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Finite difference scheme for the Willmore flow of graphs. (English) Zbl 1140.53032
Summary: 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 by T. Oberhuber [Computational study of the Willmore flow on graphs, Proc. Equadiff. 11 (2005); “Numerical solution for the Willmore flow of graphs”, in: M. Beneš, M. Kimura and T. Nakaki (eds.), Proc. Czech-Japanese Seminar in Applied Mathematics (2005), COE Lect. Note Vol. 3, Faculty of Mathematics, Kyushu Univ. Fukuoka, 126–138 (2006; Zbl 1145.65323)] 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.

MSC:
53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
74S20 Finite difference methods applied to problems in solid mechanics
35K35 Initial-boundary value problems for higher-order parabolic equations
35K55 Nonlinear parabolic equations
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References:
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