A biharmonic-modified forward time stepping method for fourth order nonlinear diffusion equations

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October 2011, 29(4): 1367-1391. doi: 10.3934/dcds.2011.29.1367

A biharmonic-modified forward time stepping method for fourth order nonlinear diffusion equations

Andrea L. Bertozzi ^1, , Ning Ju ^2, and Hsiang-Wei Lu ^3,

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

Received December 2009 Revised July 2010 Published December 2010

Abstract
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Under a Creative Commons license

We consider a class of splitting schemes for fourth order nonlinear diffusion equations. Standard backward-time differencing requires the solution of a higher order elliptic problem, which can be both computationally expensive and work-intensive to code, in higher space dimensions. Recent papers in the literature provide computational evidence that a biharmonic-modified, forward time-stepping method, can provide good results for these problems. We provide a theoretical explanation of the results. For a basic nonlinear 'thin film' type equation we prove $H^1$ stability of the method given very simple boundedness constraints of the numerical solution. For a more general class of long-wave unstable problems, we prove stability and convergence, using only constraints on the smooth solution. Computational examples include both the model of 'thin film' type problems and a quantitative model for electrowetting in a Hele-Shaw cell (Lu et al J. Fluid Mech. 2007). The methods considered here are related to 'convexity splitting' methods for gradient flows with nonconvex energies.

Keywords: Fourth order equations, convex splitting..

Mathematics Subject Classification: 65M06, 65M12, 65M7.

Citation: Andrea L. Bertozzi, Ning Ju, Hsiang-Wei Lu. A biharmonic-modified forward time stepping method for fourth order nonlinear diffusion equations. Discrete and Continuous Dynamical Systems, 2011, 29 (4) : 1367-1391. doi: 10.3934/dcds.2011.29.1367

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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