4 SARAH D. OLSON AND ANITA T. LAYTON
(2) Smooth the force density to the grid via Eq. (2.3) to determine the Carte-
sian grid force density
fijn
(3) Solve the fluid equations, Eq. (2.1a) or (2.1b) using the incompressibility
condition in Eq. (2.2) and the problem dependent fluid domain boundary
conditions to determine
un+1(xij)
at each of the Cartesian grid points
(4) Interpolate the Cartesian grid velocity to the Lagrangian structure via
Eq. (2.4b) to determine
Un+1(Xk n)
at each of the immersed boundary
points
(5) Update the location of the structure. Use Euler method or higher order
methods such as Runge-Kutta methods to determine
Xn+1
using the no-
slip condition given in Eq. (2.4a)
(6) Repeat
There are a few additional pieces of information to emphasize. In immersed bound-
ary applications, if we have an elastic structure, we want to require that Δs is
sufficiently small in order to ensure that fluid is not leaking across the immersed
structure. It has been previously shown that this can be done if we choose Δs = h/2
[138]. In general, most numerical methods handle the force term explicitly, result-
ing in a severe time step restriction due to the stiff material properties of the
immersed structure (see §3.1). Since the immersed boundary smooths or smears
a sharp interface or singular force layer with a smooth approximation to the delta
function, the interface then inherits a thickness that is equivalent to the mesh
width. Above, we have assumed a uniform Cartesian grid where the Eulerian fluid
and pressure are defined. Since the lowest accuracy is in the region of the immersed
structure, adaptive grid methods have been developed to have finer detail in the
region around the boundary to resolve boundary layers or regions of larger vortic-
ity [63]. An explicit, formally second-order accurate in space and time numerical
method, given sufficient smoothness (e.g. thicker boundaries), has been developed
for the IB method using a projection-type method [62,92]. A convergence proof has
been developed for a simplified immersed boundary problem of Stokes flow with an
external force field supported on a curve [123]. Mori is able to give pointwise error
estimates away from the immersed boundary as well as global
L∞
error estimate of
the velocity. The work of Mori [123] was a major convergence result, proving that
the velocity field solved for in the IB method converges to the true solution.
2.1.2. Delta function. The delta function δc in Eq. (2.3) and (2.4b) is replaced
by a product of one-dimensional discrete delta functions that are scaled by the mesh
width h. For example, in 3-d,
(2.5) δc(x) =
1
h3
φ
xi
h
φ
yj
h
φ
zk
h
where h is mesh width. We note that spreading is the adjoint of interpolation when
the same delta function is used in both Eqs. (2.3) and (2.4b) [138]. Now we will
consider how to determine φ(r) satisfying certain properties, where r is defined as
xi/h, yj/h, or zk/h. The goal is to have a continuous φ in order to avoid jumps in
velocity or force on the Cartesian grid. We wish to enforce, in a distributional sense,
that δc δ when h 0. To increase computational efficiency, we can require that
φ(r) has compact support, e.g. φ(r) = 0 for r 2. In order to interpolate the
velocity from the Cartesian grid to the Lagrangian structure in Eq. (2.4b), we wish
to enforce exact interpolation of linear functions and second order interpolation
for smooth functions. This can be satisfied if φ is chosen to satisfy the following
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