  In this paper, we consider a numerical method for the multi-term
Caputo-Fabrizio time-fractional diffusion equations (with orders
$\alpha_i\in(0,1)$, $i=1,2,\cdots,n$). The proposed method employs a fast
finite difference scheme to approximate multi-term fractional derivatives in
time, requiring only $O(1)$ storage and $O(N_T)$ computational complexity,
where $N_T$ denotes the total number of time steps. Then we use a Legendre
spectral collocation method for spatial discretization. The stability and
convergence of the scheme have been thoroughly discussed and rigorously
established. We demonstrate that the proposed scheme is unconditionally stable
and convergent with an order of $O(\left(\Delta t\right)^{2}+N^{-m})$, where
$\Delta t$, $N$, and $m$ represent the timestep size, polynomial degree, and
regularity in the spatial variable of the exact solution, respectively.
Numerical results are presented to validate the theoretical predictions.
