A fully implicit numerical scheme is established for solving the time fractional Swift-Hohenberg (TFSH) equation with a Caputo time derivative of order $\alpha\in(0,1)$. The variable-step L1 formula and the finite difference method are employed for the time and the space discretizations, respectively. The unique solvability of the numerical scheme is proved by the Brouwer fixed-point theorem. With the help of the discrete convolution form of L1 formula, the time-stepping scheme is shown to preserve a discrete energy dissipation law which is asymptotically compatible with the classic energy law as $\alpha\to1^-$. Furthermore, the $L^\infty$ norm boundedness of the discrete solution is obtained. Combining with the global consistency error analysis framework, the $L^2$ norm convergence order is shown rigorously. Several numerical examples are provided to illustrate the accuracy and the energy dissipation law of the proposed method. In particular, the adaptive time-stepping strategy is utilized to capture the multi-scale time behavior of the TFSH model efficiently.
翻译:用于解决时间分数 Swift-Hohenberg (TFSH) 方程式的完全隐含的数值方案, 以Caputo时间衍生物 $\ alpha\ in( 0, 1, 1, 1) 来解决时间分数 Swift-Hohenberg (TFSH) 方程式。 对时间和空间分解分别使用可变步骤 L1 公式和限制差法。 数字方案的独特可溶性由 Broewer 固定点定调调调调调调调调制来证明。 在L1 方公式的离散分变式形式帮助下, 时间步调制方案被显示保持离散的能量消散法, 与传统的能源法则基本相同, 特别是, 使用适应性时间步制战略来捕捉到TFSH 模型的多重时间行为。</s>