A Compact Finite Difference Scheme with Second-Order Temporal and Sixth-Order Spatial Accuracy for Variable-Order Time-Fractional Sub-Diffusion Equations

In this paper, a finite difference scheme is proposed for the variable-order time-fractional sub-diffusion equation, achieving second-order accuracy in time and sixth-order accuracy in space. For spatial discretization, a newly constructed operator A is employed to obtain a sixth-order compact approximation of the second derivative. Using an energy analysis method, a priori estimates of the scheme are derived, and the unconditional stability and convergence are rigorously proved. Numerical examples are provided to verify the theoretical accuracy of the scheme.