A generalized statement for advective-diffusive phenomena. Finite element model and applications

Abstract

[Abstract] Solving convective-diffusive transport problems is a frequent task in engineering, especially in convection dominated situations. Moreover, the standard statement for the transport problem leads to the result that mass can propagate at an infinite speed. This paradoxical result occurs as a consequence of using Fick's law. It seems that this fact is related to the spurious oscillations that occur in the numerical solution of the standard formulation of the transport problem when the Galerkin ¯nite element method is used for the spatial discretization. For these reasons, we propose to use Cattaneo's law instead of Fick's law for the formulation of the advective-diffusive problem. Cattaneo's law has been previously applied to pure-diffusive problems and it is a generalization of Fick's law. The formulation of the transport problem by using Cattaneo's law leads to a hyperbolic system of partial differential equations which can be written in conservative form. As a consequence of being a hyperbolic system, a finite diffusive velocity can be defined. A Taylor-Galerkin procedure can be used to solve these equations. In this paper, several problems in one and two-dimensional domains have been solved to show that this new approach can be used in real engineering problems.