise, Second-order convex splitting schemes for nonlocal Cahn-Hilliard and Allen-Cahn equations, J. Comput. Phys., 277 (2014), 48-71. doi: 10.1016/j.jcp.2014.08.001.

[17]

Z. GuanC. Wang and S. M. Wise, A convergent convex splitting scheme for the periodic nonlocal Cahn-Hilliard equation, Numer. Math., 128 (2014), 377-406. doi: 10.1007/s00211-014-0608-2.

[18]

J. GuoC. WangS. M. Wise and X. Yue, An $H^2$ convergence of a second-order convex-splitting, finite difference scheme for the three-dimensional Cahn-Hilliard equation, Commu. Math. Sci., 14 (2016), 489-515. doi: 10.4310/CMS.2016.v14.n2.a8.

[19]

R. GuoY. Xia and Y. Xu, An efficient fully-discrete local discontinuous Galerkin method for the Cahn-Hilliard-Hele-Shaw system, J. Comput. Phys., 264 (2014), 23-40. doi: 10.1016/j.jcp.2014.01.037.

[20]

D. Han, A decoupled unconditionally stable numerical scheme for the Cahn-Hilliard-Hele-Shaw system, J. Sci. Comput., 66 (2016), 1102-1121. doi: 10.1007/s10915-015-0055-y.

[21]

F. H. Harlow and J. E. Welch, Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface, Phys. Fluids, 8 (1965), 2182-2189. doi: 10.1063/1.1761178.

[22]

Z. HuS. M. WiseC. Wang and J. S. Lowengrub, Stable and efficient finite-difference nonlinear-multigrid schemes for the phase-field crystal equation, J. Comput. Phys., 228 (2009), 5323-5339. doi: 10.1016/j.jcp.2009.04.020.

[23]

H. LeeJ. S. Lowengrub and J. Goodman, Modeling pinchoff and reconnection in a Hele-Shaw cell. Ⅱ. Analysis and simulation in the nonlinear regime, Phys. Fluids, 14 (2002), 514-545. doi: 10.1063/1.1425844.

[24]

H. LeeJ. S. Lowengrub and J. Goodman, Modeling pinchoff and reconnection in a Hele-Shaw cell. Ⅰ. The models and their calibration, Phys. Fluids, 14 (2002), 492-513. doi: 10.1063/1.1425843.

[25]

D. Li and Z. Qiao, On second order semi-implicit Fourier spectral methods for 2D Cahn-Hilliard equations, J. Sci. Comput., 70 (2017), 301-341. doi: 10.1007/s10915-016-0251-4.

[26]

D. LiZ. Qiao and T. Tang, Characterizing the stabilization size for semi-implicit Fourier-spectral method to phase field equations, SIAM J. Numer. Anal., 54 (2016), 1653-1681. doi: 10.1137/140993193.

[27]

Y. LiuW. ChenC. Wang and S. M. Wise, Error analysis of a mixed finite element method for a Cahn-Hilliard-Hele-Shaw system, Numer. Math., 135 (2017), 679-709. doi: 10.1007/s00211-016-0813-2.

[28]

Z. QiaoC. WangS. Wise and Z. Zhang, Error analysis of a finite difference scheme for the epitaxial thin film growth model with slope selection with an improved convergence constant, Int. J. Numer. Anal. Model., 14 (2017), 283-305.

[29]

R. SamelsonR. TemamC. Wang and S. Wang, Surface pressure Poisson equation formulation of the primitive equations: Numerical schemes, SIAM J. Numer. Anal., 41 (2003), 1163-1194. doi: 10.1137/S0036142901396284.

[30]

J. ShenC. WangX. Wang and S. M. Wise, Second-order convex splitting schemes for gradient flows with Ehrlich-Schwoebel type energy: Application to thin film epitaxy, SIAM J. Numer. Anal., 50 (2012), 105-125. doi: 10.1137/110822839.

[31]

C. WangX. Wang and S. M. Wise, Unconditionally stable schemes for equations of thin film epitaxy, Discrete Contin. Dyn. Syst. A, 28 (2010), 405-423. doi: 10.3934/dcds.2010.28.405.

[32]

C. Wang and S. M. Wise, An energy stable and convergent finite-difference scheme for the modified phase field crystal equation, SIAM J. Numer. Anal., 49 (2011), 945-969. doi: 10.1137/090752675.

[33]

X. Wang and H. Wu, Long-time behavior of the Hele-Shaw-Cahn-Hilliard system, Asympt. Anal., 78 (2012), 217-245.

[34]

X. Wang and Z.-F. Zhang, Well-posedness of the Hele-Shaw-Cahn-Hilliard system, Ann. I. H. Poincaré CAN., 30 (2013), 367-384. doi: 10.1016/j.anihpc.2012.06.003.

[35]

S. M. Wise, Unconditionally stable finite difference, nonlinear multigrid simulation of the Cahn-Hilliard-Hele-Shaw system of equations, J. Sci. Comput., 44 (2010), 38-68. doi: 10.1007/s10915-010-9363-4.

[36]

S. M. WiseJ. S. LowengrubH. B. Frieboes and V. Cristini, Three-dimensional multispecies nonlinear tumor growth-Ⅰ: model and numerical method, J. Theor. Biol., 253 (2008), 524-543. doi: 10.1016/j.jtbi.2008.03.027.

[37]

S. M. WiseC. Wang and J. S. Lowengrub, An energy-stable and convergent finite-difference scheme for the phase field crystal equation, SIAM J. Numer. Anal., 47 (2009), 2269-2288. doi: 10.1137/080738143.

show all references

References:
[1]

A. BaskaranZ. Guan and J. S. Lowengrub, Energy stable multigrid method for local and non-local hydrodynamic models for freezing, Comput. Methods in Appl. Mech. Eng., 299 (2016), 22-56. doi: 10.1016/j.cma.2015.10.011.

[2]

A. BaskaranZ. HuJ. LowengrubC. WangS. M. Wise and P. Zhou, Energy stable and efficient finite-difference nonlinear multigrid schemes for the modified phase field crystal equation, J. Comput. Phys., 250 (2013), 270-292. doi: 10.1016/j.jcp.2013.04.024.

[3]

A. BaskaranJ. LowengrubC. Wang and S. M. Wise, Convergence analysis of a second order convex splitting scheme for the modified phase field crystal equation, SIAM J. Numer. Anal., 51 (2013), 2851-2873. doi: 10.1137/120880677.

[4]

J.W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. Ⅰ. Interfacial free energy, J. Chem. Phys., 28 (1958), 258-267.

[5]

W. ChenY. LiuC. Wang and S. M. Wise, Convergence analysis of a fully discrete finite difference scheme for Cahn-Hilliard-Hele-Shaw equation, Math. Comp., 85 (2016), 2231-2257. doi: 10.1090/mcom3052.

[6]

C. CollinsJ. Shen and S. M. Wise, An efficient, energy stable scheme for the Cahn-Hilliard-Brinkman system, Commun. Comput. Phys., 13 (2013), 929-957. doi: 10.4208/cicp.171211.130412a.

[7]

A. DiegelX. Feng and S. M. Wise, Convergence analysis of an unconditionally stable method for a Cahn-Hilliard-Stokes system of equations, SIAM J. Numer. Anal., 53 (2015), 127-152. doi: 10.1137/130950628.

[8]

A. DiegelC. WangX. Wang and S. M. Wise, Convergence analysis and error estimates for a second order accurate finite element method for the Cahn-Hilliard-Navier-Stokes system, Numer. Math., 137 (2017), 495-534. doi: 10.1007/s00211-017-0887-5.

[9]

A. DiegelC. Wang and S. M. Wise, Stability and convergence of a second order mixed finite element method for the Cahn-Hilliard equation, IMA J. Numer. Anal., 36 (2016), 1867-1897. doi: 10.1093/imanum/drv065.

[10]

W. E and J.-G. Liu, Projection method Ⅲ: Spatial discretization on the staggered grid, Math. Comp., 71 (2002), 27-47.

[11]

W. FengZ. GuoJ. Lowengrub and S. M. Wise, A mass-conservative adaptive FAS multigrid solver for cell-centered finite difference methods on block-structured, locally-cartesian grids, J. Comput. Phys., 352 (2018), 463-497. doi: 10.1016/j.jcp.2017.09.065.

[12]

W. FengA.J. SalgadoC. Wang and S. M. Wise, Preconditioned steepest descent methods for some nonlinear elliptic equations involving p-Laplacian terms, J. Comput. Phys., 334 (2017), 45-67. doi: 10.1016/j.jcp.2016.12.046.

[13]

X. Feng and S. M. Wise, Analysis of a Darcy-Cahn-Hilliard diffuse interface model for the Hele-Shaw flow and its fully discrete finite element approximation, SIAM J. Numer. Anal., 50 (2012), 1320-1343. doi: 10.1137/110827119.

[14]

H.B. FrieboesF. JinY. ChuangS. M. WiseJ. S. Lowengrub and V. Cristini, Three-dimensional multispecies nonlinear tumor growth-Ⅱ: Tumor invasion and angiogenesis, J. Theor. Biol., 264 (2010), 1254-1278. doi: 10.1016/j.jtbi.2010.02.036.

[15]

Z. GuanJ. S. Lowengrub and C. Wang, Convergence analysis for second order accurate schemes for the periodic nonlocal Allen-Cahn and Cahn-Hilliard equations, Math. Methods Appl. Sci., 40 (2017), 6836-6863. doi: 10.1002/mma.4497.

[16]

Z. GuanJ.S. LowengrubC. Wang and S. M. Wise, Second-order convex splitting schemes for nonlocal Cahn-Hilliard and Allen-Cahn equations, J. Comput. Phys., 277 (2014), 48-71. doi: 10.1016/j.jcp.2014.08.001.

[17]

Z. GuanC. Wang and S. M. Wise, A convergent convex splitting scheme for the periodic nonlocal Cahn-Hilliard equation, Numer. Math., 128 (2014), 377-406. doi: 10.1007/s00211-014-0608-2.

[18]

J. GuoC. WangS. M. Wise and X. Yue, An $H^2$ convergence of a second-order convex-splitting, finite difference scheme for the three-dimensional Cahn-Hilliard equation, Commu. Math. Sci., 14 (2016), 489-515. doi: 10.4310/CMS.2016.v14.n2.a8.

[19]

R. GuoY. Xia and Y. Xu, An efficient fully-discrete local discontinuous Galerkin method for the Cahn-Hilliard-Hele-Shaw system, J. Comput. Phys., 264 (2014), 23-40. doi: 10.1016/j.jcp.2014.01.037.

[20]

D. Han, A decoupled unconditionally stable numerical scheme for the Cahn-Hilliard-Hele-Shaw system, J. Sci. Comput., 66 (2016), 1102-1121. doi: 10.1007/s10915-015-0055-y.

[21]

F. H. Harlow and J. E. Welch, Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface, Phys. Fluids, 8 (1965), 2182-2189. doi: 10.1063/1.1761178.

[22]

Z. HuS. M. WiseC. Wang and J. S. Lowengrub, Stable and efficient finite-difference nonlinear-multigrid schemes for the phase-field crystal equation, J. Comput. Phys., 228 (2009), 5323-5339. doi: 10.1016/j.jcp.2009.04.020.

[23]

H. LeeJ. S. Lowengrub and J. Goodman, Modeling pinchoff and reconnection in a Hele-Shaw cell. Ⅱ. Analysis and simulation in the nonlinear regime, Phys. Fluids, 14 (2002), 514-545. doi: 10.1063/1.1425844.

[24]

H. LeeJ. S. Lowengrub and J. Goodman, Modeling pinchoff and reconnection in a Hele-Shaw cell. Ⅰ. The models and their calibration, Phys. Fluids, 14 (2002), 492-513. doi: 10.1063/1.1425843.

[25]

D. Li and Z. Qiao, On second order semi-implicit Fourier spectral methods for 2D Cahn-Hilliard equations, J. Sci. Comput., 70 (2017), 301-341. doi: 10.1007/s10915-016-0251-4.

[26]

D. LiZ. Qiao and T. Tang, Characterizing the stabilization size for semi-implicit Fourier-spectral method to phase field equations, SIAM J. Numer. Anal., 54 (2016), 1653-1681. doi: 10.1137/140993193.

[27]

Y. LiuW. ChenC. Wang and S. M. Wise, Error analysis of a mixed finite element method for a Cahn-Hilliard-Hele-Shaw system, Numer. Math., 135 (2017), 679-709. doi: 10.1007/s00211-016-0813-2.

[28]

Z. QiaoC. WangS. Wise and Z. Zhang, Error analysis of a finite difference scheme for the epitaxial thin film growth model with slope selection with an improved convergence constant, Int. J. Numer. Anal. Model., 14 (2017), 283-305.

[29]

R. SamelsonR. TemamC. Wang and S. Wang, Surface pressure Poisson equation formulation of the primitive equations: Numerical schemes, SIAM J. Numer. Anal., 41 (2003), 1163-1194. doi: 10.1137/S0036142901396284.

[30]

J. ShenC. WangX. Wang and S. M. Wise, Second-order convex splitting schemes for gradient flows with Ehrlich-Schwoebel type energy: Application to thin film epitaxy, SIAM J. Numer. Anal., 50 (2012), 105-125. doi: 10.1137/110822839.

[31]

C. WangX. Wang and S. M. Wise, Unconditionally stable schemes for equations of thin film epitaxy, Discrete Contin. Dyn. Syst. A, 28 (2010), 405-423. doi: 10.3934/dcds.2010.28.405.

[32]

C. Wang and S. M. Wise, An energy stable and convergent finite-difference scheme for the modified phase field crystal equation, SIAM J. Numer. Anal., 49 (2011), 945-969. doi: 10.1137/090752675.

[33]

X. Wang and H. Wu, Long-time behavior of the Hele-Shaw-Cahn-Hilliard system, Asympt. Anal., 78 (2012), 217-245.

[34]

X. Wang and Z.-F. Zhang, Well-posedness of the Hele-Shaw-Cahn-Hilliard system, Ann. I. H. Poincaré CAN., 30 (2013), 367-384. doi: 10.1016/j.anihpc.2012.06.003.

[35]

S. M. Wise, Unconditionally stable finite difference, nonlinear multigrid simulation of the Cahn-Hilliard-Hele-Shaw system of equations, J. Sci. Comput., 44 (2010), 38-68. doi: 10.1007/s10915-010-9363-4.

[36]

S. M. WiseJ. S. LowengrubH. B. Frieboes and V. Cristini, Three-dimensional multispecies nonlinear tumor growth-Ⅰ: model and numerical method, J. Theor. Biol., 253 (2008), 524-543. doi: 10.1016/j.jtbi.2008.03.027.

[37]

S. M. WiseC. Wang and J. S. Lowengrub, An energy-stable and convergent finite-difference scheme for the phase field crystal equation, SIAM J. Numer. Anal., 47 (2009), 2269-2288. doi: 10.1137/080738143.

Figure 1.  The evolutions of discrete energy and mass for the simulation depicted in Table 1 for the $h = 3.2/512$ case
Figure 2.  Snapshots of Spinodal decomposition of a binary fluid in a Hele-Shaw cell
Figure 3.  The evolutions of discrete energy with $\gamma = 0, 2, 4$
Table 1.  Errors, convergence rates, average iteration numbers and average CPU time (in seconds) for each time step
$h_c$$h_{f}$$\|{\delta_\phi}\|_{2}$ Rate #V's $T_{cpu}(h_f)$
$\frac{3.2}{16}$$\frac{3.2}{32}$ $7.6501\times 10^{-3}$-50.0012
$\frac{3.2}{32}$$\frac{3.2}{64}$ $1.8565\times 10^{-3}$2.0450.0046
$\frac{3.2}{64}$$\frac{3.2}{128}$ $4.6141\times 10^{-4}$2.0140.0160
$\frac{3.2}{128}$ $\frac{3.2}{256}$$1.1520\times 10^{-4}$2.0040.0744
$\frac{3.2}{256}$ $\frac{3.2}{512}$$2.8792\times 10^{-5}$2.0050.3818
$h_c$$h_{f}$$\|{\delta_\phi}\|_{2}$ Rate #V's $T_{cpu}(h_f)$
$\frac{3.2}{16}$$\frac{3.2}{32}$ $7.6501\times 10^{-3}$-50.0012
$\frac{3.2}{32}$$\frac{3.2}{64}$ $1.8565\times 10^{-3}$2.0450.0046
$\frac{3.2}{64}$$\frac{3.2}{128}$ $4.6141\times 10^{-4}$2.0140.0160
$\frac{3.2}{128}$ $\frac{3.2}{256}$$1.1520\times 10^{-4}$2.0040.0744
$\frac{3.2}{256}$ $\frac{3.2}{512}$$2.8792\times 10^{-5}$2.0050.3818
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Francesco Della Porta, Maurizio Grasselli. On the nonlocal Cahn-Hilliard-Brinkman and Cahn-Hilliard-Hele-Shaw systems. Communications on Pure & Applied Analysis, 2016, 15 (2) : 299-317. doi: 10.3934/cpaa.2016.15.299

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Su-Hong Jiang, Min Li. A modified strictly contractive peaceman-rachford splitting method for multi-block separable convex programming. Journal of Industrial & Management Optimization, 2018, 14 (1) : 397-412. doi: 10.3934/jimo.2017052

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