Adaptive time-stepping and maximum-principle preserving Lagrangian schemes for gradient flows

We develop in this paper an adaptive time-stepping approach for gradient flows with distinct treatments for conservative and non-conservative dynamics. For the non-conservative gradient flows in Lagrangian coordinates, we propose a modified formulation augmented by auxiliary terms to guarantee positivity of the determinant, and prove that the corresponding adaptive second-order Backward Difference Formulas (BDF2) scheme preserves energy stability and the maximum principle under the time-step ratio constraint $0<r_n\le r_{\max }\le \frac{3}{2}$. On the other hand, for the conservative Wasserstein gradient flows in Lagrangian coordinates, we propose an adaptive BDF2 scheme which is shown to be energy dissipative, and positivity preserving under the time-step ratio constraint $0<r_n\le r_{\max }\le \frac{3+\sqrt{17}}{2}$ in 1D and $0<r_n\le r_{\max }\le \frac{5}{4}$ in 2D, respectively. We also present ample numerical simulations in 1D and 2D to validate the efficiency and accuracy of the proposed schemes.