  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.
