Stabilized Exponential-SAV Schemes Preserving Energy Dissipation Law and Maximum Bound Principle for The Allen–Cahn Type Equations
Stabilized Exponential-SAV Schemes Preserving Energy Dissipation Law and Maximum Bound Principle for The Allen–Cahn Type Equations
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DOI：
10.1007/s10915-022-01921-9
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发表时间：
2022-03
影响因子：
2.5
通讯作者：
L. Ju;Xiao Li;Zhonghua Qiao
中科院分区：
文献类型：
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作者：
L. Ju;Xiao Li;Zhonghua Qiao
众所周知，Allen-Cahn 方程不仅满足能量耗散定律，而且还具有最大界限原理 (MBP)，即在适当的初始/边界条件下，其解的绝对值始终受到某个特定常数的逐点有界。近年来，标量辅助变量（SAV）方法及其许多变体由于其保留能量耗散定律的某些离散类似物的固有优势，在梯度流问题的数值求解中引起了广泛关注。然而，现有的 SAV 方案在应用于 Allen-Cahn 方程时通常无法保留 MBP。在本文中，我们为一类 Allen-Cahn 型方程开发并分析了新的一阶和二阶稳定指数 SAV 方案，该方案被证明可以在离散设置中同时保留能量耗散定律和 MBP。此外，这两种方案都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 solutions 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.