Two kinds of fractional partial differential equations are considered. One is a spacefractional convectiondiffusion equation and the other is a time-space fractional convection-diffusion equation. Based on the shifted Grünwald formula, the weighted average finite difference method is used to approximate the spatial fractional derivatives in the first equation, and its stability is studied by eigenvalue analysis. The error estimate is O(τ+h). A high order approximation for the temporal derivative is used for the second equation. The stability is given by the technique of maximum norm analysis, with the convergence order O(τ2-max｛γ1,γ2｝+h), where γ1,γ2 are the orders of the two Caputo time fractional derivatives, respectively. Numerical examples are presented to demonstrate the theoretical results.