A novel H3N3-2$_\sigma$ interpolation approximation for the Caputo fractional derivative of order $\alpha\in(1,2)$ is derived in this paper, which improves the popular L2C formula with (3-$\alpha$)-order accuracy. By an interpolation technique, the second-order accuracy of the truncation error is skillfully estimated. Based on this formula, a finite difference scheme with second-order accuracy both in time and in space is constructed for the initial-boundary value problem of the time fractional hyperbolic equation. It is well known that the coefficients' properties of discrete fractional derivatives are fundamental to the numerical stability of time fractional differential models. We prove the related properties of the coefficients of the H3N3-2$_\sigma$ approximate formula. With these properties, the numerical stability and convergence of the difference scheme are derived immediately by the energy method in the sense of $H^1$-norm. Considering the weak regularity of the solution to the problem at the starting time, a finite difference scheme on the graded meshes based on H3N3-2$_\sigma$ formula is also presented. The numerical simulations are performed to show the effectiveness of the derived finite difference schemes, in which the fast algorithms are employed to speed up the numerical computation.

翻译：本文推导了一种针对阶数$\alpha\in(1,2)$的Caputo分数阶导数的新型H3N3-2$_\sigma$插值逼近格式，该格式改进了具有(3-$\alpha$)阶精度的经典L2C公式。通过插值技术，巧妙估计了截断误差的二阶精度。基于该公式，为时间分数阶双曲型方程的初边值问题构建了时间和空间均具有二阶精度的有限差分格式。众所周知，离散分数阶导数系数性质对时间分数阶微分模型的数值稳定性至关重要。我们证明了H3N3-2$_\sigma$逼近公式系数的相关性质。利用这些性质，通过能量方法在$H^1$-范数意义下直接推导出差分格式的数值稳定性和收敛性。考虑到问题解在起始时刻的弱正则性，还提出了基于H3N3-2$_\sigma$公式的渐变网格有限差分格式。数值模拟验证了所推导有限差分格式的有效性，其中采用快速算法加速数值计算。