Huiling Jiang, Dongdong Hu, Haorong Huang, Hongliang Liu
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引用次数: 0
摘要
本文针对时间分式 Allen-Cahn 方程提出了两种高效数值方案,它们在离散设置中保留了最大约束原则和能量耗散。为此,我们利用最近一篇论文(Ju et al. in SIAM J Numer Anal 60:1905-1931, 2022)中提出的广义辅助变量方法,将治理方程重新表述为一个等价系统,该系统在每个连续级遵循修正的能量函数和最大约束原理。通过将黎曼-刘维尔分数导数的 L1 型公式与 Crank-Nicolson 方法相结合,我们分别引入了一阶和二阶稳定项,构建了两个新的线性隐式方案。在离散正交卷积技术的帮助下,这些方案在非均匀时间网格上被证明是能量稳定和最大边界原则保留的。此外,我们还获得了所提方案在没有任何时空步长比的情况下的唯一可解性。最后,我们报告了大量数值结果,以验证理论分析的正确性和所提方案在长时间模拟中的性能。
Linearly Implicit Schemes Preserve the Maximum Bound Principle and Energy Dissipation for the Time-fractional Allen–Cahn Equation
This paper presents two highly efficient numerical schemes for the time-fractional Allen–Cahn equation that preserve the maximum bound principle and energy dissipation in discrete settings. To this end, we utilize a generalized auxiliary variable approach proposed in a recent paper (Ju et al. in SIAM J Numer Anal 60:1905–1931, 2022) to reformulate the governing equation into an equivalent system that follows a modified energy functional and the maximum bound principle at each continuous level. By combining the L1-type formula of the Riemann–Liouville fractional derivative with the Crank–Nicolson method, we construct two novel linearly implicit schemes by introducing the first- and second-order stabilized terms, respectively. These schemes are proved to be energy stable and maximum bound principle preserving on nonuniform time meshes with the help of the discrete orthogonal convolution technique. In addition, we obtain the unique solvability of the proposed schemes without any time-space step ratio. Finally, we report extensive numerical results to verify the correctness of the theoretical analysis and the performance of the proposed schemes in long-time simulations.
期刊介绍:
Journal of Scientific Computing is an international interdisciplinary forum for the publication of papers on state-of-the-art developments in scientific computing and its applications in science and engineering.
The journal publishes high-quality, peer-reviewed original papers, review papers and short communications on scientific computing.