Immersed interface methods for Stokes flow with elastic boundaries or surface tension

SIAM Journal on Scientific Computing

Volumn 18, Issue 3, 1997, Pages 709-735

  Leveque, Randall J ^a   Li, Zhilin ^b  

^a UNIVERSITY OF WASHINGTON   (United States)

^b MISSISSIPPI STATE UNIVERSITY   (United States)

Author keywords

Bubbles; Cartesian grids; Creeping flow; Discontinuous coefficients; Immersed interface methods; Interface tracking; Stokes flow

Indexed keywords

CARTESIAN GRIDS; CREEPING FLOW; IMMERSED INTERFACE METHODS; STOKES FLOW;

BUBBLES (IN FLUIDS); FINITE DIFFERENCE METHOD; INCOMPRESSIBLE FLOW; INTERFACES (MATERIALS); INTERPOLATION; SURFACE TENSION;

COMPUTATIONAL METHODS;

EID: 0031129202     PISSN: 10648275     EISSN: None     Source Type: Journal    
DOI: 10.1137/S1064827595282532     Document Type: Article

References (52)

1. A. S. ALMGREN, J. B. BELL, P. COLELLA, AND T. MARTHALER, A Cartesian Grid Projection Method for the Incompressible Euler Equations in Complex Geometries, Lawrence Livermore National Lab Report UCRL-JC-118091, Livermove, CA, 1994.
2. G. P. ASTRAKANTSEV, Methods of fictitious domains for a second order elliptic equation with natural boundary conditions, U.S.S.R. Comput. Math, and Math. Phys., 18 (1978). pp. 114-121.
3. G. K. BATCHELOR, An Introduction to Fluid Dynamics, Cambridge University Press, London, 1967.
4. M. BERGER AND R. J. LEVEQUE, A rotated difference scheme for Cartesian grids in complex geometries, AIAA Conference on Computational Fluid Dynamics, CP-91-1602, Honolulu, HI, 1991.
5. R. P. BEYER, A Computational Model of the Cochlea Using the Immersed Boundary Method, Ph.D. thesis, University of Washington, Seattle, WA, 1989.
6. R. P. BEYER, A computational model of the cochlea using the immersed boundary method, J. Comput. Phys., 98 (1992), pp. 145-162.
7. R. P. BEYER AND R. J. LEVEQUE, Analysis of a one-dimensional model for the immersed boundary method, SIAM J. Numer. Anal., 29 (1992), pp. 332-364.
8. C. BÖRGERS AND O. WIDLUND, On finite element domain imbedding methods, SIAM J. Numer. Anal., 27 (1990), pp. 963-978.
9. B. L. BUZBEE, F. W. DORR, J. A. GEORGE, AND G. H. GOLUB, The direct solution of the discrete Poisson equation on irregular grids, SIAM J. Numer. Anal., 8 (1971), pp. 722-736.
10. R. CLIFT, J. R. GRACE, AND M. E. WEBER, Bubbles, Drops and Particles, Academic Press, New York, 1978.
11. H. H. D. CLARKE AND M. SALAS, Euler Calculations for Multielement Airfoils Using Cartesian Grids, AIAA Paper 85-0291, Washington, DC, 1985.
12. D. DE ZEEUW AND K. POVVELL, An Adoptively-Refined Cartesian Mesh Solver for the Euler Equations, AIAA Paper 91-1542. Washington, DC, 1991.
13. L. J. FAUCI, Interaction of oscillating filaments-a computational study, J. Comput. Phys., 86 (1990), pp. 294-313.
14. A. L. FOGELSON, A mathematical model and numerical method for studying platelet adhesion and aggregation during blood clotting, J. Comput. Phys., 56 (1984), pp. 111-134.
15. P. E. GILL, W. MURRAY, AND M. H. WRIGHT, Practical Optimization, Academic Press, New York, 1981.
16. R. GLOWINSKI, T.-S. PAN, AND J. PERIAUX, A fictitious domain method for Dirichlet problem and applications, Comput. Methods Appl. Mech. Engrg., 111 (1994), pp. 283-303.
17. R. GLOWINSKI, T.-S. PAN, AND J. PERIAUX, A fictitious domain method for external incompressible viscous flow modeled by Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg., 112 (1994), pp. 133-148.
18. T. Y. Hou, J. S. LOWENGRUB, AND M. J. SHELLEY, Removing the stiffness from interfacial flows with surface tension, J. Comput. Phys., 114 (1994), pp. 312-338.
19. O. A. LADYZHENSKAYA, The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, New York, 1963.
20. W. E. LANGLOIS, Slow Viscous Flow, Macmillan, New York, 1964.
21. R. J. LEVEQUE, High resolution finite volume methods on arbitrary grids via wave propagation, J. Comput. Phys., 78 (1988), pp. 36-63.
22. R. J. LEVEQUE AND Z. LI, The immersed interface method for elliptic equations with discon-tinuous coefficients and singular sources, SIAM J. Numer. Anal., 31 (1994), pp. 1019-1044.
23. R. J. LEVEQUE AND C. ZHANG, Immersed interface methods for wave equations with discontinuous coefficients, Wave Motion, to appear.
24. R. J. LEVEQUE AND C. ZHANG, Finite difference methods for wave equations with discontinuous coefficients, in Proc. 1995 ASCE Conference, Boulder, CO, S. Sture, ed., to appear.
25. Z. LI, The Immersed Interface Method - A Numerical Approach for Partial Differential Equations with Interfaces, Ph.D. thesis, University of Washington, Seattle, WA, 1994.
26. Z. LI, Immersed Interface Methods for One-Dimensional Moving Interface Problems, CAM Report #95-01, University of California, Los Angeles, CA, 1995; Numer. Algorithms, to appear.
27. Z. LI, A note on immersed interface methods for three dimensional elliptic equations, Comput. Math. Appl., 31 (1996), pp. 9-17.
28. Z. LI AND A. MAYO, ADI methods for heat equations with discontinuties along an arbitrary interface, Proc. Sympos. Appl. Math., 48 (1994), pp. 311-316.
29. G. I. MARCHUK, Y. A. KUZNETSOV, AND A. M. MATSOKIN, Fictitious domain and domain decomposition methods, Soviet J. Numer. Anal. Math. Modelling, 1 (1986), pp. 3-35.
30. A. MAYO, On the Rapid Evaluation of Heat Potentials on General Regions, IBM Technical Report 14305, Yorktown Heights, NY.
31. A. MAYO, Rapid Fourth Order Accurate Methods for the Solution of the Stokes Problem in the Presence of an Immersed Boundary, IBM report RC19298, Yorktown Heights, NY.
32. A. MAYO, The fast solution of Poisson's and the biharmonic equations on irregular regions, SIAM J. Numer. Anal., 21 (1984), pp. 285-299.
33. A. MAYO AND A. GREENBAUM, Fast parallel iterative solution of Poisson's and the biharmonic equations on irregular regions, SIAM J. Sci. Statist. Comput., 13 (1992), pp. 101-118.
34. A. A. MAYO AND C. S. PESKIN, An implicit numerical method for fluid dynamics problems with immersed elastic boundaries, Contemp. Math., 141 (1993), pp. 261-277.
35. M. MIKSIS, J. M. VANDEN-BROECK, AND J. B. KELLER, Axisymmetric bubbles or drops in a uniform flow, J. Fluid Mech., 108 (1981), pp. 89-100.
36. S. OSHER AND J. A. SETHIAN, Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations, J. Comput. Phys., 79 (1988), pp. 12-49.
37. R. B. PEMBER, J. B. BELL, P. COLELLA, W. Y. CRUTCHFIELD, AND M. L. WELCOME, An adaptive Cartesian grid method for unsteady compressible flow in complex geometries, J. Comput. Phys., 120 (1995), pp. 278-304.
38. C. S. PESKIN, Numerical analysis of blood flow in the heart, J. Comput. Phys., 25 (1977), pp. 220-252.
39. C. S. PESKIN, Lectures on mathematical aspects of physiology, Lectures in Appl. Math., 19 (1981), pp. 69-107.
40. C. S. PESKIN AND D. M. MCQUEEN, Modeling prosthetic heart valves for numerical analysis of blood flow in the heart, J. Comput. Phys., 37 (1980), pp. 113-132.
41. C. S. PESKIN AND B. F. PRINTZ, Improved volume conservation in the computation of flows with immersed elastic boundaries, J. Comput. Phys., 105 (1993), pp. 33-46.
42. C. POZRIKIDIS, Boundary Integral and Singularity Methods for Linearized Viscous Flow, Cambridge University Press, London, 1992.
43. W. PROSKUROWSKI AND O. WIDLUND, On the numerical solution of Helmholtz's equation by the capacitance matrix method, Math. Comp., 30 (1976), pp. 433-468.
44. J. J. QUIRK, An Alternative to Unstructured Grids for Computing Gas Dynamic Flow Around Arbitrarily Complex Two-Dimensional Bodies, ICASE Report 92-7, Hampton, VA, 1992.
45. S. S. SAMANT, J. E. BUSSOLETTI, F. T. JOHNSON, R. H. BURKHART, B. L. EVERSON, AND R. G. MELVIN, TRANAIR: A Computer Code for Transonic Analyses of Arbitrary Configurations, AIAA Paper 87-0034, Washington, DC, 1987.
46. H. STONE, Dynamics of drop deformation and breakup in viscous flows, Ann. Rev. Fluid Mech., 26 (1994), pp. 65-102.
47. M. SUSSMAN, P. SMEREKA, AND S. OSHER, A Level Set Approach for Computing Solutions to Incompressible Two-Phase Flow, University of California, CAM Report 93-18, Los Angeles, CA, 1993.
48. P. N. SWARZTRAUBER, Fast Poisson solvers, in Studies in Numerical Analysis, G. H. Golub, ed., vol. 24, The Mathematical Association of America, Washington, DC, 1984, pp. 319-370.
49. 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 (1992), pp. 1361-1376.
50. S. O. UNVERDI AND G. TRYGGVASON, Computations of multi-fluid flows, Physica D, 60 (1992), pp. 70-83.
51. S. O. UNVERDI AND G. TRYGGVASON, A front-tracking method for viscous, incompressible, multi-fluid flows, J. Comput. Phys., 100 (1992), pp. 25-37.
52. B. WEDAN AND J. SOUTH, A method for solving the transonic full-potential equations for general configurations, in Proc. AIAA Computational Fluid Dynamics Conference, Danvers, CT, 1983, pp. 515-526.