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

Wenbin Chen; Wenqiang Feng; Yuan Liu; Cheng Wang; Steven M. Wise

doi:10.3934/dcdsb.2018090

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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

Wenbin Chen ^1, , Wenqiang Feng ^2, , Yuan Liu ^1, , Cheng Wang ^3,, and Steven M. Wise ^2,

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 January 2019 Early access 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.

Keywords: Cahn-Hilliard-Hele-Shaw, Darcy's law, convex splitting, finite difference method, unconditional energy stability, nonlinear multigrid.

Mathematics Subject Classification: 65M06, 65M12, 35K55, 76D05.

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. 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. Google Scholar
[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. Google Scholar
[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. 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. 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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. 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. 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. Google Scholar
[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. 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. 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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. 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. 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. Google Scholar
[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. Google Scholar
[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. 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. 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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. 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. 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. Google Scholar
[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. Google Scholar

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. Google Scholar
[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. Google Scholar
[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. 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. 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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. 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. 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. Google Scholar
[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. 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. 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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. 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. 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. Google Scholar
[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. Google Scholar
[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. 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. 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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. 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. 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. Google Scholar
[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. Google Scholar

Table 1 for the $h = 3.2/512$ case">Figure 1. The evolutions of discrete energy and mass for the simulation depicted in Table 1 for the $h = 3.2/512$ case

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Figure 2. Snapshots of Spinodal decomposition of a binary fluid in a Hele-Shaw cell

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Figure 3. The evolutions of discrete energy with $\gamma = 0, 2, 4$

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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	$T_{cpu}(h_f)$
$\frac{3.2}{16}$	$\frac{3.2}{32}$	$7.6501\times 10^{-3}$	-	5	0.0012
$\frac{3.2}{32}$	$\frac{3.2}{64}$	$1.8565\times 10^{-3}$	2.04	5	0.0046
$\frac{3.2}{64}$	$\frac{3.2}{128}$	$4.6141\times 10^{-4}$	2.01	4	0.0160
$\frac{3.2}{128}$	$\frac{3.2}{256}$	$1.1520\times 10^{-4}$	2.00	4	0.0744
$\frac{3.2}{256}$	$\frac{3.2}{512}$	$2.8792\times 10^{-5}$	2.00	5	0.3818

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$h_c$	$h_{f}$	$\\|{\delta_\phi}\\|_{2}$	Rate	#V's	$T_{cpu}(h_f)$
$\frac{3.2}{16}$	$\frac{3.2}{32}$	$7.6501\times 10^{-3}$	-	5	0.0012
$\frac{3.2}{32}$	$\frac{3.2}{64}$	$1.8565\times 10^{-3}$	2.04	5	0.0046
$\frac{3.2}{64}$	$\frac{3.2}{128}$	$4.6141\times 10^{-4}$	2.01	4	0.0160
$\frac{3.2}{128}$	$\frac{3.2}{256}$	$1.1520\times 10^{-4}$	2.00	4	0.0744
$\frac{3.2}{256}$	$\frac{3.2}{512}$	$2.8792\times 10^{-5}$	2.00	5	0.3818

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2020 Impact Factor: 1.327

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2020 Impact Factor: 1.327

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2020 Impact Factor: 1.327

American Institute of Mathematical Sciences

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

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American Institute of Mathematical Sciences

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

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