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The numerical analysis of time fractional evolution equations with thesecond-order elliptic operator including general time-space dependent variablecoefficients is challenging, especially when the classical weak initialsingularities are taken into account. In this paper, we introduce a concisetechnique to construct efficient time-stepping schemes with variable time stepsizes for two-dimensional time fractional sub-diffusion and diffusion-waveequations with general time-space dependent variable coefficients. By means ofthe novel technique, the nonuniform Alikhanov type schemes are constructed andanalyzed for the sub-diffusion and diffusion-wave problems. For thediffusion-wave problem, our scheme is constructed by employing the recentlyestablished symmetric fractional-order reduction (SFOR) method. Theunconditional stability of proposed schemes is rigorously discussed under mildassumptions on variable coefficients and, based on reasonable regularityassumptions and weak time mesh restrictions, the second-order convergence isobtained with respect to discrete $H^1$-norm. Numerical experiments are givento demonstrate the theoretical statements.

Second-order and nonuniform time-stepping schemes for time fractional evolution equations with time-space dependent coefficients