Numerical approximation of a diffusive hyperbolic equation

In this work numerical methods for one-dimensional diffusion problems are discussed. The differential equation considered, takes into account the variation of the relaxation time of the mass flux and the existence of a potential field. Consequently, according to which values of the relaxation parameter or the potential field we assume, the equation can have properties similar to a hyperbolic equation or to a parabolic equation. The numerical schemes consist of using an inverse Laplace transform algorithm to remove the time-dependent terms in the governing equation and boundary conditions. For the spatial discretization, three different approaches are discussed and we show their advantages and disadvantages according to which values of the potential field and relaxation time parameters we choose.