The numerical analysis of time fractional evolution equations with the second-order elliptic operator including general time-space dependent variable coefficients is challenging, especially when the classical weak initial singularities are taken into account. In this paper, we introduce a concise technique to construct efficient time-stepping schemes with variable time step sizes for two-dimensional time fractional sub-diffusion and diffusion-wave equations with general time-space dependent variable coefficients. By means of the novel technique, the nonuniform Alikhanov type schemes are constructed and analyzed for the sub-diffusion and diffusion-wave problems. For the diffusion-wave problem, our scheme is constructed by employing the recently established symmetric fractional-order reduction (SFOR) method. The unconditional stability of proposed schemes is rigorously discussed under mild assumptions on variable coefficients and, based on reasonable regularity assumptions and weak time mesh restrictions, the second-order convergence is obtained with respect to discrete $H^1$-norm. Numerical experiments are given to demonstrate the theoretical statements.

翻译：在本文件中,我们引入了一种简明技术,用于为二维时间分分分分分扩散和扩散波变异方程式制定有效的时间步骤计划,为两维时间分分分分扩散和扩散波变异系数制定可变时间步骤大小计划;通过新技术,为次扩散和传播波问题建立非单向Alikhanov型计划并进行分析。关于扩散波问题,我们的计划是采用最近建立的对称分级削减法(SFOR)方法构建的。在对可变系数的轻度假设下,严格讨论了拟议计划的无条件稳定性,并根据合理的正常假设和对时间网格限制,对离散的1H1美元-norm进行了二级趋同。做了数字实验,以展示理论说明。