论文标题
稳定的指数-SAV方案保留了能量耗散定律和Allen-CAHN类型方程的最大结合原理
Stabilized exponential-SAV schemes preserving energy dissipation law and maximum bound principle for the Allen-Cahn type equations
论文作者
论文摘要
众所周知，Allen-CaHN方程不仅满足耗能法则，而且还具有最大结合原理（MBP）的意义，即在适当的初始/边界条件下，其解决方案的绝对值在某些特定常数中均以某种特定的常数为界。近年来，标量辅助变量（SAV）方法及其许多变体在数值解决方案中引起了梯度流问题的广泛关注，因为它们固有的优势保留了能量耗散定律的某些离散类似物。但是，现有的SAV方案通常无法保留在Allen-Cahn方程式上时保留MBP。在本文中，我们为一类Allen-CAHN类型方程开发和分析了新的一阶和二阶稳定的指数-SAV方案，这些方程可同时保留离散设置中的能量耗散定律和MBP。此外，对于两个方案，严格获得了数值解的最佳误差估计。还进行了广泛的数值测试和比较，以证明所提出的方案的性能。
It is well-known that the Allen-Cahn equation not only satisfies the energy dissipation law but also possesses the maximum bound principle (MBP) in the sense that the absolute value of its solution is pointwise bounded for all time by some specific constant under appropriate initial/boundary conditions. In recent years, the scalar auxiliary variable (SAV) method and many of its variants have attracted much attention in numerical solution for gradient flow problems due to their inherent advantage of preserving certain discrete analogues of the energy dissipation law. However, existing SAV schemes usually fail to preserve the MBP when applied to the Allen-Cahn equation. In this paper, we develop and analyze new first- and second-order stabilized exponential-SAV schemes for a class of Allen-Cahn type equations, which are shown to simultaneously preserve the energy dissipation law and MBP in discrete settings. In addition, optimal error estimates for the numerical solutions are rigorously obtained for both schemes. Extensive numerical tests and comparisons are also conducted to demonstrate the performance of the proposed schemes.
