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

TL;DR

This paper investigates the one-dimensional Cattaneo equation for scalar transport, revealing how its stochastic interpretation constrains boundary conditions to ensure positivity, with implications for higher-dimensional problems.

Contribution

It derives boundary condition constraints from wave formulation and energetic inequalities, linking stochastic interpretation with physical admissibility in the Cattaneo model.

Findings

Negative concentrations can occur with certain boundary conditions.

Wave formulation provides constraints on admissible boundary conditions.

Energetic inequalities support the stochastic constraints.

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.