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Error 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. [ 21 ] for solving time fractional partial differential equation. We prove that this method has the convergence order  O ( τ  3− α  ) for all  α  ∈ (0, 1) when the first and second derivatives of the solution are vanish at  t  = 0, where  τ  is the time step size and  α  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. [ 21 ]. We show that this new method also has the convergence order  O ( τ  3− α  ) for all  α  ∈ (0, 1). The proofs of the error estimates are based on the energy method developed recently by Lv and Xu [ 26 ]. We also consider the space discretization by using the finite element method. Error estimates with convergence order  O ( τ  3− α   +  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.

Error estimates of high-order numerical methods for solving time fractional partial differential equations