Skip to main content Accessibility help
×
Home

A Projection Preconditioner for Solving the Implicit Immersed Boundary Equations

  • Qinghai Zhang (a1), Robert D. Guy (a2) and Bobby Philip (a3)

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.

Copyright

Corresponding author

* Corresponding author. Email address: qinghai@math.utah.edu

References

Hide All
[1] Bell, J. B., Colella, P., and Glaz, H. M.. A second-order projection method for the incompressible Navier-Stokes equations. J. Comput. Phys., 85:257283, 1989.
[2] Benzi, M., Golub, G. H., and Liesen, J.. Numerical solution of saddle point problems. Acta Numerica, pages 1137, 2005. doi:10.1017/S0962492904000212.
[3] Ceniceros, H. D. and Fisher, J. E.. A fast, robust, and non-stiff immersed boundary method. J. Comput. Phys., 230:51335153, 2011. doi:10.1016/j.jcp.2011.03.037.
[4] Ceniceros, H. D., Fisher, J. E., and Roma, A. M.. Efficient solutions to robust, semi-implicit discretizations of the immersed boundary method. J. Comput. Phys., 228(19):71377158, 2009.
[5] Chen, K.-Y., Feng, K.-A., Kim, Y., and Lai, M.-C.. A note on pressure accuracy in immersed boundary method for Stokes flow. J. Comput. Phys., 230:43774383, 2011.
[6] Chorin, A. J.. Numerical solution of the Navier-Stokes equations. Math. Comput., 22(104):745762, 1968.
[7] Liu, W. E. and Liu, J.-G.. Gauge method for viscous incompressible flows. Comm. Math. Sci., 1(2):317, 2003.
[8] Elman, H. C., Silvester, D. J., and Wathen, A. J.. Finite Elements and Fast Iterative Solvers with Applications in Incompressible Fluid Dynamics. Oxford University Press, USA, New York, 2005. isbn: 978-0198528685.
[9] Griffith, B. E.. An accurate and efficient method for the incompressible Navier-Stokes equations using the projection method as a preconditioner. J. Comput. Phys., 228(20):75657595, 2009. doi:10.1016/j.jcp.2009.07.001.
[10] Guy, R. D. and Fogelson, A. L.. A wave propagation algorithm for viscoelastic fluids with spatially and temporally varying properties. Comput. Methods Appl. Mech. Engrg., 197:22502264, 2008. doi:10.1016/j.cma.2007.11.022.
[11] Guy, R. D. and Philip, B.. A multigrid method for a model of the implicit immersed boundary equations. Commun. Comput. Phys., 12(2):378400, 2012.
[12] Guy, R. D., Philip, B., and Griffith, B. E.. Geometric multigrid for an implicit-time immersed boundary method. Advances in Computational Mathematics, in press, 2014. doi:10.1007/s10444-014-9380-1.
[13] Hou, T. Y. and Shi, Z.. An efficient semi-implicit immersed boundary method for the Navier-Stokes equations. J. Comput. Phys., 227:89688991, 2008.
[14] Kim, J. and Moin, P.. Application of a fractional-step method to incompressible Navier-Stokes equations. J. Comput. Phys., 59(2):308323, 1985.
[15] Mayo, A. A. and Peskin, C. S.. An implicit numerical method for fluid dynamics problems with immersed elastic boundaries. In Fluid dynamics in biology (Seattle, WA, 1991), volume 141 of Contemp. Math., pages 261277. Amer. Math. Soc., Providence, RI, 1993.
[16] Mori, Y. and Peskin, C. S.. Implicit second-order immersed boundary methods with boundary mass. Comput. Methods Appl. Mech. Engrg., 197(25-28):20492067, 2008.
[17] Newren, E. P., Fogelson, A. L., Guy, R. D., and Kirby, R. M.. Unconditionally stable discretizations of the immersed boundary equations. J. Comput. Phys., 222:702719, 2007. doi:10.1016/j.jcp.2006.08.004.
[18] Newren, E. P., Fogelson, A. L., Guy, R. D., and Kirby, R. M.. A comparison of implicit solvers for the immersed boundary equations. Comput. Methods Appl. Mech. Engrg., 197:22902304, 2009. doi:10.1016/j.cma.2007.11.030.
[19] Peskin, C. S.. Numerical analysis of blood flow in the heart. J. Comput. Phys., 25:220252, 1977.
[20] Peskin, C. S.. The immersed boundary method. Acta Numerica, pages 479517, 2002. doi:10.1017/S0962492902000077.
[21] Peskin, C. S. and McQueen, D. M.. A three-dimensional computational method for blood flow in the heart: I. immersed elastic fibers in a viscous incompressible fluid. J. Comput. Phys., 81:372405, 1989.
[22] Stockie, J. M. and Wetton, B. R.. Analysis of stiffness in the immersed boundary method and implications for time-stepping schemes. J. Comput. Phys., 154(1):4164, 1999.
[23] Trottenberg, U., Oosterlee, C., and Schuller, A.. Multigrid. Academic Press, San Diego, CA, 2001. ISBN:0-12-701070-X.
[24] Tu, C. and Peskin, C. S.. Stability and instability in the computation of flows with moving immersed boundaries: A comparison of three methods. SIAM J. Sci. Stat. Comput., 13(6):13611376, 1992.
[25] Zhang, Q.. A fourth-order approximate projection method for the incompressible Navier-Stokes equations on locally-refined periodic domains. Appl. Numer. Math., 77(C):1630, 2014.

Keywords

Related content

Powered by UNSILO

A Projection Preconditioner for Solving the Implicit Immersed Boundary Equations

  • Qinghai Zhang (a1), Robert D. Guy (a2) and Bobby Philip (a3)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.