  The multi-term time-fractional mixed diffusion-wave equations (TFMDWEs) are
considered and the numerical method with its error analysis is presented in
this paper. First, a $L2$ approximation is proved with first order accuracy to
the Caputo fractional derivative of order $\beta \in (1,2).$ Then the
approximation is applied to solve a one-dimensional TFMDWE and an implicit,
compact difference scheme is constructed. Next, a rigorous error analysis of
the proposed scheme is carried out by employing the energy method, and it is
proved to be convergent with first order accuracy in time and fourth order in
space, respectively. In addition, some results for the distributed order and
two-dimensional extensions are also reported in this work. Subsequently, a
practical fast solver with linearithmic complexity is provided with partial
diagonalization technique. Finally, several numerical examples are given to
demonstrate the accuracy and efficiency of proposed schemes.
