One-dimensional hyperbolic transport: Positivity and admissible boundary conditions derived from the wave formulation

Abstract

We consider the one-dimensional Cattaneo equation for transport of scalar fields such as solute concentration and temperature in mass and heat transport problems, respectively. Although the Cattaneo equation admits a stochastic interpretation–at least in the one-dimensional case–negative concentration values can occur in boundary-value problems on a finite interval. This phenomenon stems from the probabilistic nature of this model: the stochastic interpretation provides constraints on the admissible boundary conditions, as can be deduced from the wave formulation here presented. Moreover, as here shown, energetic inequalities and the dissipative nature of the equation provide an alternative way to derive the same constraints on the boundary conditions derived by enforcing positivity. The analysis reported is also extended to transport problems in the presence of a biasing velocity field. Several general conclusions are drawn from this analysis that could be extended to the higher-dimensional case.