In consideration of the initial singularity of the solution, a temporally second-order fast compact difference scheme with unequal time-steps is presented and analyzed for simulating the subdiffusion problems in several spatial dimensions. On the basis of sum-of-exponentials technique, a fast Alikhanov formula is derived on general nonuniform meshes to approximate the Caputo’s time derivative. Meanwhile, the spatial derivatives are approximated by the fourth-order compact difference operator, which can be implemented by a fast discrete sine transform via the FFT algorithm. So the proposed algorithm is computationally efficient with the computational cost about \(O(MN\log M\log N)\) and the storage requirement \(O(M\log N)\), where M and N are the total numbers of grids in space and time, respectively. With the aids of discrete fractional Grönwall inequality and global consistency analysis, the unconditional stability and sharp H¹-norm error estimate reflecting the regularity of solution are established rigorously by the discrete energy approach. Three numerical experiments are included to confirm the sharpness of our analysis and the effectiveness of our fast algorithm.

考虑到解的初始奇异性，提出并分析了具有不等时步的时间二阶快速紧致差分格式，并在多个空间维度上对子扩散问题进行了仿真分析。基于指数和技术，在一般的非均匀网格上导出了一个快速的Alikhanov公式，以近似Caputo的时间导数。同时，空间导数由四阶紧致差分算子近似，可以通过FFT算法通过快速离散正弦变换来实现。因此，所提出的算法在计算上具有约\（O（MN \ log M \ log N）\）和存储需求\（O（M \ log N）\）的计算效率，其中M和N是分别在空间和时间上的网格总数。借助离散分数Grönwall不等式和全局一致性分析，通过离散能量方法严格建立了无条件稳定性和反映解决方案规律性的H ^1-范数误差估计。包括三个数值实验，以确认我们分析的敏锐性和快速算法的有效性。