Fully Implicit Spectral Boundary Integral Computation of Red Blood Cell Flow

TL;DR

This paper introduces an implicit spectral boundary integral method for simulating red blood cell flow, enabling larger time steps and efficient computation through advanced numerical techniques.

Contribution

It develops a novel implicit time integration scheme for boundary integral models of red blood cell flow using spectral discretization and Jacobian computations.

Findings

Larger time steps are feasible with the implicit method.

Number of matrix-vector products is comparable to explicit methods.

The approach improves computational efficiency for red blood cell flow simulations.

Abstract

An approach is presented for implicit time integration in computations of red blood cell flow by a spectral boundary integral method. The flow of a red cell in ambient fluid is represented as a boundary integral equation (BIE), whose structure is that of an implicit ordinary differential equation (IODE). The cell configuration and velocity field are discretized with spherical harmonics. The IODE is integrated in time using a multi-step implicit method based on backward difference formulas, with variable order and adaptive time-stepping controlled by local truncation error and convergence of Newton iterations. Jacobians of the IODE, required for Newton's method, are implemented as Jacobian matrix-vector products that are nothing but directional derivatives. Their computation is facilitated by the weakly singular format of the BIE, and these matrix-vector products themselves amount to computing a second BIE. Numerical examples show that larger time steps are possible and that the number of matrix-vector products is comparable to explicit methods.