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\title{Numerical Analysis of a 1-Dimensional
Immersed-Boundary\\ Method}
 
\abstract{We present the numerical analysis of a simplified,
one-dimensional version  
of Peskin's immersed boundary method, which has been used to
solve 
the two- and three-dimensional Navier-Stokes equations in the 
presence of immersed boundaries. We consider the heat
equation  
 in a finite domain with a moving source term.  
We denote the solution as $u(x,t)$ and the location of the
source 
 term as $X(t)$.  The source term is a moving delta function  
whose strength is a function of u at the location of the
delta function. 
The p.d.e. is coupled to an ordinary differential equation
whose  
solution gives the location of the source term. 
 The o.d.e. is $X'(t) = u(X(t),t)$, which can be interpreted as 
saying the source term moves at the local velocity.   
The accuracy the numerical method of solution depends on how
the  
delta function is discretized when the delta function is not
at 
a grid point and on how the solution, u, is represented 
 at locations between grid points. We present results showing 
the effect of different choices of spreading the source to
the grid and 
of restricting the solution to the source location. 
The problem we analyze is also similar to the Stefan problem
and 
the immersed-boundary method has features in common with
particle-in-cell 
 methods.}
\author{\underbar{Richard P. Beyer, Jr.}\\
University of Washington, Seattle, WA\\
Randall J. LeVeque\\
University of Washington, Seattle, WA}

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