  We present an adaptive arbitrary-order accurate time-stepping numerical
scheme for the flow of vesicles suspended in Stokesian fluids. Our scheme can
be summarized as an approximate implicit spectral deferred correction (SDC)
method. Applying a textbook fully implicit SDC scheme to vesicle flows is
prohibitively expensive. For this reason we introduce several approximations.
Our scheme is based on a semi-implicit linearized low-order time stepping
method. (Our discretization is spectrally accurate in space.) We also use
invariant properties of vesicle flows, constant area and boundary length in two
dimensions, to reduce the computational cost of error estimation for adaptive
time stepping. We present results in two dimensions for single-vesicle flows,
constricted geometry flows, converging flows, and flows in a Couette apparatus.
We experimentally demonstrate that the proposed scheme enables automatic
selection of the step size and high-order accuracy.
