Complementary finite volume scheme for the anisotropic surface diffusion flow. (English) Zbl 1172.65375

Oberhuber, Tomáš

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

Keywords:

numerical examples; surface-diffusion flow; anisotropy; finite volume method; method of lines; motion of hypersurfaces; graph formulation; Runge-Kutta method