**Analysis of a one-dimensional model for the immersed boundary method
**

by R. P. Beyer and R. J. LeVeque
*SIAM J. Numer. Anal.* 29(1992), pp. 332-364.

**Abstract.**
Numerical methods are studied for the one-dimensional heat equation with a
singular forcing term, $u_t = u_{xx} + c(t)\delta (x - \alpha (t)).$ The
delta function $\delta (x)$ is replaced by a discrete approximation $d_h
(x)$ and the resulting equation is solved by a CrankâNicolson method on a
uniform grid. The accuracy of this method is analyzed for various choices of
$d_h $. The case where $c(t)$ is specified and also the case where $c$ is
determined implicitly by a constraint on the solution at the point a are
studied. These problems serve as a model for the immersed boundary method of
Peskin for incompressible flow problems in irregular regions. Some insight
is gained into the accuracy that can be achieved and the importance of
choosing appropriate discrete delta functions

**Keywords.**
numerical analysis, immersed-boundary method, error analysis, discrete delta
function

Note: SIAM allows authors to post published papers on their website.

**bibtex entry:**

cle{beyer:332, author = {R. P. Beyer and R. J. LeVeque}, collaboration = {}, title = {Analysis of a One-Dimensional Model for the Immersed Boundary Method}, publisher = {SIAM}, year = {1992}, journal = {SIAM Journal on Numerical Analysis}, volume = {29}, number = {2}, pages = {332-364}, keywords = {numerical analysis; immersed-boundary method; error analysis; discrete delta function}, url = {http://link.aip.org/link/?SNA/29/332/1}, doi = {10.1137/0729022} }