Efficient and stable numerical methods for the two-dimensional fractional Cattaneo equation

New numerical methods are presented for the solution of a two-dimensional Cattaneo equation with the time fractional derivative. A difference scheme combining the compact difference approach for the spatial discretization and alternating direction implicit (ADI) method in the time stepping is proposed and analyzed. The compact ADI scheme is constructed by adding a small term, the unconditional stability and the global convergence of the scheme are proved rigorously. In addition, an ADI scheme is presented and the corresponding error estimate is also established. Numerical experiments are included to support the theoretical results, and the comparison with the ADI scheme is presented to show the effectiveness of the compact ADI scheme.