Fully discrete second-order energy-stable HDG method for Carreau-type blood flows with applications to three-dimensional aneurysm hemodynamics

Accurate simulation of incompressible Carreau-type blood flow in aneurysmal vessels is challenging due to the nonlinear coupling between viscosity and shear rate and tortuous vascular geometries. In this paper, we develop a fully discrete second-order-in-time numerical method for the incompressible Carreau model relevant to large-artery hemodynamics. In space, we use a hybridizable discontinuous Galerkin (HDG) discretization with numerical traces on element interfaces, yielding locally conservative fluxes and allowing efficient static condensation of unknowns. In time, we employ a BDF2-based pressure-correction scheme together with a discrete gradient reconstruction and lifting operator for the nonlinear diffusion term, designed so that the fully discrete scheme satisfies a discrete analogue of the continuous energy law. We prove that the scheme is well posed and unconditionally energy stable. Numerical experiments verify second-order convergence in time and space, and include idealized fusiform and saccular aneurysm configurations that probe jets and recirculation regions, as well as patient-specific cerebral aneurysm geometries. These results demonstrate that the proposed numerical framework provides an accurate, robust tool for computing non-Newtonian blood flow fields and wall-shear-stress distributions in complex three-dimensional vascular domains.