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Fourth Order Partial Differential Equations

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Fourth order partial differential equations on general geometriesIntroductionBasics, challenges, and goalsExample: The heat equationExample: Linear fourth order diffusionBasic observations and challengesLevel set methods for fourth order geometric motionsGoalsThe embedded equationComputational domain: a band containing SExtension of initial surface dataBoundary conditionsRe-extension of surface dataTime steppingConvexity splittingExample: Linear fourth order diffusionSolving the linear systemsIterative solver: conjugate gradient methodADI schemesExample: Linear fourth order diffusionNumerical implementationData structuresSurface complexityVisualizationNumerical examples: linear fourth order diffusionDiffusion on the unit circleDiffusion on the unit sphereDiffusion on the Stanford BunnyNumerical examples: Cahn-Hilliard equationNumerical examples: thin film fluid flowExample: Thin film on an ellipseSphere with gravity: fingering instabilityConclusionsAcknowledgmentsSpatial discretizationDegeneracy and ADIADI without convexity splittingFourth order problemReferencesFourth order partial differential equations on general geometriesJohn B. Greera,*, Andrea L. Bertozzib, Guillermo SapirocaDepartment of Mathematics, Courant Institute for Mathematical Sciences, New York University, 251 Mercer Street,New York, NY 10012-1185, United StatesbDepartment of Mathematics, UCLA, Los Angeles, CA 90095, United StatescElectrical and Computer Engineering, University of Minnesota, Minneapolis, MN 55455, United StatesReceived 22 March 2005; received in revised form 3 November 2005; accepted 30 November 2005Available online 23 January 2006AbstractWe extend a recently introduced method for numerically solving partial differential equations on implicit surfaces [M.Bertalmı´o, L.T. Cheng, S. Osher, G. Sapiro. Variational problems and partial differential equations on implicit surfaces,J. Comput. Phys. 174 (2) (2001) 759–780] to fourth order PDEs including the Cahn–Hilliard equation and a lubricationmodel for curved surfaces. By representing a surface in RNas the level set of a smooth function, /, we compute thePDE using only finite differences on a standard Cartesian mesh in RN. The higher order equations introduce a numberof challenges that are of less concern when applying this method to first and second order PDEs. Many of these problems,such as time-stepping restrictions and large stencil sizes, are shared by standard fourth order equations in Euclideandomains, but others are caused by the extreme degeneracy of the PDEs that result from this method and the general geom-etry. We approach these difficulties by applying convexity splitting methods, ADI schemes, and iterative solvers. We dis-cuss in detail the differences between computing these fourth order equations and computing the first and second orderPDEs considered in earlier work. We explicitly derive schemes for the linear fourth order diffusion, the Cahn–Hilliardequation for phase transition in a binary alloy, and surface tension driven flows on complex geometries. Numerical exam-ples validating our methods are presented for these flows for data on general surfaces. 2005 Elsevier Inc. All rights reserved.Keywords: Nonlinear partial differential equations; Level set method; Implicit surfaces; Higher order equations; Lubrication theory;Cahn–Hilliard equation; ADI methods1. IntroductionPartial differential equations (PDEs) defined on surfaces embedded in R3arise in a wide range of applica-tions, including fluid dynamics, biology (e.g., fluids on the lungs), materials science (e.g., ice formation), elec-tromagnetism, image processing (e.g., images on manifolds and inverse problems such as EEG ), computergraphics (e.g., water flowing on a surface), computer aided geometric design (e.g., special curves on surfaces),0021-9991/$ - see front matter  2005 Elsevier Inc. All rights reserved.doi:10.1016/j.jcp.2005.11.031*Corresponding author. Tel.: +1 646 291 6411; fax: +1 212 995 4121.E-mail addresses: [email protected] (J.B. Greer), [email protected] (A.L. Bertozzi), [email protected] (G. Sapiro).Journal of Computational Physics 216 (2006) 216–246www.elsevier.com/locate/jcpand pattern formation. The work in this paper is concerned with fourth order differential equations, whichhave interests in all the areas mentioned abov e. Examples of physical flows modeled by fourth order PDEsinclude ice formation [42,43], fluids on lungs [27], brain warping [39,58], and designing special curves on sur-faces [28, 39]. In this paper we extend the work in [8] to these high order flows. We represent the surface witharbitrary geometry implicitly, as the level set of a smooth function defined in all of the embedding space R3,and rewrite the relevant equatio ns in terms of Euclidean coordinates and derivatives of the level set function(see Section 2). This method has been used for solving first and second order equations on static, [8,33,37,38],as well as dynamic, [2,62], surfaces. In [8], the authors introduced the general framework and showed how tosolve second ord er linear and nonlinear diffusions and reaction-diffusion equations on implicitly defined sur-faces. In [33,37], the authors solved the Eikonal equation on surfaces like those in [8] (while in the first paperthe work is for triangulated surfaces, in the second implicit representations are used). In these works, staticsurfaces wer e considered. The authors of [2,62] solve second order diffusion equations on interfaces thatare deforming subject to an extrinsic flow. As discussed in the above papers in detail, implicit representationsprovide a natural means for addressing these flows on arbitrary surfaces.Solving PDEs and variational problems with polynomial meshes involves the non-trivial discretization ofthe equations in general polygonal grids, as well as the difficult numerical computation of other quantities likeprojections onto the discretized surface (when computing gradients and Laplaci ans for example). Althoughthe use of triangulated surfaces is quite popular, there is still no consensus on how to compute simple differ-ential characteristics such as tangents, normals, principal directions, and curvatures. On the other hand, it iscommonly accepted that computing these objects for iso-surfaces (implicit representations) is simpler an dmore accurate and robust. This problem becomes even more significant when we not only have to computethese first and


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