Time-fractional parabolic equations with a Caputo time derivative are considered. For such equations, we explore and further develop the new methodology of the a-posteriori error estimation and adaptive time stepping proposed in [7]. We improve the earlier time stepping algorithm based on this theory, and specifically address its stable and efficient implementation in the context of high-order methods. The considered methods include an L1-2 method and continuous collocation methods of arbitrary order, for which adaptive temporal meshes are shown to yield optimal convergence rates in the presence of solution singularities.

翻译：考虑含有Caputo时间导数的分数阶抛物型方程。针对此类方程，我们探索并进一步发展了文献[7]提出的后验误差估计与自适应时间步进新方法。基于该理论，我们改进了原有的时间步进算法，并着重解决其在高阶方法框架下的稳定高效实现问题。所研究的数值方法包括L1-2方法及任意阶连续配置法，数值实验表明，采用自适应时间网格可在解存在奇异性时获得最优收敛阶。