A finite element formulation for a convection–diffusion equation based on Cattaneo’s law Original Research Article

A number of contributions have been made during the last decades to model pure-diffusive transport problems by using the so-called hyperbolic diffusion equations [M. Zakari, D. Jou, Equations of state and transport equations in viscous cosmological models, Phys. Rev. D 48 (1993) 1597–1601; T. Ruggeri, A. Muracchini, L. Seccia, Shock waves and second sound in a rigid heat conductor: a critical temperature for NaF and Bi, Phys. Rev. Lett. 64 (1990) 2640–2643]. These equations are used for both mass and heat transport. The hyperbolic diffusion equations are obtained by substituting the classic time-independent constitutive equation (Fick’s [A. Fick, Uber diffusion, Poggendorff’s Annal. Phys. Chem. 94 (1855) 59–86] and Fourier’s [J.B. Fourier, Théorie analytique de la chaleur, Jacques Gabay, 1822] law, respectively) by a more general time-dependent equation, due to Cattaneo [M.C. Cattaneo, Sur une forme de l’equation de la chaleur liminant le paradoxe d’une propagation instantane, Comptes Rendus de L’Academie des Sciences: Series I-Mathematics 247 (1958) 431–433]. In some applications the use of a parabolic model for diffusive processes is accurate enough in spite of predicting an infinite speed of propagation (Cattaneo, 1958). However, the use of a wave-like equation that predicts a finite velocity of propagation is necessary in many other calculations [A. Compte, The generalized Cattaneo equation for the description of anomalous transport processes, J. Phys. A: Math. General 30 (1997) 7277–7289].