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.

翻译：本文借助降阶技术，针对时间分数阶扩散波动方程构建了一种时间非均匀L1型差分格式。严格证明了非均匀差分格式在L²范数下的无条件收敛性。主要工具是基于L1型公式系数核的离散互补卷积核。互补卷积核的正定性对于稳定性和收敛性至关重要。据我们所知，该性质首次在非均匀时间网格上得到证明。通过两个数值实验验证了所提数值方法的精度和效率。