Convergence proof of the velocity field for a stokes flow immersed boundary method

Communications on Pure and Applied Mathematics

Volumn 61, Issue 9, 2008, Pages 1213-1263

  Mori, Yoichiro ^a  

^a UNIVERSITY OF BRITISH COLUMBIA   (Canada)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 52349091971     PISSN: 00103640     EISSN: 00103640     Source Type: Journal    
DOI: 10.1002/cpa.20233     Document Type: Article

References (42)

1. Beyer, R. P. A computational model of the cochlea using the immersed boundary method. J. Comput. Phys. 98 (1992), no. 1, 145-162.
2. Beyer, R. P.; Leveque, R. J. Analysis of a one-dimensional model for the immersed boundary method. SIAM J. Numer. Anal. 29 (1992), no. 2, 332-364.
3. Boffi, D.; Gastaldi, L.; Heltai, L. A finite element approach to the immersed boundary method. Progress in Engineering Computational Technology, 271-298. Saxe-Coburg, Stirling, Scotland, 2004.
4. Cortez, R.; Cowen, N.; Dillon, R.; Fauci, L. J. Simulation of swimming organisms: Coupling internal mechanics with external fluid dynamics. Comput. Sci. Eng. 6 (2004), no. 3, 38-45.
5. Cortez, R.; Peskin, C. S.; Stockie, J. M.; Varela, D. Parametric resonance in immersed elastic boundaries. SIAM J. Appl. Math. 65 (2004/05), no. 2, 494-520 (electronic).
6. Fauci, L. J. Interaction of oscillating filaments: a computational study. J. Comput. Phys. 86 (1990), no. 2, 294-313.
7. Fauci, L. J.; Fogelson, A. L. Truncated Newton methods and the modeling of complex immersed elastic structures. Comm. Pure Appl. Math. 46 (1993), no. 6, 787-816.
8. Fauci, L. J.; Peskin, C. S. A computational model of aquatic animal locomotion. J. Comput. Phys. 77 (1988), no. 1, 85-108.
9. Fogelson, A. L.; Peskin, C. S. A fast numerical method for solving three-dimensional Stokes equations in the presence of suspended particles. J. Comput. Phys. 79 (1988), no. 1, 50-69.
10. Givelberg, E. Modeling elastic shells immersed in fluid. Doctoral dissertation, Courant Institute of Mathematical Sciences, 1997.
11. Griffith, B. E.; Peskin, C. S. On the order of accuracy of the immersed boundary method: higher order convergence rates for sufficiently smooth problems. J. Comput. Phys. 208 (2005), no. 1, 75-105.
12. Hasimoto, H. On the periodic fundamental solutions of the Stokes equations and their application to viscous flow past a cubic array of spheres. J. Fluid Mech. 5 (1956), no. 2, 317-328.
13. Heltai, L. The finite element immersed boundary method. Doctoral dissertation, Università di Pavia, Italy, 2006. Available online at: http://www-dimat.unipv.it/heltai/papers.php
14. Hopkins, M.; Fauci, L. J. A computational model of the collective fluid dynamics of motile microorganisms. J. Fluid Mech. 455 (2002), 149-174.
15. Jung, E.; Peskin, C. S. Two-dimensional simulations of valveless pumping using the immersed boundary method. SIAM J. Sci. Comput. 23 (2001), no. 1, 19-45.
16. Lim, S.; Peskin, C. S. Simulations of the whirling instability by the immersed boundary method. SIAM J. Sci. Comput. 25 (2004), no. 6, 2066-2083 (electronic).
17. McQueen, D. M.; Peskin, C. S. Heart simulation by an immersed boundary method with formal second order accuracy and reduced numerical viscosity, 429-444. Mechanics for a New Millennium, Proceedings of the International Conference on Theoretical and Applied Mechanics (ICTAM) 2000. Kluwer, Dordrecht, The Netherlands, 2001.
18. Miller, L. A.; Peskin, C. S. When vortices stick: an aerodynamic transition in tiny insects. J. Exp. Biol. 207 (2004), 3073-3088.
19. Miller, L. A.; Peskin, C. S. A computational fluid dynamics of "clap and fling" in small insects. J. Exp. Biol. 208 (2005), 195-212.
20. Möller, T.; Machiraju, R.; Mueller, K.; Yagel, R. Evaluation and design of filters using a Taylor series expansion. IEEE Transactions on Visualization and Computer Graphics 3 (1997), no. 2, 184-199.
21. Monaghan, J. J. Extrapolating B splines for interpolation. J. Comput. Phys. 60 (1985), 253-262.
22. Nirenberg, L. Topics in nonlinear functional analysis. Chapter 6 by E. Zehnder. Notes by R. A. Artino. Revised reprint of the 1974 original. Courant Lecture Notes in Mathematics, 6. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, R.I., 2001.
23. Osher, S.; Fedkiw, R. Level set methods and dynamic implicit surfaces. Applied Mathematical Sciences, 153. Springer, New York, 2002.
24. Peskin, C. S. Numerical analysis of blood flow in the heart. J. Computational Phys. 25 (1977), no. 3, 220-252.
25. Peskin, C. S. The immersed boundary method. Acta Numer. 11 (2002), 479-517.
26. Peskin, C. S.; Printz, B. F. Improved volume conservation in the computation of flows with immersed elastic boundaries. J. Comput. Phys. 105 (1993), no. 1, 33-46.
27. Pozrikidis, C. Boundary integral and singularity methods for linearized viscous flow. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge, 1992.
28. Pozrikidis, C. Computation of periodic Green's functions of Stokes flow. J. Engrg. Math. 30 (1996), no. 1-2, 79-96.
29. Rosar, M. E.; Peskin, C. S. Fluid flow in collapsible elastic tubes: a three-dimensional numerical method. New York J. Math. 7 (2001), 281-302 (electronic).
30. Sethian, J. A. Level set methods and fast marching methods. Evolving interfaces in computational geometry, fluid mechanics, computer vision, and materials science. 2nd ed. Cambridge Monographs on Applied and Computational Mathematics, 3. Cambridge University Press, Cambridge, 1999.
31. Stockie, J. M.; Wetton, B. T. R. Stability analysis for the immersed fiber problem. SIAM J. Appl. Math. 55 (1995), no. 6, 1577-1591.
32. Stockie, J. M. Analysis and computation of immersed boundaries, with application to pulp fibres. Doctoral dissertation, Institute of Applied Mathematics, University of British Columbia, Vancouver, 1997.
33. Stockie, J. M.; Green, S. I. Simulating the motion of flexible pulp fibres using the immersed boundary method. J. Comput. Phys. 147 (1998), no. 1, 147-165.
34. Stockie, J. M.; Wetton, B. R. Analysis of stiffness in the immersed boundary method and implications for time-stepping schemes. J. Comput. Phys. 154 (1999), 41-64.
35. Stoker, J. J. Water waves: The mathematical theory with applications. Pure and Applied Mathematics, 6. Interscience, New York-London, 1957.
36. Tornberg, A.-K.; Engquist, B. Numerical approximations of singular source terms in differential equations. J. Comput. Phys. 200 (2004), no. 2, 462-488.
37. Tryggvason, G.; Bunner, B.; Esmaeeli, A.; Juric, D.; Al-Rawahi, N.; Tauber, W.; Han, J.; Nas, S.; Jan, Y.-J. A front-tracking method for the computations of multiphase flow. J. Comput. Phys. 169 (2001), 708-759.
38. Waldén, J. On the approximation of singular source terms in differential equations. Numer. Methods for Partial Differential Equations 15 (1999), no. 4, 503-520.
39. Wang, N. T.; Fogelson, A. L. Computational methods or continuum models of platelet aggregation. J. Comput. Phys. 151 (1999), no. 2, 649-675.
40. Wang, X. S. From immersed boundary method to immersed continuum method. Int. J. Multi-scale Comput. Eng. 4 (2006), 127-146.
41. Wang, X.; Liu, W. K. Extended immersed boundary method using FEM and RKPM. Comput. Methods Appl. Mech. Engrg. 193 (2004), no. 12-14, 1305-1321.
42. Yu, H. Three-dimensional computational modeling and simulation of platelet aggregation on parallel computers. Doctoral dissertation, University of Utah, Salt Lake City, 2000.