The discrete gradient structure and the positive definiteness of discrete fractional integrals or derivatives are fundamental to the numerical stability in long-time simulation of nonlinear integro-differential models. We build up a discrete gradient structure for a class of second-order variable-step approximations of fractional Riemann-Liouville integral and fractional Caputo derivative. Then certain variational energy dissipation laws at discrete levels of the corresponding variable-step Crank-Nicolson type methods are established for time-fractional Allen-Cahn and time-fractional Klein-Gordon type models. They are shown to be asymptotically compatible with the associated energy laws of the classical Allen-Cahn and Klein-Gordon equations in the associated fractional order limits.Numerical examples together with an adaptive time-stepping procedure are provided to demonstrate the effectiveness of our second-order methods.

翻译：离散梯度结构与离散分数阶积分或导数的正定性对于非线性积分微分模型长时间模拟中的数值稳定性至关重要。针对分数阶Riemann-Liouville积分和分数阶Caputo导数的一类二阶变步长近似，我们构建了其离散梯度结构。进而，对于时间分数阶Allen-Cahn和时间分数阶Klein-Gordon型模型，建立了相应变步长Crank-Nicolson型方法在离散层面的变分能量耗散律。这些耗散律在相关分数阶极限下与经典Allen-Cahn和Klein-Gordon方程对应的能量律渐近相容。结合自适应时间步进程序，通过数值算例验证了所提二阶方法的有效性。