Stability and Convergence Analysis of Compact Finite Difference Method for High-Dimensional Time-Fractional Diffusion Equations with High-Order Accuracy in Time

Based on the spatial compact finite difference (SCFD) method, an improved high-order temporal accuracy scheme for high-dimensional time-fractional diffusion equations (TFDEs) is presented in this work. Combining the temporal piecewise quadratic interpolation and the high-dimensional SCFD method, the proposed numerical method is described. In order to establish the stability and convergence analysis, we introduce a norm ||·||H˜1, which is rigorously proved equivalent to the standard H1-norm. Considering that the coefficients of high-order numerical schemes are not entirely positive, we introduce an appropriate parameter to transform the numerical scheme into an equivalent form with positive coefficients. Based on the equivalent form, we prove that the temporal and spatial convergence orders are (3−γ) and 4 by applying the convergence of geometric progression. The proposed scheme ensures that the theoretical convergence accuracy at each time step is of order (3−γ) without requiring any additional processing techniques. Ultimately, the convergence of the proposed high-order accurate scheme is verified through numerical experiments involving (non-)linear high-dimensional TFDEs.