论文标题
基于标量辅助变量和梯度流量不变二次化方法的逐步解决方案
Step-by-step solving schemes based on scalar auxiliary variable and invariant energy quadratization approaches for gradient flows
论文作者
论文摘要
在本文中，我们提出了几种新型的数值技术来处理梯度流中的非线性术语。这些分步求解方案，称为3S-SAV和3S-IEQ方案，基于最近流行的标量辅助变量（SAV）和不变能量倍率化（IEQ）方法。在这些构建的数值方法中，可以逐步计算相位函数$ ϕ $和辅助变量。与传统的SAV/IEQ方法相比，新颖的3S-SAV/3S-IEQ方案有许多优势。首先，我们不需要从非线性自由能势/密度函数下方限制的限制。其次，可以在3S-SAV/3S-IEQ方法中完全明确处理与非线性函数的辅助变量。特别是，为了基于IEQ方法求解离散方案，线性系统通常涉及在每个时间步骤都会改变的可变系数。但是，基于3S-IEQ方法的离散方案导致具有恒定系数的线性方程。传统的SAV/IEQ和3S-SAV/3S-IEQ方法的两项比较研究被认为显示了准确性和效率。最后，我们提供各种2D数值模拟，以证明稳定性和准确性。
In this paper, we propose several novel numerical techniques to deal with nonlinear terms in gradient flows. These step-by-step solving schemes, termed 3S-SAV and 3S-IEQ schemes, are based on recently popular scalar auxiliary variable (SAV) and invariant energy quadratization (IEQ) approaches. In these constructed numerical methods, the phase function $ϕ$ and auxiliary variable can be calculated step-by-step. Compared with the traditional SAV/IEQ approaches, there are many advantages for the novel 3S-SAV/3S-IEQ schemes. Firstly, we do not need the restriction of the bounded from below of the nonlinear free energy potential/density function. Secondly, the auxiliary variable combined with nonlinear function can be treated totally explicitly in the 3S-SAV/3S-IEQ approaches. Specially, for solving the discrete scheme based on IEQ approach, the linear system usually involves variable coefficients which change at each time step. However, the discrete scheme based on 3S-IEQ approach leads to linear equation with constant coefficients. Two comparative studies of traditional SAV/IEQ and 3S-SAV/3S-IEQ approaches are considered to show the accuracy and efficiency. Finally, we present various 2D numerical simulations to demonstrate the stability and accuracy.
