Efficiently energy-dissipation-preserving ADI methods for solving two-dimensional nonlinear Allen-Cahn equation
Efficiently energy-dissipation-preserving ADI methods for solving two-dimensional nonlinear Allen-Cahn equation
复制标题
求解二维非线性 Allen-Cahn 方程的高效能量耗散守恒 ADI 方法
DOI：
10.1016/j.camwa.2022.10.023
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发表时间：
2022
期刊：
影响因子：
--
通讯作者：
Zilin Zhao
中科院分区：
文献类型：
--
作者：
Dingwen Deng;Zilin Zhao
近年来，非线性耗能系统的能量耗散保持数值方法（EDP NMs）因其良好的长期计算性能而受到广泛关注。最近，杨晓峰和他的合作者提出了不变能量二次化方法（IEQMs），用于构造非线性能量耗散系统（如梯度流）的线性化EDP-NM。然而，系统的线性代数方程组对应于所获得的EDP-NM涉及的变系数矩阵一般，从而导致在复杂的计算。此外，大多数有效的交替方向隐式（ADI）方法不能保持连续问题的结构，如能量耗散、能量守恒等。为了克服这些缺点，以二维非线性Allen-Cahn方程为例，利用IEQM方法发展了两种保能量耗散的ADI有限差分方法。对它们进行了详细的数值分析，如离散的能量耗散规律、误差估计和稳定性等。数值结果支持了理论分析，并证实了所提出的算法的效率和准确性。
Over the years, energy-dissipation-preserving (EDP) numerical methods (EDP-NMs) for the nonlinear system of energy dissipation have attracted considerable attention because they are good at long-term computations. More recently, the invariant energy quadratization methods (IEQMs) have been proposed by Xiaofeng Yang and his collaborators to construct linearized EDP-NMs for the nonlinear system of energy dissipation, such as gradient flows. However, the system of the linear algebraic equations corresponding to the obtained EDP-NMs involves the variable coefficient matrices in general, thus resulting in complex computations. Besides, most of efficient alternating direction implicit (ADI) methods are not able to preserve the structures of continuous problem, such as, energy dissipation, energy conservation. To overcome these shortcomings, taking two-dimensional (2D) nonlinear Allen-Cahn equation for example, two energy-dissipation-preserving ADI finite difference methods (FDMs) have been developed by using IEQMs. Numerical analyses of them, such as, the discrete energy-dissipation laws, error estimations and stabilities, have been derived in detail. Numerical results support the theoretical analysis, and confirm the efficiency and accuracy of the proposed algorithms.