Numerical Ricci-DeTurck flow

作者：Fritz Hans^*

来源：Numerische Mathematik, 2015, 131(2): 241-271.

DOI：10.1007/s00211-014-0690-5

We present a numerical method for the computation of the -dimensional Ricci-DeTurck flow. The Ricci flow is a partial differential equation (PDE) deforming a time-dependent metric on a closed Riemannian manifold in proportion to its Ricci curvature. The Ricci-DeTurck flow is a reparametrization of this flow using the harmonic map flow in order to get a strictly parabolic PDE. Our numerical method is based on the assumption that the manifold is embeddable into as a differentiable manifold. By this means, it is possible to do computations in the Euclidean coordinates of the ambient space. A weak formulation of the Ricci-DeTurck flow is derived such that it only contains tangential gradients. A spatial discretization of this formulation with finite elements on polyhedral hypersurfaces and a semi-implicit time discretization lead to an algorithm for computing the Ricci-DeTurck flow. We have performed numerical tests for two- and three-dimensional hypersurfaces using piecewise linear finite elements. The generalization to non-orientable hypersurfaces of higher codimensions is still open.

• 出版日期2015-10