Fully implicit spectral boundary integral computation of red blood cell flow

Abstract

This paper is on an implicit time integration scheme for simulation of red blood cell (RBC) flow in an ambient fluid. The intra- and extracellular plasmas are modeled as Stokes flows and represented by boundary integral equations (BIE) written in a weakly singular form. The cell membrane is modeled as a thin elastic shell. Expressed in this way, the RBC flow model constitutes an implicit ordinary differential equation (IODE) in the cell shape. The cell shape and velocity field are discretized spatially by a spectral approach using spherical harmonic basis functions. It is then convenient to express the BIE in the Galerkin form with the spherical harmonics themselves as test functions. The key aspect in this paper is the recognition of the IODE structure of the RBC flow model and consequent application of a multi-step implicit solver for time integration. As with any implicit solver, a nonlinear equation in the cell shape is solved at each time step, for which Newton's method is applied. This requires the Jacobian of the IODE, or equivalently computation of Jacobian-vector products. An important contribution is the formulation of such Jacobian-vector products as evaluating a second BIE. The original weakly singular form is crucial in facilitating this formulation. The implicit solver employs variable order and adaptive time stepping controlled by truncation error and convergence of Newton iterations. Numerical examples show that larger time steps are possible and that the number of matrix-vector products is comparable to explicit methods. Source code is provided in the online supplementary material.