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