An alternating direction implicit method for a second-order hyperbolic diffusion equation with convection

A numerical method is presented to solve a two-dimensional hyperbolic diffusion problem where is assumed that both convection and diffusion are responsible for flow motion. Since direct solutions based on implicit schemes for multidimensional problems are computationally inefficient, we apply an alternating direction method which is second order accurate in time and space. The stability of the alternating direction method is analyzed using the energy method. Numerical results are presented to illustrate the performance in different cases.