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A dimension splitting and characteristic projection method for three-dimensional incompressible flow
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 |
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.
References:
[1] |
A. Baskaran, Z. 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. Baskaran, Z. Hu, J. Lowengrub, C. Wang, S. 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. Baskaran, J. Lowengrub, C. 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. Chen, Y. Liu, C. 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. Collins, J. 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. Diegel, X. 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. Diegel, C. Wang, X. 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. Diegel, C. 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. Feng, Z. Guo, J. 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. Feng, A.J. Salgado, C. 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. Frieboes, F. Jin, Y. Chuang, S. M. Wise, J. 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. Guan, J. 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. Guan, J.S. Lowengrub, C. 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. Guan, C. 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. Guo, C. Wang, S. 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. Guo, Y. 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. Hu, S. M. Wise, C. 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. Lee, J. 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. Lee, J. 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. Li, Z. 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. Liu, W. Chen, C. 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. Qiao, C. Wang, S. 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. Samelson, R. Temam, C. 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. Shen, C. Wang, X. 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. Wang, X. 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. Wise, J. S. Lowengrub, H. 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. Wise, C. 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. Baskaran, Z. 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. Baskaran, Z. Hu, J. Lowengrub, C. Wang, S. 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. Baskaran, J. Lowengrub, C. 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. Chen, Y. Liu, C. 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. Collins, J. 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. Diegel, X. 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. Diegel, C. Wang, X. 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. Diegel, C. 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. Feng, Z. Guo, J. 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. Feng, A.J. Salgado, C. 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. Frieboes, F. Jin, Y. Chuang, S. M. Wise, J. 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. Guan, J. 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. Guan, J.S. Lowengrub, C. 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. Guan, C. 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. Guo, C. Wang, S. 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. Guo, Y. 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. Hu, S. M. Wise, C. 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. Lee, J. 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. Lee, J. 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. Li, Z. 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. Liu, W. Chen, C. 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. Qiao, C. Wang, S. 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. Samelson, R. Temam, C. 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. Shen, C. Wang, X. 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. Wang, X. 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. Wise, J. S. Lowengrub, H. 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. Wise, C. 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. |


| Rate | #V's | | ||
| | - | 5 | 0.0012 | |
| 2.04 | 5 | 0.0046 | ||
| | 2.01 | 4 | 0.0160 | |
| | 2.00 | 4 | 0.0744 | |
| | 2.00 | 5 | 0.3818 |
| Rate | #V's | | ||
| | - | 5 | 0.0012 | |
| 2.04 | 5 | 0.0046 | ||
| | 2.01 | 4 | 0.0160 | |
| | 2.00 | 4 | 0.0744 | |
| | 2.00 | 5 | 0.3818 |
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