@Article{EAJAM-1-155,
author = {},
title = {New Finite Difference Methods Based on IIM for Inextensible Interfaces in Incompressible Flows},
journal = {East Asian Journal on Applied Mathematics},
year = {2018},
volume = {1},
number = {2},
pages = {155--171},
abstract = {<p style="text-align: justify;">In this paper, new finite difference methods based on the augmented immersed
interface method (IIM) are proposed for simulating an inextensible moving
interface in an incompressible two-dimensional flow. The mathematical models arise
from studying the deformation of red blood cells in mathematical biology. The governing
equations are incompressible Stokes or Navier-Stokes equations with an unknown
surface tension, which should be determined in such a way that the surface divergence
of the velocity is zero along the interface. Thus, the area enclosed by the interface
and the total length of the interface should be conserved during the evolution process.
Because of the nonlinear and coupling nature of the problem, direct discretization by
applying the immersed boundary or immersed interface method yields complex nonlinear
systems to be solved. In our new methods, we treat the unknown surface tension
as an augmented variable so that the augmented IIM can be applied. Since finding the
unknown surface tension is essentially an inverse problem that is sensitive to perturbations,
our regularization strategy is to introduce a controlled tangential force along the
interface, which leads to a least squares problem. For Stokes equations, the forward
solver at one time level involves solving three Poisson equations with an interface. For
Navier-Stokes equations, we propose a modified projection method that can enforce the
pressure jump condition corresponding directly to the unknown surface tension. Several
numerical experiments show good agreement with other results in the literature
and reveal some interesting phenomena.</p>},
issn = {2079-7370},
doi = {https://doi.org/10.4208/eajam.030510.250910a},
url = {http://global-sci.org/intro/article_detail/eajam/10901.html}
}