Volume 7, Issue 4
A Projection Preconditioner for Solving the Implicit Immersed Boundary Equations

Qinghai Zhang, Robert D. Guy & Bobby Philip

Numer. Math. Theor. Meth. Appl., 7 (2014), pp. 473-498.

Published online: 2014-07

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  • Abstract

This paper presents a method for solving the linear semi-implicit immersed boundary equations which avoids the severe time step restriction presented by explicit-time 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 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.

  • Keywords

Fluid-structure interaction, immersed boundary method, projection method, preconditioning.

  • AMS Subject Headings

65M55, 65F08, 76M20, 76D99

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{NMTMA-7-473, author = {}, title = {A Projection Preconditioner for Solving the Implicit Immersed Boundary Equations}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2014}, volume = {7}, number = {4}, pages = {473--498}, abstract = {

This paper presents a method for solving the linear semi-implicit immersed boundary equations which avoids the severe time step restriction presented by explicit-time 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 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.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.2014.1304si}, url = {http://global-sci.org/intro/article_detail/nmtma/5885.html} }
TY - JOUR T1 - A Projection Preconditioner for Solving the Implicit Immersed Boundary Equations JO - Numerical Mathematics: Theory, Methods and Applications VL - 4 SP - 473 EP - 498 PY - 2014 DA - 2014/07 SN - 7 DO - http://doi.org/10.4208/nmtma.2014.1304si UR - https://global-sci.org/intro/article_detail/nmtma/5885.html KW - Fluid-structure interaction, immersed boundary method, projection method, preconditioning. AB -

This paper presents a method for solving the linear semi-implicit immersed boundary equations which avoids the severe time step restriction presented by explicit-time 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 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.

Qinghai Zhang, Robert D. Guy & Bobby Philip. (2020). A Projection Preconditioner for Solving the Implicit Immersed Boundary Equations. Numerical Mathematics: Theory, Methods and Applications. 7 (4). 473-498. doi:10.4208/nmtma.2014.1304si
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