Li, ZhiqiangYan, Yubin2018-08-212018-08-212018-07-12Li, Z., Yan, Y. (2018). Error estimates of high-order numerical methods for solving time fractional partial differential equations. Fractional Calculus and Applied Analysis, 21(3), 746-774. https://doi.org/10.1515/fca-2018-003910.1515/fca-2018-0039http://hdl.handle.net/10034/621346Error estimates of some high-order numerical methods for solving time fractional partial differential equations are studied in this paper. We first provide the detailed error estimate of a high-order numerical method proposed recently by Li et al. \cite{liwudin} for solving time fractional partial differential equation. We prove that this method has the convergence order $O(\tau^{3- \alpha})$ for all $\alpha \in (0, 1)$ when the first and second derivatives of the solution are vanish at $t=0$, where $\tau$ is the time step size and $\alpha$ is the fractional order in the Caputo sense. We then introduce a new time discretization method for solving time fractional partial differential equations, which has no requirements for the initial values as imposed in Li et al. \cite{liwudin}. We show that this new method also has the convergence order $O(\tau^{3- \alpha})$ for all $\alpha \in (0, 1)$. The proofs of the error estimates are based on the energy method developed recently by Lv and Xu \cite{lvxu}. We also consider the space discretization by using the finite element method. Error estimates with convergence order $O(\tau^{3- \alpha} + h^2)$ are proved in the fully discrete case, where $h$ is the space step size. Numerical examples in both one- and two-dimensional cases are given to show that the numerical results are consistent with the theoretical results.enhttps://creativecommons.org/licenses/by/4.0/time fractional partial differential equationsstabilityerror estimatesArticle1314-2224Fractional Calculus and Applied Analysis