@UNPUBLISHED{publication1, title = "Approximating coupled hyperbolic-parabolic systems arising in enhanced drug delivery", author = "{FERREIRA, Jos{\'{e}} Augusto} and {JORD{\~{A}}O, Daniela} and {PINTO, Lu{\ยด{i}}s}", year = "2017-11-21", abstract = " In this paper we study a system of partial differential equations defined by a hyperbolic equation and a parabolic equation. The convective term of the parabolic equation depends on the solution and eventually on the gradient of the solution of the hyperbolic equation. This system arises in the mathematical modeling of several physical processes as for instance ultrasound enhanced drug delivery. In this case the propagation of the acoustic wave, which is described by a hyperbolic equation, induces an active drug transport that depends on the acoustic pressure. Consequently the drug diffusion process is governed by a hyperbolic and a convection-diffusion equation. Here, we propose a numerical method that allows us to compute second-order accurate approximations to the solution of the hyperbolic and the parabolic equation. The method can be seen as a fully discrete piecewise linear finite element method or as a finite difference method. The convergence rates for both approximations are unexpected. In fact we prove that the error for the approximation of the pressure and concentration is of second-order with respect to discrete versions of the H1-norm and L2-norm, respectively. ", url = "http://www.mat.uc.pt/preprints/eng\_2017.html", note = "" }