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 computa-tionally 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. Hyperbolic diffusion models have been widely discussed in literature since they overcome the unphysical property of infinite speed of propagation that is specific to parabolic models [10,14,15,24]. There are experimental evidences which prove that diffusive processes take place with finite velocity inside matter [4,12,13]. In some applications, this issue can be ignored but in many others it is necessary to take into account the wave nature of diffusive processes [22]. We consider a two dimensional hyperbolic transport equation that assumes that both convection and diffusion are responsible for flow motion, which can be seen as a more general telegrapher's equation [31]. Similar equations have been appearing in several works for different applications, such as, diffusive processes in the presence of a potential field [2,3,6], physical models with transport memory and nonlinear damping [21], hyperbolic models for convection–diffusion problems in computational fluid dynamics [11] and various heat transfer models [1,16,18,19,26,27]. Despite its great relevance in practical applications, the incorporation of a convection term has not been exhaustively studied. In this work we derive a two-level alternating direction implicit (ADI) scheme to solve the two-dimensional problem. There are a great variety of applications of ADI methods based on the finite difference methods. When implicit methods are applied in one dimension, usually extensions to two dimensions require approaches such as the ADI, since they reduce the solution of a multidimensional problem to a set of independent one-dimensional problems and thus we obtain a more efficient method than the implicit schemes. For these reasons special attention has been given to these type of methods, when trying to solve multidimensional problems.