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A Multigrid Method for a Model of the Implicit Immersed Boundary Equations

Published online by Cambridge University Press:  20 August 2015

Robert D. Guy  and
Bobby Philip
Robert D. Guy*
Affiliation:
Department of Mathematics, University of California Davis, Davis, CA 95616, USA
Bobby Philip*
Affiliation:
Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA
*
Corresponding author.Email:[email protected]
Email:[email protected]
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Abstract

Explicit time stepping schemes for the immersed boundary method require very small time steps in order to maintain stability. Solving the equations that arise from an implicit discretization is difficult. Recently, several different approaches have been proposed, but a complete understanding of this problem is still emerging. A multigrid 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 forces on an immersed boundary. This model does not include the inertial terms or the incompressibility constraint. The method is more efficient than an explicit method, but the efficiency gain is limited. The multigrid method alone may not be an effective solver, but when used as a preconditioner for Krylov methods, the speed-up over the explicit-time method is substantial. For example, depending on the constitutive law for 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. A very 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 boundary points.

Keywords

65M5565F0876M2076D99Preconditioningimplicit methodsfluid-structure interaction
Type
Research Article
Information
Communications in Computational Physics , Volume 12 , Issue 2 , August 2012 , pp. 378 - 400
Copyright
Copyright © Global Science Press Limited 2012

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