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

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  • 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.
    Mathematics Subject Classification: 65M06, 65M12, 65M70.


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  • [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.


    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.


    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.


    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.


    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.


    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.


    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.


    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.


    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.


    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.


    A. L. Bertozzi, The mathematics of moving contact lines in thin liquid films, Notices of the American Math. Soc., 45 (1998), 689-697.


    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.


    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.


    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.


    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.


    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.


    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.


    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.


    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.


    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.


    D. Eyre, An unconditionally stable one-step scheme for gradient systems, Unpublished paper, June 9, 1998.


    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.


    K. Glasner, Nonlinear preconditioning for diffuse interfaces, J. Comp. Phys., 174 (2001), 695-71.doi: 10.1006/jcph.2001.6933.


    K. B. Glasner and T. P. Witelski, Coarsening dynamics of dewetting films, Phys. Rev. E, 67 (2003), 016302.doi: 10.1103/PhysRevE.67.016302.


    K. Glasner, A diffuse interface approach to Hele-Shaw flow, Nonlinearity, 16 (2003), 49-66.doi: 10.1088/0951-7715/16/1/304.


    K. B. Glasner and T. P. Witelski, Coarsening dynamics of dewetting films, Phys. Rev. E, 67 (2003), 016302.doi: 10.1103/PhysRevE.67.016302.


    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.


    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.


    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.


    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.


    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.


    G. Grün and M. Rumpf, Nonnegativity preserving convergent schemes for the thin film equation, Num. Math., 87 (2000), 113-152.doi: 10.1007/s002110000197.


    L. M. Hocking, A moving fluid interface on a rough surface, Journal of Fluid Mechanics, 76 (1976), 801-817.doi: 10.1017/S0022112076000906.


    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.


    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.


    L. M. Hocking, Rival contact-angle models and the spreading of drops, J. Fluid. Mech., 239 (1992), 671-68.doi: 10.1017/S0022112092004579.


    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.


    M. G. Lippman, Relations entre les phènoménes électriques et capillaires, Ann. Chim. Phys., 5 (1875), 494-548.


    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.


    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.


    T. G. Myers, Thin films with high surface tension, SIAM Rev., 40 (1998), 441-462 (electronic).doi: 10.1137/S003614459529284X.


    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.


    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.


    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.


    C.-B. Schoenlieb and A. BertozziUnconditionally stable schemes for higher order inpainting, 2010.


    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.


    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.


    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.


    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.


    T. P. Witelski and M. Bowen, Adi methods for high order parabolic equations, Appl. Num. Anal., 45 (2003), 331-35.


    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.


    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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