Dynamic-stabilization-based linear schemes for the Allen-Cahn equation with degenerate mobility: MBP and energy stability

Abstract

In this paper, we investigate linear first- and second-order numerical schemes for the Allen--Cahn equation with a general (possibly degenerate) mobility. Compared with existing numerical methods, our schemes employ a novel dynamic stabilization approach that guarantees unconditional preservation of the maximum bound principle (MBP) and energy stability. A key advance is that the discrete energy stability remains valid even in the presence of degenerate mobility-a property we refer to as mobility robustness. Rigorous maximum-norm error estimates are also established. In particular, for the second-order scheme, we introduce a new prediction strategy with a cut-off preprocessing procedure on the extrapolation solution, and only one linear system needs to be solved per time level. Representative numerical examples are provided to validate the theoretical findings and performance of the proposed schemes.