Geodesic
Finite difference scheme for the Willmore flow of graphs
Kybernetika, Tome 43 (2007) no. 6, pp. 855-867.

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In this article we discuss numerical scheme for the approximation of the Willmore flow of graphs. The scheme is based on the finite difference method. We improve the scheme we presented in Oberhuber [Obe-2005-2,Obe-2005-1] which is based on combination of the forward and the backward finite differences. The new scheme approximates the Willmore flow by the central differences and as a result it better preserves symmetry of the solution. Since it requires higher regularity of the solution, additional numerical viscosity is necessary in some cases. We also present theorem showing stability of the scheme together with the EOC and several results of the numerical experiments.
Classification : 35K35, 35K55, 53C44, 65M12, 65M20, 74S20
Keywords: Willmore flow; method of lines; curvature minimization; gradient flow; Laplace–Beltrami operator; Gauss curvature; central differences; numerical viscosity 
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     author = {Oberhuber, Tom\'a\v{s}},
     title = {Finite difference scheme for the {Willmore} flow of graphs},
     journal = {Kybernetika},
     pages = {855--867},
     publisher = {mathdoc},
     volume = {43},
     number = {6},
     year = {2007},
     mrnumber = {2388399},
     zbl = {1140.53032},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/KYB_2007__43_6_a9/}
}
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Oberhuber, Tomáš. Finite difference scheme for the Willmore flow of graphs. Kybernetika, Tome 43 (2007) no. 6, pp. 855-867. http://geodesic.mathdoc.fr/item/KYB_2007__43_6_a9/