论文标题
有效移动的分数梯形规则，用于均匀网格上非平滑溶液的次扩散问题
Efficient shifted fractional trapezoidal rule for subdiffusion problems with nonsmooth solutions on uniform meshes
论文作者
论文摘要
本文致力于开发适用但简单的校正技术和一类二阶阶梯方法的有效算法，即转移的分数梯形规则（SFTR），用于解决初始奇点性和非局限性。列出了稳定性分析和急剧误差估计，从初始数据和源项的平滑度方面进行了估计。作为数值测试中的副产品，我们令人惊讶地发现，没有初始校正的曲柄 - 尼科尔森方案（$θ= \ frac {1} {2} $）可以恢复平滑的初始数据和源术语的亚缩放问题的最佳收敛率。为了处理非局部性，快速算法被认为将计算成本从$ O（n^2）$减少到$ O（n \ log n）$，并将内存存储从$ O（n）$保存到$ O（n）$（\ log log n）$，其中$ n $表示$ o（$ n $）表示时间级。进行数值测试以验证理论结果的清晰度，并确认初始校正和快速算法的效率和准确性。
This article devotes to developing robust but simple correction techniques and efficient algorithms for a class of second-order time stepping methods, namely the shifted fractional trapezoidal rule (SFTR), for subdiffusion problems to resolve the initial singularity and nonlocality. The stability analysis and sharp error estimates in terms of the smoothness of the initial data and source term are presented. As a byproduct in numerical tests, we find amazingly that the Crank-Nicolson scheme ($θ=\frac{1}{2}$) without initial corrections can restore the optimal convergence rate for the subdiffusion problem with smooth initial data and source terms. To deal with the nonlocality, fast algorithms are considered to reduce the computational cost from $O(N^2)$ to $O(N \log N)$ and save the memory storage from $O(N)$ to $O(\log N)$, where $N$ denotes the number of time levels. Numerical tests are performed to verify the sharpness of the theoretical results and confirm the efficiency and accuracy of initial corrections and the fast algorithms.
