Author(s)

H. G´omez, I. Colominas, F. Navarrina & M. Casteleiro

Abstract

Besides the computational cost of solving advective-diffusive problems in convection dominated situations, the standard statement for this phenomenon leads to the result that mass can propagate at an infinite speed. This paradoxical result occurs as a consequence of using Fick’s law [1] and it is related to the appearance of boundary layers on outflow borders when convection dominates diffusion. For these reasons we propose to use Cattaneo’s law [2] instead of Fick’s law as the constitutive equation of the problem. This approach leads to a totally hyperbolic system of partial differential equations. A finite diffusive velocity can be defined by using this approach. Several problems have been solved to show that the proposed formulation leads to stable results in convection dominated situations. Keywords: advection-diffusion, Taylor–Galerkin, Cattaneo’s equation. 1 Introduction Transport problems involving convective and diffusive processes in fluid media have a great applicability in engineering. This kind of phenomena can be modeled by using the so-called advective-diffusive equation. Unfortunately, it is very complicated to obtain an accurate and stable solution for this equation when the convective term becomes important. To overcome this problem, many interesting numerical methods have been introduced, but completely satisfactory results have not been achieved (for a detailed presentation of most of these methods see reference [3]). In this paper we review the formulation of the advective-diffusive equations applied to the spillage of a pollutant into a fluid medium. In particular, we notice that Fick’s law leads to the result that mass can propagate at an infinite speed. This fact is related to the appearance of spurious oscillations in the numerical

Keywords

advection-diffusion, Taylor–Galerkin, Cattaneo’s equation.