A boundary integral method for simulating the dynamics of inextensible vesicles suspended in a viscous fluid in 2D

We present a new method for the evolution of inextensible vesicles immersed in a Stokesian fluid. We use a standard boundary integral formulation for the fluid, which results in a set of nonlinear integro-differential equations for the vesicle dynamics. This dynamics is driven by hydrodynamic and elastic forces. In 2D, the elastic forces are given by the Euler-Bernoulli elasticity. Fluid-structure interaction problems of this type are challenging to simulate. On one hand, explicit time-stepping schemes suffer from a severe stability constraint due to the stiffness related to high-order spatial derivatives and a milder constraint due to a transport-like stability condition. On the other hand, an implicit scheme can be expensive because it requires the solution of a set of nonlinear equations at each time step. We present a semi-implicit scheme that circumvents the severe stability constraints and whose computational cost per time step is comparable to that of an explicit scheme. We discretize the equations by using a spectral method in space, and a multistep scheme in time. We use the fast multipole method (FMM) to efficiently compute vesiclevesicle interaction forces in a suspension with a large number of vesicles. We report several numerical experiments that demonstrate the convergence and algorithmic complexity properties of our scheme.