In this paper, we propose a fast second-order approximation to the variable-order (VO) Caputo fractional derivative, which is developed based on $L2$-$1_\sigma$ formula and the exponential-sum-approximation technique. The fast evaluation method can achieve the second-order accuracy and further reduce the computational cost and the acting memory for the VO Caputo fractional derivative. This fast algorithm is applied to construct a relevant fast temporal second-order and spatial fourth-order scheme ($FL2$-$1_{\sigma}$ scheme) for the multi-dimensional VO time-fractional sub-diffusion equations. Theoretically, $FL2$-$1_{\sigma}$ scheme is proved to fulfill the similar properties of the coefficients as those of the well-studied $L2$-$1_\sigma$ scheme. Therefore, $FL2$-$1_{\sigma}$ scheme is strictly proved to be unconditionally stable and convergent. A sharp decrease in the computational cost and the acting memory is shown in the numerical examples to demonstrate the efficiency of the proposed method.
翻译:在本文中,我们提议对可变顺序(VO)Caputo分数衍生物快速进行二阶近似,该办法是根据2美元-1美元/西格玛元公式和指数和准比技术开发的,快速评估方法可以达到第二阶准确度,并进一步降低Vo Caputo分数衍生物的计算成本和动作记忆;这一快速算法用于为多维Vo分时间分数子扩散方程式构建一个相关的快速时间第二阶和空间第四阶(FLF2美元-1美元/西格玛元)方案。理论上,2美元-1美元/西格玛元方案已证明能够实现与经过仔细研究的2美元-1美元/西格玛元方案的系数的类似特性。因此,严格地证明,$LF2美元-1美元/西格玛元方案是无条件稳定和一致的。计算成本的急剧下降和动作记忆在数字示例中展示了拟议方法的效率。