A Projection Preconditioner for Solving the Implicit Immersed Boundary Equations

This paper presents a method for solving the linear semi-implicitimmersed boundary equations which avoids the severe time step restrictionpresented by explicit-time methods. The Lagrangian variables areeliminated via a Schur complement to form a purely Eulerian saddlepoint system, which is preconditioned by a projection operator andthen solved by a Krylov subspace method. From the viewpoint ofprojection methods, we derive an ideal preconditioner for thesaddle point problem and compare the efficiency of a number of simplerpreconditioners that approximate this perfect one.For low Reynolds number and high stiffness, one particular projection preconditioner yields an efficiency improvement of the explicit IB method by a factor around thirty.Substantial speed-ups over explicit-time method are achieved for Reynolds number below 100.This speedup increases as the Eulerian grid size and/or the Reynolds number are further reduced.