题目：Discrete Maximum Principle for Allen-Cahn Equations: Monotone Schemes and Cut-Off Post-Processing

报告人：杨将 副教授 （南方科技大学）

地点：致远楼101室

时间：2023年04月16日（星期日）下午3:00-4:00

摘要：Maximum principle is one intrinsic property of Allen-Cahn equations. Numerically preserving this structure is necessary for the consistency with phase-field modeling, particularly for the singular logarithmic potentials. In this talk, we present two classes of numerical schemes to preserve discrete maximum principle for Allen-Cahn equations. In the first part, we establish a framework of monotone schemes, in which only several concise and reasonable conditions are assumed. These conditions can guarantee both the unique solvability and the maximum principle. We apply this framework to some well-studied numerical schemes. It is found that the framework is very effective to check whether a scheme can preserve the discrete maximum principle. In the second part, we focus on the cut-off post-processing to preserve the discrete maximum principle. With suitable integrators in time, a lumped mass finite element method in space, and a cut-off operation acting at the finite element nodal points at each time level, we successfully establish an optimal error bound of $O(\tau^{k+1}+h^r)$. The accuracy can be made arbitrarily high-order by choosing large $k$ and $r$. Moreover, combining the cut-off strategy with the scalar auxiliary value (SAV) technique, we develop a class of energy-stable and maximum bound preserving schemes.

欢迎各位参加！