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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[33]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[33]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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