Commun. Comput. Phys.,
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Volume 3.


Stability Analysis of the Immersed Boundary Method for a Two-Dimensional Membrane with Bending Rigidity

Zhaoxin Gong 1, Huaxiong Huang 2*, Chuanjing Lu 1

1 Department of Mechanics, Shanghai Jiaotong University, Shanghai 200240, China.
2 Department of Mechanics, Shanghai Jiaotong University, Shanghai 200240, China; and Department of Mathematics and Statistics, York University, Toronto, Ontario M3J 1P3, Canada.

Received 19 January 2007; Accepted (in revised version) 5 September 2007
Available online 27 November 2007

Abstract

In this paper, we analyze the stability of the Immersed Boundary Method applied to a membrane-fluid system with a plasma membrane immersed in an incompressible viscous fluid. We show that for small deformations, the planar rest state is stable for a membrane with bending rigidity. The smoothed version, using a standard regularization technique for the singular force, is also shown to be stable. Furthermore, we show that the coupled fluid-membrane system is stiff and smoothing helps to reduce the stiffness. Compared to the system of elastic fibers immersed in an incompressible fluid, membrane with bending rigidity consist of a wider range of decay rates. Therefore numerical instability could occur more easily for an explicit method when the time step size is not sufficiently small, even though the continuous problem is stable.

AMS subject classifications: 74K15, 76D05, 35Q30, 76D27, 76M25, 65M06

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Key words: Bending rigidity, immersed boundary method, membrane, moving interface, stability.

*Corresponding author.
Email: jackyff@sjtu.edu.cn (Z. X. Gong), hhuang@mathstat.yorku.ca (H. X. Huang), cjlu@sjtu.edu.cn (C. J. Lu)
 

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