Analysis of Stiffness in the Immersed Boundary Method and Implications for Time-Stepping Schemes

Journal of Computational Physics

Volumn 154, Issue 1, 1999, Pages 41-64

  Stockie, John M ^a   Wetton, Brian R ^b  

^a SIMON FRASER UNIVERSITY   (Canada)

^b UNIVERSITY OF BRITISH COLUMBIA   (Canada)

Author keywords

Immersed boundary method; Linear stability; Time stepping schemes

Indexed keywords

COMPUTATIONAL FLUID DYNAMICS; DELTA FUNCTIONS; EQUATIONS OF MOTION; ITERATIVE METHODS; NUMERICAL METHODS; RUNGE KUTTA METHODS; STIFFNESS; TURBULENT FLOW;

ANALYTICAL RESULTS; DISCRETE SETS; ELASTIC FIBRES; EQUATION OF MOTION; FIBER MODE; IMMERSED BOUNDARY METHODS; LINEAR ANALYSIS; LINEAR STABILITY; PROBLEM PARAMETERS; TIME-STEPPING SCHEMES;

FIBERS;

EID: 0009001404     PISSN: 00219991     EISSN: None     Source Type: Journal    
DOI: 10.1006/jcph.1999.6297     Document Type: Article

References (26)

1. U. M. Ascher, S. J. Ruuth, and R. J. Spiteri, Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations, Appl. Numer. Math. (Special Issue on Innovative Time Integrators) 25(2/3), 151 (1997).
2. R. P. Beyer and R. J. LeVeque, Analysis of a one-dimensional model for the immersed boundary method, SIAM J. Numer. Anal. 29(2), 332 (1992).
3. Y. C. Chang, T. Y. Hou, B. Merriman, and S. Osher, A level set formulation of Eulerian interface capturing methods for incompressible fluid flows, J. Comput. Phys. 124, 449 (1996).
4. B. W. Char et al., Maple VLanguage Reference Manual (Springer-Verlag, Berlin/New York, 1991).
5. R. Cortez and D. A. Varela, The dynamics of an elastic membrane using the impulse method, J. Comput. Phys. 138, 224 (1997).
6. L. J. Fauci and C. S. Peskin, A computational model of aquatic animal locomotion, J. Comput. Phys. 77, 85 (1988).
7. A. L. Fogelson, Continuum models of platelet aggregation: Formulation and mechanical properties, SIAM J. Appl. Math. 52(4), 1089 (1992).
8. A. L. Fogelson and C. S. Peskin, A fast numerical method for solving the three-dimensional Stokes' equations in the presence of suspended particles, J. Comput. Phys. 79, 50 (1988).
9. T. Y Hou, J. S. Lowengrub, and M. J. Shelley, Removing the stiffness from interfacial flows with surface tension, J. Comput. Phys. 114, 312 (1994).
10. G. Lapenta, F. Iinoya, and J. U. Brackbill, Particle-in-cell simulation of glow discharges in complex geometries, IEEE Trans. Plasma Sci. 23(4), 769 (1995).
11. R. J. LeVeque and Z. Li, Immersed interface methods for Stokes flow with elastic boundaries or surface tension, SIAM J. Sci. Comput. 18(3), 709 (1997).
12. R. J. LeVeque, C. S. Peskin, and P. D. Lax, Solution of a two-dimensional cochlea model with fluid viscosity, SIAM J. Appl. Math. 48(1), 191 (1988).
13. A. A. Mayo and C. S. Peskin, An implicit numerical method for fluid dynamics problems with immersed elastic boundaries, in Fluid Dynamics in Biology: Proceedings, AMS-IMS-SIAM Joint Summer Research Conference on Biofluiddynamics, edited by A. Y Cheer and C. P. van Dam, Contemporary Mathematics (Am. Math. Soc. Providence, RI, 1993), Vol. 141, pp. 261-277.
14. D. M. McQueen and C. S. Peskin, Shared-memory parallel vector implementation of the immersed boundary method for the computation of blood flow in the beating mammalian heart, J. Supercomputing 11, 213 (1997).
15. C. S. Peskin, Numerical analysis of blood flow in the heart, J. Comput. Phys. 25, 220 (1977).
16. C. S. Peskin and D. M. McQueen, A three-dimensional computational model for blood flow in the heart. I. Immersed elastic fibers in a viscous incompressible fluid, J. Comput. Phys. 81, 372 (1989).
17. C. S. Peskin and D. M. McQueen, A general method for the computer simulation of biological systems interacting with fluids, in Biological Fluid Dynamics, edited by C. P. Ellington and T. J. Pedley, Symposia of the Society for Experimental Biology (Soc. Exper. Biol., Leeds, 1995), Vol. 49, pp. 265-276.
18. C. S. Peskin and B. F. Printz, Improved volume conservation in the computation of flows with immersed elastic boundaries, J. Comput. Phys. 105, 33 (1993).
19. A. M. Roma, A Multilevel Self Adaptive Version of the Immersed Boundary Method, Ph.D. thesis, New York University (1996).
20. M. E. Rosar, A Three Dimensional Model for Fluid Flow through a Collapsible Tube, Ph.D. thesis, New York University (1994).
21. J. M. Stockie, Analysis and Computation of Immersed Boundaries, with Application to Pulp Fibres, Ph.D. thesis, Institute of Applied Mathematics, University of British Columbia, Vancouver, BC, Canada (1997). Available at http : //www.math.sfu.ca/~jms/. [unpublished]
22. J. M. Stockie and S. I. Green, Simulating the motion of pulp fibres using the immersed boundary method, J. Comput. Phys. 147(1), 147 (1998).
23. J. M. Stockie and B. T. R. Wetton, Stability analysis for the immersed fiber problem, SIAM J. Appl. Math. 55(6), 1577 (1995).
24. A.-K. Tornberg, R. W. Metcalfe, L. R. Scott, and B. Bagheri, A front-tracking method for simulating fluid particle motion using high-order finite element methods, in Proceedings, 1997 ASME Fluids Engineering Division Summer Meeting, Vancouver, Canada, June 22-26, 1997.
25. C. Tu and C. S. Peskin, Stability and instability in the computation of flows with moving immersed boundaries: A comparison of three methods, SIAM J. Sci. Statist. Comput. 13(6), 1361 (1992).
26. S. O. Unverdi and G. Tryggvason, A front-tracking method for viscous, incompressible, multi-fluid flows, J. Comput. Phys. 100, 25 (1992).