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：考虑使用Caputo时间衍生物的不折不扣的抛物线方程式。对于这种方程式,我们探索并进一步发展[7] 中提议的新的外在误差估计和适应性时间阶梯方法。我们改进以这一理论为基础的早期时间阶梯算法,并具体处理在高阶方法背景下稳定有效地实施这一算法。考虑的方法包括L1-2法和任意顺序连续合用的方法,在这些方法中,在有解决方案特性的情况下,经调整的时空模子可以产生最佳的趋同率。