Advection-Diffusion-Reaction Equations: Hyperbolization and High-Order ADER Discretizations

The purpose of this paper is twofold. First, we extend the applicability of Cattaneo's relaxation approach, one of the currently known relaxation approaches, to reformulate time-dependent advection-diffusion-reaction equations, which may include stiff reactive terms, as hyperbolic balance laws with stiff source terms. We show that the approach works well for second-order partial differential equations. However, it does not for the higher-order case, as proved here for third-order partial differential equations, such as equations of the Korteweg--deVries--Burgers type. In this paper we restrict ourselves to selected second-order model equations. Second, we extend the applicability of existent high-order schemes to approximating this class of partial differential equations numerically. We start from linear problems, proceeding to nonlinear problems, and ending up with an application to an atherosclerosis model consisting of a system of time-dependent diffusion-reaction equations. Several scalar examples are considered, and exact solutions are given for some of these. The proposed hyperbolization procedures turn out to give a generous stability range for the choice of the time step. This results in more efficient schemes than some methods with the parabolic restriction reported in the current literature. Implementations of our numerical schemes and convergence-rate assessments are carried out for methods of up to seventh order of accuracy in both space and time.