# American Institute of Mathematical Sciences

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

 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

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.
Citation: Andrea L. Bertozzi, Ning Ju, Hsiang-Wei Lu. A biharmonic-modified forward time stepping method for fourth order nonlinear diffusion equations. Discrete & Continuous Dynamical Systems - A, 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. doi: 10.1007/s002110050276. Google Scholar [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. doi: 10.1093/imanum/18.2.287. Google Scholar [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. doi: 10.1142/S0218202599000336. Google Scholar [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. doi: 10.1090/S0025-5718-99-01015-7. Google Scholar [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. doi: 10.1007/s002110050377. Google Scholar [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. doi: 10.1137/S0036142997331669. Google Scholar [7] F. Bernis and A. Friedman, Higher order nonlinear degenerate parabolic equations,, J. Diff. Equations, 83 (1990), 179. doi: 10.1016/0022-0396(90)90074-Y. Google Scholar [8] A. L. Bertozzi and M. C. Pugh, Long-wave instabilities and saturation in thin film equations,, Comm. Pure Appl. Math., 51 (1998), 625. doi: 10.1002/(SICI)1097-0312(199806)51:6<625::AID-CPA3>3.0.CO;2-9. Google Scholar [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. doi: 10.1002/(SICI)1097-0312(199602)49:2<85::AID-CPA1>3.0.CO;2-2. Google Scholar [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. Google Scholar [11] A. L. Bertozzi, The mathematics of moving contact lines in thin liquid films,, Notices of the American Math. Soc., 45 (1998), 689. Google Scholar [12] A. L. Bertozzi, M. P. Brenner, T. F. Dupont and L. P. Kadanoff, Singularities and similarities in interface flow,, in, 100 (1994), 155. Google Scholar [13] A. L. Bertozzi, G. Grün and T. P. Witelski, Dewetting films: bifurcations and concentrations,, Nonlinearity, 14 (2001), 1569. doi: 10.1088/0951-7715/14/6/309. Google Scholar [14] M. Brenner and A. Bertozzi, Spreading of droplets on a solid surface,, Phys. Rev. Lett., 71 (1993), 593. doi: 10.1103/PhysRevLett.71.593. Google Scholar [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. doi: 10.1103/PhysRevE.47.4169. Google Scholar [16] J. Douglas, Jr. and T. Dupont, Alternating-direction Galerkin methods on rectangles,, in, (1971), 133. Google Scholar [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. doi: 10.1103/PhysRevE.47.4182. Google Scholar [18] P. Ehrhard and S. H. Davis, Non-isothermal spreading of liquid drops on horizontal plates,, J. Fluid. Mech., 229 (1991), 365. doi: 10.1017/S0022112091003063. Google Scholar [19] C. M. Elliott and A. M. Stuart, The global dynamics of discrete semilinear parabolic equations,, SIAM J. Numer. Anal., 30 (1993), 1622. doi: 10.1137/0730084. Google Scholar [20] C. M. Elliott and H. Garke, On the cahn hilliard equation with degenerate mobility,, SIAM J. Math. Anal., 27 (1996), 404. doi: 10.1137/S0036141094267662. Google Scholar [21] D. Eyre, An unconditionally stable one-step scheme for gradient systems,, Unpublished paper, (1998). Google Scholar [22] R. Ferreira and F. Bernis, Source-type solutions to thin-film equations in higher dimensions,, Euro. J. Appl. Math., 9 (1997), 507. doi: 10.1017/S0956792597003197. Google Scholar [23] K. Glasner, Nonlinear preconditioning for diffuse interfaces,, J. Comp. Phys., 174 (2001), 695. doi: 10.1006/jcph.2001.6933. Google Scholar [24] K. B. Glasner and T. P. Witelski, Coarsening dynamics of dewetting films,, Phys. Rev. E, 67 (2003). doi: 10.1103/PhysRevE.67.016302. Google Scholar [25] K. Glasner, A diffuse interface approach to Hele-Shaw flow,, Nonlinearity, 16 (2003), 49. doi: 10.1088/0951-7715/16/1/304. Google Scholar [26] K. B. Glasner and T. P. Witelski, Coarsening dynamics of dewetting films,, Phys. Rev. E, 67 (2003). doi: 10.1103/PhysRevE.67.016302. Google Scholar [27] R. E. Goldstein, A. I. Pesci and M. J. Shelley, Topology transitions and singularities in viscous flows,, Physical Review Letters, 70 (1993), 3043. doi: 10.1103/PhysRevLett.70.3043. Google Scholar [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. doi: 10.1103/PhysRevLett.75.3665. Google Scholar [29] H. P. Greenspan, On the motion of a small viscous droplet that wets a surface,, J. Fluid Mech., 84 (1978), 125. doi: 10.1017/S0022112078000075. Google Scholar [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. Google Scholar [31] J. Greer, A. Bertozzi and G. Sapiro, Fourth order partial differential equations on general geometries,, J. Computational Physics, 216 (2006), 216. doi: 10.1016/j.jcp.2005.11.031. Google Scholar [32] G. Grün and M. Rumpf, Nonnegativity preserving convergent schemes for the thin film equation,, Num. Math., 87 (2000), 113. doi: 10.1007/s002110000197. Google Scholar [33] L. M. Hocking, A moving fluid interface on a rough surface,, Journal of Fluid Mechanics, 76 (1976), 801. doi: 10.1017/S0022112076000906. Google Scholar [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. doi: 10.1017/S0022112077000123. Google Scholar [35] L. M. Hocking, Sliding and spreading of thin two-dimensional drops,, Q. J. Mech. Appl. Math., 34 (1981), 37. doi: 10.1093/qjmam/34.1.37. Google Scholar [36] L. M. Hocking, Rival contact-angle models and the spreading of drops,, J. Fluid. Mech., 239 (1992), 671. doi: 10.1017/S0022112092004579. Google Scholar [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. doi: 10.1006/jcph.1994.1170. Google Scholar [38] M. G. Lippman, Relations entre les phènoménes électriques et capillaires,, Ann. Chim. Phys., 5 (1875), 494. Google Scholar [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. doi: 10.1017/S0022112007008154. Google Scholar [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. doi: 10.1063/1.858006. Google Scholar [41] T. G. Myers, Thin films with high surface tension,, SIAM Rev., 40 (1998), 441. doi: 10.1137/S003614459529284X. Google Scholar [42] P. Neogi and C. A. Miller, Spreading kinetics of a drop on a smooth solid surface,, J. Colloid Interface Sci., 86 (1982), 525. doi: 10.1016/0021-9797(82)90097-2. Google Scholar [43] A. Oron, S. H. Davis and S. George Bankoff, Long-scale evolution of thin liquid films,, Rev. Mod. Phys., 69 (1997), 931. doi: 10.1103/RevModPhys.69.931. Google Scholar [44] W. H. Press, S. A. Teukolsky, W. T. Vetterling and B. P. Flannery, "Numerical Recipes in C,", Second Edition, (1993). Google Scholar [45] C.-B. Schoenlieb and A. Bertozzi, Unconditionally stable schemes for higher order inpainting,, 2010., (). Google Scholar [46] M. J. Shelley, R. E. Goldstein and A. I. Pesci, Topological transitions in Hele-Shaw flow,, in, (1993), 167. Google Scholar [47] P. Smereka, Semi-implicit level set methods for curvature flow and for motion by surface diffusion,, J. Sci. Comp., 19 (2003), 439. doi: 10.1023/A:1025324613450. Google Scholar [48] S. M. Troian, E. Herbolzheimer, S. A. Safran and J. F. Joanny, Fingering instabilities of driven spreading films,, Europhys. Lett., 10 (1989), 25. doi: 10.1209/0295-5075/10/1/005. Google Scholar [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. doi: 10.1103/PhysRevE.68.066703. Google Scholar [50] T. P. Witelski and M. Bowen, Adi methods for high order parabolic equations,, Appl. Num. Anal., 45 (2003), 331. Google Scholar [51] T. P. Witelski, Equilibrium solutions of a degenerate singular Cahn-Hilliard equation,, Applied Mathematics Letters, 11 (1998), 127. doi: 10.1016/S0893-9659(98)00092-5. Google Scholar [52] L. Zhornitskaya and A. L. Bertozzi, Positivity-preserving numerical schemes for lubrication-type equations,, SIAM J. Numer. Anal., 37 (2000), 523. doi: 10.1137/S0036142998335698. Google Scholar

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##### 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. doi: 10.1007/s002110050276. Google Scholar [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. doi: 10.1093/imanum/18.2.287. Google Scholar [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. doi: 10.1142/S0218202599000336. Google Scholar [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. doi: 10.1090/S0025-5718-99-01015-7. Google Scholar [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. doi: 10.1007/s002110050377. Google Scholar [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. doi: 10.1137/S0036142997331669. Google Scholar [7] F. Bernis and A. Friedman, Higher order nonlinear degenerate parabolic equations,, J. Diff. Equations, 83 (1990), 179. doi: 10.1016/0022-0396(90)90074-Y. Google Scholar [8] A. L. Bertozzi and M. C. Pugh, Long-wave instabilities and saturation in thin film equations,, Comm. Pure Appl. Math., 51 (1998), 625. doi: 10.1002/(SICI)1097-0312(199806)51:6<625::AID-CPA3>3.0.CO;2-9. Google Scholar [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. doi: 10.1002/(SICI)1097-0312(199602)49:2<85::AID-CPA1>3.0.CO;2-2. Google Scholar [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. Google Scholar [11] A. L. Bertozzi, The mathematics of moving contact lines in thin liquid films,, Notices of the American Math. Soc., 45 (1998), 689. Google Scholar [12] A. L. Bertozzi, M. P. Brenner, T. F. Dupont and L. P. Kadanoff, Singularities and similarities in interface flow,, in, 100 (1994), 155. Google Scholar [13] A. L. Bertozzi, G. Grün and T. P. Witelski, Dewetting films: bifurcations and concentrations,, Nonlinearity, 14 (2001), 1569. doi: 10.1088/0951-7715/14/6/309. Google Scholar [14] M. Brenner and A. Bertozzi, Spreading of droplets on a solid surface,, Phys. Rev. Lett., 71 (1993), 593. doi: 10.1103/PhysRevLett.71.593. Google Scholar [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. doi: 10.1103/PhysRevE.47.4169. Google Scholar [16] J. Douglas, Jr. and T. Dupont, Alternating-direction Galerkin methods on rectangles,, in, (1971), 133. Google Scholar [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. doi: 10.1103/PhysRevE.47.4182. Google Scholar [18] P. Ehrhard and S. H. Davis, Non-isothermal spreading of liquid drops on horizontal plates,, J. Fluid. Mech., 229 (1991), 365. doi: 10.1017/S0022112091003063. Google Scholar [19] C. M. Elliott and A. M. Stuart, The global dynamics of discrete semilinear parabolic equations,, SIAM J. Numer. Anal., 30 (1993), 1622. doi: 10.1137/0730084. Google Scholar [20] C. M. Elliott and H. Garke, On the cahn hilliard equation with degenerate mobility,, SIAM J. Math. Anal., 27 (1996), 404. doi: 10.1137/S0036141094267662. Google Scholar [21] D. Eyre, An unconditionally stable one-step scheme for gradient systems,, Unpublished paper, (1998). Google Scholar [22] R. Ferreira and F. Bernis, Source-type solutions to thin-film equations in higher dimensions,, Euro. J. Appl. Math., 9 (1997), 507. doi: 10.1017/S0956792597003197. Google Scholar [23] K. Glasner, Nonlinear preconditioning for diffuse interfaces,, J. Comp. Phys., 174 (2001), 695. doi: 10.1006/jcph.2001.6933. Google Scholar [24] K. B. Glasner and T. P. Witelski, Coarsening dynamics of dewetting films,, Phys. Rev. E, 67 (2003). doi: 10.1103/PhysRevE.67.016302. Google Scholar [25] K. Glasner, A diffuse interface approach to Hele-Shaw flow,, Nonlinearity, 16 (2003), 49. doi: 10.1088/0951-7715/16/1/304. Google Scholar [26] K. B. Glasner and T. P. Witelski, Coarsening dynamics of dewetting films,, Phys. Rev. E, 67 (2003). doi: 10.1103/PhysRevE.67.016302. Google Scholar [27] R. E. Goldstein, A. I. Pesci and M. J. Shelley, Topology transitions and singularities in viscous flows,, Physical Review Letters, 70 (1993), 3043. doi: 10.1103/PhysRevLett.70.3043. Google Scholar [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. doi: 10.1103/PhysRevLett.75.3665. Google Scholar [29] H. P. Greenspan, On the motion of a small viscous droplet that wets a surface,, J. Fluid Mech., 84 (1978), 125. doi: 10.1017/S0022112078000075. Google Scholar [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. Google Scholar [31] J. Greer, A. Bertozzi and G. Sapiro, Fourth order partial differential equations on general geometries,, J. Computational Physics, 216 (2006), 216. doi: 10.1016/j.jcp.2005.11.031. Google Scholar [32] G. Grün and M. Rumpf, Nonnegativity preserving convergent schemes for the thin film equation,, Num. Math., 87 (2000), 113. doi: 10.1007/s002110000197. Google Scholar [33] L. M. Hocking, A moving fluid interface on a rough surface,, Journal of Fluid Mechanics, 76 (1976), 801. doi: 10.1017/S0022112076000906. Google Scholar [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. doi: 10.1017/S0022112077000123. Google Scholar [35] L. M. Hocking, Sliding and spreading of thin two-dimensional drops,, Q. J. Mech. Appl. Math., 34 (1981), 37. doi: 10.1093/qjmam/34.1.37. Google Scholar [36] L. M. Hocking, Rival contact-angle models and the spreading of drops,, J. Fluid. Mech., 239 (1992), 671. doi: 10.1017/S0022112092004579. Google Scholar [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. doi: 10.1006/jcph.1994.1170. Google Scholar [38] M. G. Lippman, Relations entre les phènoménes électriques et capillaires,, Ann. Chim. Phys., 5 (1875), 494. Google Scholar [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. doi: 10.1017/S0022112007008154. Google Scholar [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. doi: 10.1063/1.858006. Google Scholar [41] T. G. Myers, Thin films with high surface tension,, SIAM Rev., 40 (1998), 441. doi: 10.1137/S003614459529284X. Google Scholar [42] P. Neogi and C. A. Miller, Spreading kinetics of a drop on a smooth solid surface,, J. Colloid Interface Sci., 86 (1982), 525. doi: 10.1016/0021-9797(82)90097-2. Google Scholar [43] A. Oron, S. H. Davis and S. George Bankoff, Long-scale evolution of thin liquid films,, Rev. Mod. Phys., 69 (1997), 931. doi: 10.1103/RevModPhys.69.931. Google Scholar [44] W. H. Press, S. A. Teukolsky, W. T. Vetterling and B. P. Flannery, "Numerical Recipes in C,", Second Edition, (1993). Google Scholar [45] C.-B. Schoenlieb and A. Bertozzi, Unconditionally stable schemes for higher order inpainting,, 2010., (). Google Scholar [46] M. J. Shelley, R. E. Goldstein and A. I. Pesci, Topological transitions in Hele-Shaw flow,, in, (1993), 167. Google Scholar [47] P. Smereka, Semi-implicit level set methods for curvature flow and for motion by surface diffusion,, J. Sci. Comp., 19 (2003), 439. doi: 10.1023/A:1025324613450. Google Scholar [48] S. M. Troian, E. Herbolzheimer, S. A. Safran and J. F. Joanny, Fingering instabilities of driven spreading films,, Europhys. Lett., 10 (1989), 25. doi: 10.1209/0295-5075/10/1/005. Google Scholar [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. doi: 10.1103/PhysRevE.68.066703. Google Scholar [50] T. P. Witelski and M. Bowen, Adi methods for high order parabolic equations,, Appl. Num. Anal., 45 (2003), 331. Google Scholar [51] T. P. Witelski, Equilibrium solutions of a degenerate singular Cahn-Hilliard equation,, Applied Mathematics Letters, 11 (1998), 127. doi: 10.1016/S0893-9659(98)00092-5. Google Scholar [52] L. Zhornitskaya and A. L. Bertozzi, Positivity-preserving numerical schemes for lubrication-type equations,, SIAM J. Numer. Anal., 37 (2000), 523. doi: 10.1137/S0036142998335698. Google Scholar
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