A fast second-order block-centered finite difference method for the fractional Cattaneo equation

In this paper, a fast second-order block-centered finite difference method based on the $ \mathcal {FL} $2-$ 1_{\sigma} $ formula and a weighted approach was proposed for the time fractional Cattaneo equation. Using the special properties of the discrete coefficients and mathematical induction method, we derived the unconditional stability and convergence of the method rigorously. Second-order superconvergence in time and space for the velocity and pressure in discrete $ L^{\infty}(L^{2}) $-norms was established on non-uniform rectangular grids. Numerical examples were provided to verify our theoretical results.