论文标题
Allen-CAHN类型梯度流的广义SAV-EDPERTAINT
Generalized SAV-exponential integrator schemes for Allen-Cahn type gradient flows
论文作者
论文摘要
能量耗散定律和最大结合原理（MBP）是众所周知的Allen-Cahn方程的两个重要物理特征。尽管某些常用的一阶时间步进计划已成为方程式无条件地保留能量耗散法和MBP，但对于现有的二阶甚至高阶方案仍然需要对时间步长的限制，以便具有这种同时保存。在本文中，我们为一类Allen-CAHN型梯度流的新型一阶线性数值方案开发和分析。我们的方案结合了具有稳定项的广义标量辅助变量（SAV）方法和指数时间积分器，而标准中央差异模板则用于离散空间差分运算符。我们不仅证明了它们在离散环境中对耗能法和MBP的无条件保存，而且还会在固定空间网格下得出其最佳的时间误差估计。还进行了数值实验以证明所提出方案的性质和性能。
The energy dissipation law and the maximum bound principle (MBP) are two important physical features of the well-known Allen-Cahn equation. While some commonly-used first-order time stepping schemes have turned out to preserve unconditionally both energy dissipation law and MBP for the equation, restrictions on the time step size are still needed for existing second-order or even higher-order schemes in order to have such simultaneous preservation. In this paper, we develop and analyze novel first- and second-order linear numerical schemes for a class of Allen-Cahn type gradient flows. Our schemes combine the generalized scalar auxiliary variable (SAV) approach and the exponential time integrator with a stabilization term, while the standard central difference stencil is used for discretization of the spatial differential operator. We not only prove their unconditional preservation of the energy dissipation law and the MBP in the discrete setting, but also derive their optimal temporal error estimates under fixed spatial mesh. Numerical experiments are also carried out to demonstrate the properties and performance of the proposed schemes.
