On the non-linear Maxwell–Cattaneo equation with non-constant diffusivity: Shock and discontinuity waves

A numerical study on a non-linear hyperbolic diffusion equation is proposed. The Hartree hybrid method combining finite difference techniques with the method of characteristics is used in the presence of discontinuities between initial and boundary conditions. The technique proved to be an useful tool to overcome oscillation problems and spurious solutions in case of strong non-linearities related to both attractive or repulsive interactions between diffusing species. Two different expressions for the diffusion coefficient are used in order to compare our results with the ones obtained in previous studies relying upon the Laplace transform technique and the MacCormack predictor–corrector method. Finally, an analytic approach based on the singular surface theory is proposed to motivate the numerical results and to clarify some controversial aspects concerning the penetration depth of a diffusive front in the presence of interactions.