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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:guy@math.ucdavis.edu
Email:philipb@ornl.gov
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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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References

[1]Brandt, A. and Dinar, N., Multigrid solutions to elliptic flow problems, in Numerical Methods for Partial Differential Equations, Parter, S., ed., Academic Press, New York, 1979, pp. 53–147.Google Scholar
[2]Ceniceros, H. D. and Fisher, J. E., A fast, robust, and non-stiff immersed boundary method, J. Comput. Phys., 230 (2011), 5133–5153.Google Scholar
[3]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 (2009), 7137–7158.CrossRefGoogle Scholar
[4]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, New York, 2005.Google Scholar
[5]Gong, Z., Huang, H., and Lu, C., Stability analysis of the immersed boundary method for a two-dimensional membrane with bending rigidity, Commun. Comput. Phys., 3 (2008), 704–723.Google Scholar
[6]Griffith, B. E., Hornung, R. D., McQueen, D. M., and Peskin, C. S., An adaptive, formally second order accurate version of the immersed boundary method, J. Comput. Phys., 223 (2007), 10–49.Google Scholar
[7]Hou, T. Y. and Shi, Z., An efficient semi-implicit immersed boundary method for the navier-stokes equations, J. Comput. Phys., 227 (2008), 8968–8991.Google Scholar
[8]Hou, T. Y. and Shi, Z., Removing the stiffness of elastic force from the immersed boundary method for the 2d stokes equations, J. Comput. Phys., 227 (2008), 9138–9169. Special Issue Celebrating Tony Leonard’s 70th Birthday.CrossRefGoogle Scholar
[9]Le, D., White, J., Peraire, J., Lim, K., and Khoo, B., An implicit immersed boundary method for three-dimensional fluid-membrane interactions, J. Comput. Phys., 228 (2009), 8427–8445.Google Scholar
[10]Lee, L. and LeVeque, R. J., An immersed interface method for incompressible navier–stokes equations, SIAM J. Sci. Comput. 25 (2003), 832–856.Google Scholar
[11]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), Vol. 141 of Contemp. Math., Amer. Math. Soc., Providence, RI, 1993, pp. 261–277.Google Scholar
[12]Mittal, R. and Iaccarino, G., Immersed boundary methods, Annual Review of Fluid Mechanics, 37 (2005), 239–261.CrossRefGoogle Scholar
[13]Mori, Y. and Peskin, C. S., Implicit second-order immersed boundary methods with boundary mass, Computer Methods in Applied Mechanics and Engineering, 197 (2008), 2049–2067. Immersed Boundary Method and Its Extensions.Google Scholar
[14]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 (2007), 702–719.Google Scholar
[15]Newren, E. P., Fogelson, A. L., Guy, R. D., and Kirby, R. M., A comparison of implicit solvers for the immersed boundary equations, Computer Methods in Applied Mechanics and Engineering, 197 (2008), 2290–2304. Immersed Boundary Method and Its Extensions.Google Scholar
[16]Oosterlee, C. W. and Washio, T., An evaluation of parallel multigrid as a solver and a precon-ditioner for singularly perturbed problems, SIAM J. Sci. Comput., 19 (1998), 87–110.Google Scholar
[17]Pacull, F. and Garbey, M., The multigrid/tau-extrapolation technique applied to the immersed boundary method, in Domain Decomposition Methods in Science and Engineering XVI, Widlund, O. B. and Keyes, D. E., eds., Vol. 55 of Lecture Notes in Computational Science and Engineering, Springer Berlin Heidelberg, 2007, pp. 707–714.Google Scholar
[18]Peskin, C. S., Numerical analysis of blood flow in the heart, J. Comput. Phys., 25 (1977), 220–252.Google Scholar
[19]Roma, A. M., Peskin, C. S., and Berger, M. J., An adaptive version of the immersed boundary method, J. Comput. Phys., 153 (1999), 509–534.Google Scholar
[20]Saad, Y. and Schultz, M. H., Gmres: A generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Stat. Comput., 7 (1986), 856–869.Google Scholar
[21]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 (1999), 41–64.CrossRefGoogle Scholar
[22]Sussman, M., Almgren, A. S., Bell, J. B., Colella, P., Howell, L. H., and Welcome, M. L., An adaptive level set approach for incompressible two-phase flows, J. Comput. Phys., 148 (1999), 81–124.Google Scholar
[23]Tatebe, O., The multigrid preconditioned conjugate gradient method, in Proceedings of the Sixth Copper Mountain Conference on Multigrid Methods, 1993, pp. 621–634.Google Scholar
[24]Trottenberg, U., Oosterlee, C. W., and Schuller, A., Multigrid, Academic Press, London, 2000.Google Scholar
[25]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 (1992), 1361–1376.Google Scholar
[26]Vanka, S. P., Block-implicit multigrid solution of navier-stokes equations in primitive variables, J. Comput. Phys., 65 (1986), 138–158.Google Scholar
[27]Wright, G. B., Guy, R. D., and Fogelson, A. L., An efficient and robust method for simulating two-phase gel dynamics, SIAM J. Sci. Comput., 30 (2008), 2535–2565.Google Scholar
[28]Zhu, L. and Peskin, C. S., Simulation of a flapping flexible filament in a flowing soap film by the immersed boundary method, J. Comput. Phys., 179 (2002), 452–468.Google Scholar