A projection preconditioner for solving the implicit immersed boundary equations
Abstract
This study presents a method for solving the linear semiimplicit immersed boundary equations which avoids the severe time step restriction presented by explicittime methods. The Lagrangian variables are eliminated via a Schur complement to form a purely Eulerian saddle point system, which is preconditioned by a projection operator and then solved by a Krylov subspace method. From the viewpoint of projection methods, we derive an ideal preconditioner for the saddle point problem and compare the efficiency of a number of simpler preconditioners 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 speedups over explicittime method are achieved for Reynolds number below 100. In conclusion, this speedup increases as the Eulerian grid size and/or the Reynolds number are further reduced.
 Authors:

 Univ. of Utah, Salt Lake City, UT (United States). Department of Mathematics
 University of California Davis, Davis, CA (United States). Department ofMathematics
 Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States)
 Publication Date:
 Research Org.:
 Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States)
 Sponsoring Org.:
 USDOE; USDOE Office of Management, Budget, and Evaluation; Work for Others (WFO)
 OSTI Identifier:
 1265332
 DOE Contract Number:
 AC0500OR22725
 Resource Type:
 Journal Article
 Journal Name:
 Numerical Mathematics: Theory, Methods and Applications
 Additional Journal Information:
 Journal Volume: 7; Journal Issue: 4; Journal ID: ISSN 10048979
 Publisher:
 Global Science Press
 Country of Publication:
 United States
 Language:
 English
 Subject:
 97 MATHEMATICS AND COMPUTING; 42 ENGINEERING
Citation Formats
Zhang, Qinghai, Guy, Robert D., and Philip, Bobby. A projection preconditioner for solving the implicit immersed boundary equations. United States: N. p., 2014.
Web.
Zhang, Qinghai, Guy, Robert D., & Philip, Bobby. A projection preconditioner for solving the implicit immersed boundary equations. United States.
Zhang, Qinghai, Guy, Robert D., and Philip, Bobby. Sat .
"A projection preconditioner for solving the implicit immersed boundary equations". United States.
@article{osti_1265332,
title = {A projection preconditioner for solving the implicit immersed boundary equations},
author = {Zhang, Qinghai and Guy, Robert D. and Philip, Bobby},
abstractNote = {This study presents a method for solving the linear semiimplicit immersed boundary equations which avoids the severe time step restriction presented by explicittime methods. The Lagrangian variables are eliminated via a Schur complement to form a purely Eulerian saddle point system, which is preconditioned by a projection operator and then solved by a Krylov subspace method. From the viewpoint of projection methods, we derive an ideal preconditioner for the saddle point problem and compare the efficiency of a number of simpler preconditioners 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 speedups over explicittime method are achieved for Reynolds number below 100. In conclusion, this speedup increases as the Eulerian grid size and/or the Reynolds number are further reduced.},
doi = {},
journal = {Numerical Mathematics: Theory, Methods and Applications},
issn = {10048979},
number = 4,
volume = 7,
place = {United States},
year = {2014},
month = {11}
}