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Complementary finite volume scheme for the anisotropic surface diffusion flow. (English) Zbl 1172.65375
Handlovičová, Angela (ed.) et al., Algoritmy 2009. 18th conference on scientific computing, Vysoké Tatry – Podbsanské, Slovakia, March 15–20, 2009. Proceedings of contributed papers and posters. Bratislava: Slovak University of Technology, Faculty of Civil Engineering, Department of Mathematics and Descriptive Geometry (ISBN 978-80-227-3032-7/pbk). 153-164 (2009).
Summary: We study anisotropic surface diffusion flow of hypersurfaces in $$\mathbb R^n$$. We present a finite volume numerical scheme for the graph and the level-set formulation. The graph formulation is applied on surfaces in $$\mathbb R^3$$ and with the level-set formulation we evolve curves in $$\mathbb R^2$$. The discretisation in time is done by mean of the method of lines which gives us a system of ordinary differential equations. To solve this system we apply the Merson modification of the fourth order Runge-Kutta method. We show several qualitative results for both, the graph and the level-set formulation.
For the entire collection see [Zbl 1158.65005].

##### MSC:
 65M20 Method of lines for initial value and initial-boundary value problems involving PDEs 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 35K55 Nonlinear parabolic equations 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs 53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010) 58J35 Heat and other parabolic equation methods for PDEs on manifolds