2008 Stability Analysis of the Immersed Boundary Method for a Two-Dimensional Membrane with Bending Rigidity

In this paper,we analyze the stability of the Immersed Boundary Methodapplied to a membrane-fluid system with a plasma membrane immersed in an incompressibleviscous fluid.We show that for small deformations,the planar rest state isstable for a membrane with bending rigidity.The smoothed version,using a standardregularization technique for the singular force,is also shown to be stable.Furthermore,we show that the coupled fluid-membrane system is stiff and smoothing helpsto reduce the stiffness.Compared to the system of elastic fibers immersed in an incompressiblefluid,membrane with bending rigidity consist of a wider range of decayrates.Therefore numerical instability could occur more easily for an explicit methodwhen the time step size is not sufficiently small,even though the continuous problemis stable.