| Abstract: |
| It is well known that the Allen-Cahn equation is a gradient flow.
In 1993, Elliott and Stuart proved that the k-step backward differentiation formula (BDFk) applied to the Allen-Cahn equation preserves its gradient structure for k=1, 2 and 3, if the time step is small enough. In 1996, Stuart and Humphries generalized this result for the BDFk method applied to the gradient flow of a semiconvex function and they left open the question for k=4, 5 and 6. In this talk, we show that the BDF4 and BDF5 schemes are gradient stable and we give a negative answer to the question for the BDF6 scheme. We also give some applications of these results to the Allen-Cahn and Cahn-Hilliard equations. |
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