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A biharmonic-modified forward time stepping method for fourth order nonlinear diffusion equations
1. | Department of Mathematics, UCLA, Los Angeles, CA, 90095 |
2. | Department of Mathematics, Oklahoma State University, 401 Mathematical Sciences, Stillwater, OK 74078 |
3. | Department of Mechanical Engineering, University of California, Los Angeles, CA, 90095-1555, United States |
References:
[1] |
J. W. Barrett and J. F. Blowey, Finite element approximation of a model for phase seperation of muti-component alloy with non-smooth free energy. Numer. Math., 77 (1997), 1-34.
doi: 10.1007/s002110050276. |
[2] |
J. W. Barrett and J. F. Blowey, Finite element approximation of a model for phase separation of a multi-component alloy with a concentration-dependent mobility matrix, IMA Journal of Numerical Analysis, 18 (1998), 287-328.
doi: 10.1093/imanum/18.2.287. |
[3] |
J. W. Barrett and J. F. Blowey, Finite element approximation of a model for phase separation of a multi-component alloy with nonsmooth free energy and a concentration dependent mobility matrix, Math. Models Methods Appl. Sci., 9 (1999), 627-663.
doi: 10.1142/S0218202599000336. |
[4] |
J. W. Barrett and J. F. Blowey, Finite element approximation of the Cahn-Hilliard equation with concentration dependent mobility, Mathematics of Computation, 68 (1999), 487-517.
doi: 10.1090/S0025-5718-99-01015-7. |
[5] |
J. W. Barrett, J. F. Blowey and H. Garcke, Finite element approximation of a fourth order nonlinear degenerate parabolic equation, Numer. Math., 80 (1998), 525-556.
doi: 10.1007/s002110050377. |
[6] |
J. W. Barrett, J. F. Blowey and H. Garcke, Finite element approximation of the Cahn-Hilliard equation with degenerate mobility, SIAM J. Num. Anal., 37 (1999), 286-318.
doi: 10.1137/S0036142997331669. |
[7] |
F. Bernis and A. Friedman, Higher order nonlinear degenerate parabolic equations, J. Diff. Equations, 83 (1990), 179-206.
doi: 10.1016/0022-0396(90)90074-Y. |
[8] |
A. L. Bertozzi and M. C. Pugh, Long-wave instabilities and saturation in thin film equations, Comm. Pure Appl. Math., 51 (1998), 625-666.
doi: 10.1002/(SICI)1097-0312(199806)51:6<625::AID-CPA3>3.0.CO;2-9. |
[9] |
A. L. Bertozzi and M. Pugh, The lubrication approximation for thin viscous films: Regularity and long time behavior of weak solutions, Comm. Pur. Appl. Math., 49 (1996), 85-123.
doi: 10.1002/(SICI)1097-0312(199602)49:2<85::AID-CPA1>3.0.CO;2-2. |
[10] |
A. Bertozzi and M. Pugh, Finite-time blow-up of solutions of some long-wave unstable thin film equations, Indiana Univ. Math. J., 32 (2000), 1323-1366. |
[11] |
A. L. Bertozzi, The mathematics of moving contact lines in thin liquid films, Notices of the American Math. Soc., 45 (1998), 689-697. |
[12] |
A. L. Bertozzi, M. P. Brenner, T. F. Dupont and L. P. Kadanoff, Singularities and similarities in interface flow, in "Trends and Perspectives in Applied Mathematics" (L. Sirovich, editor), volume 100 of Applied Mathematical Sciences, Springer-Verlag, New York, (1994), 155-208. |
[13] |
A. L. Bertozzi, G. Grün and T. P. Witelski, Dewetting films: bifurcations and concentrations, Nonlinearity, 14 (2001), 1569-1592.
doi: 10.1088/0951-7715/14/6/309. |
[14] |
M. Brenner and A. Bertozzi, Spreading of droplets on a solid surface, Phys. Rev. Lett., 71 (1993), 593-596.
doi: 10.1103/PhysRevLett.71.593. |
[15] |
P. Constantin, T. F. Dupont, R. E. Goldstein, L. P. Kadanoff, M. J. Shelley and S.-M. Zhou, Droplet breakup in a model of the Hele-Shaw cell, Physical Review E, 47 (1993), 4169-418.
doi: 10.1103/PhysRevE.47.4169. |
[16] |
J. Douglas, Jr. and T. Dupont, Alternating-direction Galerkin methods on rectangles, in "Numerical Solution of Partial Differential Equations, II (SYNSPADE 1970) (Proc. Sympos., Univ. of Maryland, College Park, Md., 1970)," Academic Press, New York, (1971), 133-214. |
[17] |
T. F. Dupont, R. E. Goldstein, L. P. Kadanoff and Su-Min Zhou, Finite-time singularity formation in Hele Shaw systems, Physical Review E, 47 (1993), 4182-4196.
doi: 10.1103/PhysRevE.47.4182. |
[18] |
P. Ehrhard and S. H. Davis, Non-isothermal spreading of liquid drops on horizontal plates, J. Fluid. Mech., 229 (1991), 365-388.
doi: 10.1017/S0022112091003063. |
[19] |
C. M. Elliott and A. M. Stuart, The global dynamics of discrete semilinear parabolic equations, SIAM J. Numer. Anal., 30 (1993), 1622-1663.
doi: 10.1137/0730084. |
[20] |
C. M. Elliott and H. Garke, On the cahn hilliard equation with degenerate mobility, SIAM J. Math. Anal., 27 (1996), 404-423.
doi: 10.1137/S0036141094267662. |
[21] |
D. Eyre, An unconditionally stable one-step scheme for gradient systems, Unpublished paper, June 9, 1998. |
[22] |
R. Ferreira and F. Bernis, Source-type solutions to thin-film equations in higher dimensions, Euro. J. Appl. Math., 9 (1997), 507-524.
doi: 10.1017/S0956792597003197. |
[23] |
K. Glasner, Nonlinear preconditioning for diffuse interfaces, J. Comp. Phys., 174 (2001), 695-71.
doi: 10.1006/jcph.2001.6933. |
[24] |
K. B. Glasner and T. P. Witelski, Coarsening dynamics of dewetting films, Phys. Rev. E, 67 (2003), 016302.
doi: 10.1103/PhysRevE.67.016302. |
[25] |
K. Glasner, A diffuse interface approach to Hele-Shaw flow, Nonlinearity, 16 (2003), 49-66.
doi: 10.1088/0951-7715/16/1/304. |
[26] |
K. B. Glasner and T. P. Witelski, Coarsening dynamics of dewetting films, Phys. Rev. E, 67 (2003), 016302.
doi: 10.1103/PhysRevE.67.016302. |
[27] |
R. E. Goldstein, A. I. Pesci and M. J. Shelley, Topology transitions and singularities in viscous flows, Physical Review Letters, 70 (1993), 3043-3046.
doi: 10.1103/PhysRevLett.70.3043. |
[28] |
R. E. Goldstein, A. I. Pesci and M. J. Shelley, An attracting manifold for a viscous topology transition, Physical Review Letters, 75 (1995), 3665-3668.
doi: 10.1103/PhysRevLett.75.3665. |
[29] |
H. P. Greenspan, On the motion of a small viscous droplet that wets a surface, J. Fluid Mech., 84 (1978), 125-143.
doi: 10.1017/S0022112078000075. |
[30] |
H. P. Greenspan and B. M. McCay, On the wetting of a surface by a very viscous fluid, Studies in Applied Math., 64 (1981), 95-112. |
[31] |
J. Greer, A. Bertozzi and G. Sapiro, Fourth order partial differential equations on general geometries, J. Computational Physics, 216 (2006), 216-246.
doi: 10.1016/j.jcp.2005.11.031. |
[32] |
G. Grün and M. Rumpf, Nonnegativity preserving convergent schemes for the thin film equation, Num. Math., 87 (2000), 113-152.
doi: 10.1007/s002110000197. |
[33] |
L. M. Hocking, A moving fluid interface on a rough surface, Journal of Fluid Mechanics, 76 (1976), 801-817.
doi: 10.1017/S0022112076000906. |
[34] |
L. M. Hocking, A moving fluid interface. Part 2. The removal of the force singularity by a slip flow, Journal of Fluid Mechanics, 79 (1977), 209-229.
doi: 10.1017/S0022112077000123. |
[35] |
L. M. Hocking, Sliding and spreading of thin two-dimensional drops, Q. J. Mech. Appl. Math., 34 (1981), 37-55.
doi: 10.1093/qjmam/34.1.37. |
[36] |
L. M. Hocking, Rival contact-angle models and the spreading of drops, J. Fluid. Mech., 239 (1992), 671-68.
doi: 10.1017/S0022112092004579. |
[37] |
T. Hou, J. S. Lowengrub and M. J. Shelly, Removing the stiffness from interfacial flow with surface-tension, J. Comp. Phys., 114 (1994), 312-338.
doi: 10.1006/jcph.1994.1170. |
[38] |
M. G. Lippman, Relations entre les phènoménes électriques et capillaires, Ann. Chim. Phys., 5 (1875), 494-548. |
[39] |
H. W. Lu, K. Glasner, C. J. Kim and A. L. Bertozzi, A diffuse interface model for electrowetting droplets in a Hele-Shaw cell, Journal of Fluid Mechanics, 590 (2007), 411-435.
doi: 10.1017/S0022112007008154. |
[40] |
J. A. Moriarty, L. W. Schwartz and E. O Tuck, Unsteady spreading of thin liquid films with small surface tension, Phys. Fluids A, 3 (1991), 733-742.
doi: 10.1063/1.858006. |
[41] |
T. G. Myers, Thin films with high surface tension, SIAM Rev., 40 (1998), 441-462 (electronic).
doi: 10.1137/S003614459529284X. |
[42] |
P. Neogi and C. A. Miller, Spreading kinetics of a drop on a smooth solid surface, J. Colloid Interface Sci., 86 (1982), 525-538.
doi: 10.1016/0021-9797(82)90097-2. |
[43] |
A. Oron, S. H. Davis and S. George Bankoff, Long-scale evolution of thin liquid films, Rev. Mod. Phys., 69 (1997), 931-980.
doi: 10.1103/RevModPhys.69.931. |
[44] |
W. H. Press, S. A. Teukolsky, W. T. Vetterling and B. P. Flannery, "Numerical Recipes in C," Second Edition, Cambridge University Press, New York, 1993. |
[45] |
C.-B. Schoenlieb and A. Bertozzi, Unconditionally stable schemes for higher order inpainting,, 2010., ().
|
[46] |
M. J. Shelley, R. E. Goldstein and A. I. Pesci, Topological transitions in Hele-Shaw flow, in "Singularities in Fluids, Plasmas, and Optics" (R. E. Caflisch and G. C. Papanicolaou, editors), Kluwer Academic Publishers, The Netherlands, (1993), 167-188. |
[47] |
P. Smereka, Semi-implicit level set methods for curvature flow and for motion by surface diffusion, J. Sci. Comp., 19 (2003), 439-456.
doi: 10.1023/A:1025324613450. |
[48] |
S. M. Troian, E. Herbolzheimer, S. A. Safran and J. F. Joanny, Fingering instabilities of driven spreading films, Europhys. Lett., 10 (1989), 25-30.
doi: 10.1209/0295-5075/10/1/005. |
[49] |
B. P. Vollmayr-Lee and A. D. Rutenberg, Fast and accurate coarsening simulation with an unconditionally stable time step, Physical Review E, 68 (2003), 1-13.
doi: 10.1103/PhysRevE.68.066703. |
[50] |
T. P. Witelski and M. Bowen, Adi methods for high order parabolic equations, Appl. Num. Anal., 45 (2003), 331-35. |
[51] |
T. P. Witelski, Equilibrium solutions of a degenerate singular Cahn-Hilliard equation, Applied Mathematics Letters, 11 (1998), 127-133.
doi: 10.1016/S0893-9659(98)00092-5. |
[52] |
L. Zhornitskaya and A. L. Bertozzi, Positivity-preserving numerical schemes for lubrication-type equations, SIAM J. Numer. Anal., 37 (2000), 523-555 (electronic).
doi: 10.1137/S0036142998335698. |
show all references
References:
[1] |
J. W. Barrett and J. F. Blowey, Finite element approximation of a model for phase seperation of muti-component alloy with non-smooth free energy. Numer. Math., 77 (1997), 1-34.
doi: 10.1007/s002110050276. |
[2] |
J. W. Barrett and J. F. Blowey, Finite element approximation of a model for phase separation of a multi-component alloy with a concentration-dependent mobility matrix, IMA Journal of Numerical Analysis, 18 (1998), 287-328.
doi: 10.1093/imanum/18.2.287. |
[3] |
J. W. Barrett and J. F. Blowey, Finite element approximation of a model for phase separation of a multi-component alloy with nonsmooth free energy and a concentration dependent mobility matrix, Math. Models Methods Appl. Sci., 9 (1999), 627-663.
doi: 10.1142/S0218202599000336. |
[4] |
J. W. Barrett and J. F. Blowey, Finite element approximation of the Cahn-Hilliard equation with concentration dependent mobility, Mathematics of Computation, 68 (1999), 487-517.
doi: 10.1090/S0025-5718-99-01015-7. |
[5] |
J. W. Barrett, J. F. Blowey and H. Garcke, Finite element approximation of a fourth order nonlinear degenerate parabolic equation, Numer. Math., 80 (1998), 525-556.
doi: 10.1007/s002110050377. |
[6] |
J. W. Barrett, J. F. Blowey and H. Garcke, Finite element approximation of the Cahn-Hilliard equation with degenerate mobility, SIAM J. Num. Anal., 37 (1999), 286-318.
doi: 10.1137/S0036142997331669. |
[7] |
F. Bernis and A. Friedman, Higher order nonlinear degenerate parabolic equations, J. Diff. Equations, 83 (1990), 179-206.
doi: 10.1016/0022-0396(90)90074-Y. |
[8] |
A. L. Bertozzi and M. C. Pugh, Long-wave instabilities and saturation in thin film equations, Comm. Pure Appl. Math., 51 (1998), 625-666.
doi: 10.1002/(SICI)1097-0312(199806)51:6<625::AID-CPA3>3.0.CO;2-9. |
[9] |
A. L. Bertozzi and M. Pugh, The lubrication approximation for thin viscous films: Regularity and long time behavior of weak solutions, Comm. Pur. Appl. Math., 49 (1996), 85-123.
doi: 10.1002/(SICI)1097-0312(199602)49:2<85::AID-CPA1>3.0.CO;2-2. |
[10] |
A. Bertozzi and M. Pugh, Finite-time blow-up of solutions of some long-wave unstable thin film equations, Indiana Univ. Math. J., 32 (2000), 1323-1366. |
[11] |
A. L. Bertozzi, The mathematics of moving contact lines in thin liquid films, Notices of the American Math. Soc., 45 (1998), 689-697. |
[12] |
A. L. Bertozzi, M. P. Brenner, T. F. Dupont and L. P. Kadanoff, Singularities and similarities in interface flow, in "Trends and Perspectives in Applied Mathematics" (L. Sirovich, editor), volume 100 of Applied Mathematical Sciences, Springer-Verlag, New York, (1994), 155-208. |
[13] |
A. L. Bertozzi, G. Grün and T. P. Witelski, Dewetting films: bifurcations and concentrations, Nonlinearity, 14 (2001), 1569-1592.
doi: 10.1088/0951-7715/14/6/309. |
[14] |
M. Brenner and A. Bertozzi, Spreading of droplets on a solid surface, Phys. Rev. Lett., 71 (1993), 593-596.
doi: 10.1103/PhysRevLett.71.593. |
[15] |
P. Constantin, T. F. Dupont, R. E. Goldstein, L. P. Kadanoff, M. J. Shelley and S.-M. Zhou, Droplet breakup in a model of the Hele-Shaw cell, Physical Review E, 47 (1993), 4169-418.
doi: 10.1103/PhysRevE.47.4169. |
[16] |
J. Douglas, Jr. and T. Dupont, Alternating-direction Galerkin methods on rectangles, in "Numerical Solution of Partial Differential Equations, II (SYNSPADE 1970) (Proc. Sympos., Univ. of Maryland, College Park, Md., 1970)," Academic Press, New York, (1971), 133-214. |
[17] |
T. F. Dupont, R. E. Goldstein, L. P. Kadanoff and Su-Min Zhou, Finite-time singularity formation in Hele Shaw systems, Physical Review E, 47 (1993), 4182-4196.
doi: 10.1103/PhysRevE.47.4182. |
[18] |
P. Ehrhard and S. H. Davis, Non-isothermal spreading of liquid drops on horizontal plates, J. Fluid. Mech., 229 (1991), 365-388.
doi: 10.1017/S0022112091003063. |
[19] |
C. M. Elliott and A. M. Stuart, The global dynamics of discrete semilinear parabolic equations, SIAM J. Numer. Anal., 30 (1993), 1622-1663.
doi: 10.1137/0730084. |
[20] |
C. M. Elliott and H. Garke, On the cahn hilliard equation with degenerate mobility, SIAM J. Math. Anal., 27 (1996), 404-423.
doi: 10.1137/S0036141094267662. |
[21] |
D. Eyre, An unconditionally stable one-step scheme for gradient systems, Unpublished paper, June 9, 1998. |
[22] |
R. Ferreira and F. Bernis, Source-type solutions to thin-film equations in higher dimensions, Euro. J. Appl. Math., 9 (1997), 507-524.
doi: 10.1017/S0956792597003197. |
[23] |
K. Glasner, Nonlinear preconditioning for diffuse interfaces, J. Comp. Phys., 174 (2001), 695-71.
doi: 10.1006/jcph.2001.6933. |
[24] |
K. B. Glasner and T. P. Witelski, Coarsening dynamics of dewetting films, Phys. Rev. E, 67 (2003), 016302.
doi: 10.1103/PhysRevE.67.016302. |
[25] |
K. Glasner, A diffuse interface approach to Hele-Shaw flow, Nonlinearity, 16 (2003), 49-66.
doi: 10.1088/0951-7715/16/1/304. |
[26] |
K. B. Glasner and T. P. Witelski, Coarsening dynamics of dewetting films, Phys. Rev. E, 67 (2003), 016302.
doi: 10.1103/PhysRevE.67.016302. |
[27] |
R. E. Goldstein, A. I. Pesci and M. J. Shelley, Topology transitions and singularities in viscous flows, Physical Review Letters, 70 (1993), 3043-3046.
doi: 10.1103/PhysRevLett.70.3043. |
[28] |
R. E. Goldstein, A. I. Pesci and M. J. Shelley, An attracting manifold for a viscous topology transition, Physical Review Letters, 75 (1995), 3665-3668.
doi: 10.1103/PhysRevLett.75.3665. |
[29] |
H. P. Greenspan, On the motion of a small viscous droplet that wets a surface, J. Fluid Mech., 84 (1978), 125-143.
doi: 10.1017/S0022112078000075. |
[30] |
H. P. Greenspan and B. M. McCay, On the wetting of a surface by a very viscous fluid, Studies in Applied Math., 64 (1981), 95-112. |
[31] |
J. Greer, A. Bertozzi and G. Sapiro, Fourth order partial differential equations on general geometries, J. Computational Physics, 216 (2006), 216-246.
doi: 10.1016/j.jcp.2005.11.031. |
[32] |
G. Grün and M. Rumpf, Nonnegativity preserving convergent schemes for the thin film equation, Num. Math., 87 (2000), 113-152.
doi: 10.1007/s002110000197. |
[33] |
L. M. Hocking, A moving fluid interface on a rough surface, Journal of Fluid Mechanics, 76 (1976), 801-817.
doi: 10.1017/S0022112076000906. |
[34] |
L. M. Hocking, A moving fluid interface. Part 2. The removal of the force singularity by a slip flow, Journal of Fluid Mechanics, 79 (1977), 209-229.
doi: 10.1017/S0022112077000123. |
[35] |
L. M. Hocking, Sliding and spreading of thin two-dimensional drops, Q. J. Mech. Appl. Math., 34 (1981), 37-55.
doi: 10.1093/qjmam/34.1.37. |
[36] |
L. M. Hocking, Rival contact-angle models and the spreading of drops, J. Fluid. Mech., 239 (1992), 671-68.
doi: 10.1017/S0022112092004579. |
[37] |
T. Hou, J. S. Lowengrub and M. J. Shelly, Removing the stiffness from interfacial flow with surface-tension, J. Comp. Phys., 114 (1994), 312-338.
doi: 10.1006/jcph.1994.1170. |
[38] |
M. G. Lippman, Relations entre les phènoménes électriques et capillaires, Ann. Chim. Phys., 5 (1875), 494-548. |
[39] |
H. W. Lu, K. Glasner, C. J. Kim and A. L. Bertozzi, A diffuse interface model for electrowetting droplets in a Hele-Shaw cell, Journal of Fluid Mechanics, 590 (2007), 411-435.
doi: 10.1017/S0022112007008154. |
[40] |
J. A. Moriarty, L. W. Schwartz and E. O Tuck, Unsteady spreading of thin liquid films with small surface tension, Phys. Fluids A, 3 (1991), 733-742.
doi: 10.1063/1.858006. |
[41] |
T. G. Myers, Thin films with high surface tension, SIAM Rev., 40 (1998), 441-462 (electronic).
doi: 10.1137/S003614459529284X. |
[42] |
P. Neogi and C. A. Miller, Spreading kinetics of a drop on a smooth solid surface, J. Colloid Interface Sci., 86 (1982), 525-538.
doi: 10.1016/0021-9797(82)90097-2. |
[43] |
A. Oron, S. H. Davis and S. George Bankoff, Long-scale evolution of thin liquid films, Rev. Mod. Phys., 69 (1997), 931-980.
doi: 10.1103/RevModPhys.69.931. |
[44] |
W. H. Press, S. A. Teukolsky, W. T. Vetterling and B. P. Flannery, "Numerical Recipes in C," Second Edition, Cambridge University Press, New York, 1993. |
[45] |
C.-B. Schoenlieb and A. Bertozzi, Unconditionally stable schemes for higher order inpainting,, 2010., ().
|
[46] |
M. J. Shelley, R. E. Goldstein and A. I. Pesci, Topological transitions in Hele-Shaw flow, in "Singularities in Fluids, Plasmas, and Optics" (R. E. Caflisch and G. C. Papanicolaou, editors), Kluwer Academic Publishers, The Netherlands, (1993), 167-188. |
[47] |
P. Smereka, Semi-implicit level set methods for curvature flow and for motion by surface diffusion, J. Sci. Comp., 19 (2003), 439-456.
doi: 10.1023/A:1025324613450. |
[48] |
S. M. Troian, E. Herbolzheimer, S. A. Safran and J. F. Joanny, Fingering instabilities of driven spreading films, Europhys. Lett., 10 (1989), 25-30.
doi: 10.1209/0295-5075/10/1/005. |
[49] |
B. P. Vollmayr-Lee and A. D. Rutenberg, Fast and accurate coarsening simulation with an unconditionally stable time step, Physical Review E, 68 (2003), 1-13.
doi: 10.1103/PhysRevE.68.066703. |
[50] |
T. P. Witelski and M. Bowen, Adi methods for high order parabolic equations, Appl. Num. Anal., 45 (2003), 331-35. |
[51] |
T. P. Witelski, Equilibrium solutions of a degenerate singular Cahn-Hilliard equation, Applied Mathematics Letters, 11 (1998), 127-133.
doi: 10.1016/S0893-9659(98)00092-5. |
[52] |
L. Zhornitskaya and A. L. Bertozzi, Positivity-preserving numerical schemes for lubrication-type equations, SIAM J. Numer. Anal., 37 (2000), 523-555 (electronic).
doi: 10.1137/S0036142998335698. |
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