High resolution methods for scalar transport problems in compliant systems of arteries

In this paper a new semi-implicit high resolution scheme for the simulation of advection–diffusion problems in compliant arterial systems is proposed. Such transport problems are not only of great importance for the modeling of drug delivery processes, but also for the simulation of continuous processes occurring in the human body such as the exchange of oxygen, carbon dioxide, nutrients and toxics. Assuming cylindrical geometry and axially symmetric blood flow, a finite volume scheme for scalar transport on unstructured staggered grids is derived. It is shown how both mass conservation and maximum principle can be assured by the present method. Since the discrete maximum principle imposes a CFL type restriction on the time step, the scalar transport equation is solved using a consistent local time-stepping approach in order to not affect the unconditional stability of the underlying semi-implicit scheme for the hydrodynamics. It is a key feature of the present approach that the radial profiles of axial velocity and scalar concentration are computed directly from first principles and that no heuristic model for the velocity profile is needed as in classical one-dimensional approaches, which are still frequently used for the simulation of artery trees. The knowledge of radial velocity and concentration gradients is fundamental for the exchange processes happening across the vessel walls. The accuracy of the proposed approach is validated on one- and two-dimensional test problems with exact solution. An example for scalar transport in a model artery tree with 55 branches rounds off the numerical test problems discussed in this paper.