Fast explicit operator splitting method and time-step adaptivity for fractional non-local Allen-Cahn model

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• 作者：Shuying Zhai ; ^a ; ^{zhaishuying123456@163.com" class="auth_mail" title="E-mail the corresponding author} ; Zhifeng Weng^a ; ^{zfwmath@163.com" class="auth_mail" title="E-mail the corresponding author} ; Xinlong Feng^b ; ^{fxlmath@xju.edu.cn" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Non-local Allen&ndash ; Cahn equation ; Fractional Laplacian ; Operator splitting method ; Spectral method ; Adaptive time stepping
• 刊名：Applied Mathematical Modelling
• 出版年：2016
• 出版时间：15 January 2016
• 年：2016
• 卷：40
• 期：2
• 页码：1315-1324
• 全文大小：1570 K

文摘

The main purpose of this paper is to solve the fractional non-local Allen–Cahn model via a super fast explicit operator splitting spectral (FEOSS) method, which is based on the Strang splitting scheme for time discretization and the Fourier spectral method for space discretization. For a problem of size N, this method reduces the amount of storage from O(N²) to O(N) and cuts down the computational cost from O(N³) to O(Nlog₂N). Moreover, significant computational gains can be obtained by applying a new adaptive time-stepping algorithm based on the local error method. This allows accessing time-scales that vary by several orders of magnitude. Numerical results demonstrate that the FEOSS method with adaptive time-stepping algorithm achieves a remarkable resolution and accuracy in a very efficient manner.