The numerical integration of phase-field equations is a delicate task which needs to recover at the discrete level intrinsic properties of the solution such as energy dissipation and maximum principle. Although the theory of energy dissipation for classical phase field models is well established, the corresponding theory for time-fractional phase-field models is still incomplete. In this article, we study certain nonlocal-in-time energies using the first-order stabilized semi-implicit L1 scheme. In particular, we will establish a discrete fractional energy law and a discrete weighted energy law. The extension for a $(2-{\alpha})$-order L1 scalar auxiliary variable scheme will be investigated. Moreover, we demonstrate that the energy bound is preserved for the L1 schemes with nonuniform time steps. Several numerical experiments are carried to verify our theoretical analysis.
翻译:阶段- 实地方程式的数值整合是一项微妙的任务,需要在诸如能源消散和最大原则等解决方案的离散层面内在特性中加以恢复。 虽然古典阶段字段模型的能源消散理论已经确立,但时间偏差阶段字段模型的相应理论仍然不完整。 在本篇文章中,我们使用一级稳定半隐含L1计划研究某些非当地时能。特别是,我们将制定离散分分能法和离散加权能源法。将调查一个$(2- phalpha}) $- order L1 卡拉尔辅助变量计划的扩展。此外,我们证明,L1 计划所约束的能源保留在不统一的时间步骤中。我们进行了数项实验,以核实我们的理论分析。