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

Andrea L. Bertozzi; Ning Ju; Hsiang-Wei Lu

doi:10.3934/dcds.2011.29.1367

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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
Related Papers

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 & 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

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. 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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American Institute of Mathematical Sciences

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

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American Institute of Mathematical Sciences

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

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