Numerical Considerations for Advection-Diffusion Problems in Cardiovascular Hemodynamics

TL;DR

This paper introduces numerical methods to improve the stability and accuracy of advection-diffusion simulations in cardiovascular flows, addressing challenges like backflow and oscillations near concentration fronts.

Contribution

It proposes flux-based boundary conditions and discontinuity capturing techniques within a stabilized finite element framework for better cardiovascular mass transport simulations.

Findings

The proposed flux prescription improves boundary condition stability.

Consistent flux outflow boundary condition outperforms traditional methods.

DC stabilization effectively reduces oscillations near concentration fronts.

Abstract

Numerical simulations of cardiovascular mass transport pose significant challenges due to the wide range of P\'eclet numbers and backflow at Neumann boundaries. In this paper we present and discuss several numerical tools to address these challenges in the context of a stabilized finite element computational framework. To overcome numerical instabilities when backflow occurs at Neumann boundaries, we propose an approach based on the prescription of the total flux. In addition, we introduce a "consistent flux" outflow boundary condition and demonstrate its superior performance over the traditional zero diffusive flux boundary condition. Lastly, we discuss discontinuity capturing (DC) stabilization techniques to address the well-known oscillatory behavior of the solution near the concentration front in advection-dominated flows.We present numerical examples in both idealized and patient-specific geometries to demonstrate the efficacy of the proposed procedures. The three contributions dis-cussed in this paper enable to successfully address commonly found challenges when simulating mass transport processes in cardiovascular flows.