Time discretizations of anisotropic Allen-Cahn equations (Q2856619)

Four different time discretizations are investigated for anisotropic Allen-Cahn equations, two nonlinear time discretizations and two linearized variants. In the nonlinear time discretization, the anisotropic gradient terms are taken fully implcitly. In the linearized version, they are replaced by an isotropic bilinear form weighted by a certain positive factor. The nonlinear semiimplicit scheme is unconditionally stable and involves no further parameters while the stability of the linearized version requires a selection of the weighting factor. Both the nonlinear and the linearized schemes require the solution of large-scale nonlinear algebraic systems in each time step. Some numerical experiments are presented to compare the accuracy of all time-discretization schemes using a simple two-dimensional model problem with known execution time. A three-dimensional example illustrates the influence of rigid body motions of the initial configuration on the occurrence of pinch-off.