In this paper, a new derived method is developed for a known numerical differential formula of the Caputo fractional derivative of order \(\gamma \in (1,2)\) (Li and Zeng in Numerical methods for fractional calculus. Chapman & Hall/CRC numerical analysis and scientific computing, CRC Press, Boca Raton, 2015) by means of the quadratic interpolation polynomials, and a concise expression of the truncation error is given. This new method will be called as the H2N2 method because of the application of the quadratic Hermite and Newton interpolation polynomials. A finite difference scheme with a second order accuracy in space and a \((3-\gamma )\)-th order accuracy in time based on the H2N2 method is constructed for the initial boundary value problem of time-fractional wave equations. The stability and convergence of the difference scheme are proved. Furthermore, in order to increase computational efficiency, using the sum-of-exponentials to approximate the kernel \(t^{1-\gamma }\), a fast difference scheme is presented. The problem with weak regularity at the initial time is also discussed with the help of the graded meshes. At each time level, the difference scheme is solved with a fast Poisson solver. Numerical results show the effectiveness of the two difference schemes and confirm our theoretical analysis.

在本文中，针对阶数\（\ gamma \ in（1,2）\）的Caputo分数导数的已知数值微分公式，开发了一种新的导出方法（Li和Zeng在分数微积分的数值方法中。Chapman＆Hall / CRC数值分析和科学计算，CRC Press，Boca Raton，2015年），通过二次插值多项式给出截断误差的简洁表示。由于应用了二次Hermite和Newton插值多项式，因此将这种新方法称为H2N2方法。具有空间二阶精度和\（（3- \ gamma）\）的有限差分方案基于H2N2方法，建立了时间分数阶波动方程初始边界值问题的二阶时间精度。证明了差分方案的稳定性和收敛性。此外，为了提高计算效率，使用指数和来逼近内核\（t ^ {1- \ gamma} \），提出了一种快速差分方案。借助渐变网格还讨论了初始时规则性较弱的问题。在每个时间级别，差分方案都使用快速的泊松求解器进行求解。数值结果表明了两种不同方案的有效性，并证实了我们的理论分析。