Hydrodynamics simulation of red blood cells: Employing a penalty method with double jump composition of lower order time integrator

We propose a numerical framework tailored for simulating the dynamics of vesicles with inextensible membranes, which mimic red blood cells, immersed in a non-Newtonian fluid environment. A penalty method is proposed to handle the inextensibility constraint by relaxation, allowing a simple computer implementation and an affordable computational load compared to the full mixed formulation. To handle the high-order derivatives in the stress jump across the membrane, which arise due to the high geometric order of the Helfrich functional, we employ higher degree finite elements for spatial discretization. The time integration scheme relies on the double composition of the Crank–Nicolson scheme to achieve faster fourth-order convergence behavior. Additionally, an adaptive time-stepping strategy based on a third-order temporal integration error estimation is implemented. We address the main features of the proposed method, which is benchmarked against existing numerical and experimental results. Furthermore, we investigate the influence of non-Newtonian rheology on the system dynamics. © 2023 The Authors. Mathematical Methods in the Applied Sciences published by John Wiley & Sons Ltd.