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 coefficient 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 is 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：本文针对阶数 $\alpha\in(1,2)$ 的 Caputo 分数阶导数，提出了一种新型 H3N3-2$_\sigma$ 插值逼近，改进了具有 (3-$\alpha$) 阶精度的经典 L2C 公式。通过插值技术，截断误差的二阶精度得到巧妙估计。基于该公式，我们为时间分数阶双曲型方程的初边值问题构造了一个时间和空间均具有二阶精度的有限差分格式。众所周知，离散分数阶导数的系数性质是时间分数阶微分模型数值稳定性的基础。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 $H^1$ 范数意义下的能量方法，可立即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$_\sigma$ 公式的渐近网格有限差分格式。通过数值模拟验证了所构造有限差分格式的有效性，其中采用了快速算法加速数值计算。