A Multigrid Method for a Model of the Implicit Immersed Boundary Equations (Communications in Computational Physics, 2020)
Explicit time stepping schemes for the immersed boundary method requirevery small time steps in order to maintain stability. Solving the equations that arisefrom an implicit discretization is difficult. Recently, several different approaches havebeen proposed, but a complete understanding of this problem is still emerging. Amultigrid method is developed and explored for solving the equations in an implicit-time discretization of a model of the immersed boundary equations. The model problem consists of a scalar Poisson equation with conformation-dependent singular forceson an immersed boundary. This model does not include the inertial terms or the incompressibility constraint. The method is more efficient than an explicit method, butthe efficiency gain is limited. The multigrid method alone may not be an effectivesolver, but when used as a preconditioner for Krylov methods, the speed-up over theexplicit-time method is substantial. For example, depending on the constitutive lawfor the boundary force, with a time step 100 times larger than the explicit method,the implicit method is about 15-100 times more efficient than the explicit method. Avery attractive feature of this method is that the efficiency of the multigrid preconditioned Krylov solver is shown to be independent of the number of immersed boundarypoints.