  In this paper, a temporal nonuniform $L1$ type difference scheme is built up
for the time fractional diffusion-wave equation with the help of the order
reduction technique. The unconditional convergence of the nonuniform difference
scheme is proved rigorously in $L^2$ norm. Our main tool is the discrete
complementary convolution kernels with respect to the coefficient kernels of
the L1 type formula. The positive definiteness of the complementary convolution
kernels is shown to be vital to the stability and convergence. To the best of
our knowledge, this property is proved at the first time on the nonuniform time
meshes. Two numerical experiments are presented to verify the accuracy and the
efficiency of the proposed numerical methods.
