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\author{Juri Kandilarov\\
Department of Mathematics, Rousse University, 7017 Rousse, Bulgaria\\
e:mail juri@ami.ru.acad.bg}
\title{The immersed interface method for a nonlinear chemical diffusion equation
with local sites of reactions }
\date{}
\maketitle

We extend the immersed interface method of R.J.LeVeque and Z. Li, {\it The
immersed interface method for elliptic equations with discontinuous
coefficients and singular forces}, SIAM J. Numer. Anal. 31 (1994), pp.
1019-1044 to find numerical solution of the parabolic problem:

\begin{eqnarray*}
u_t-\Delta u &=&\delta _\Gamma (x)f(x,t,u(x,t)),\quad (x,t)\in \Omega
\times (0,T); \\
u(x,t) &=&0,\quad (x,t)\in \partial \Omega \times [0,T]; \\
u(x,0) &=&u_0(x),\quad x\in \Omega .
\end{eqnarray*}
Here $\Gamma $ is a $C^1$-curve in a bounded $\Omega \subset R^2$ and $\delta _\Gamma
(x)$ is the Dirac-delta function concentrated on the curve $\Gamma $. At each time step, a large, weakly
nonlinear system is set up using a finite difference scheme which is
standard away from $\Gamma $ and which is derived for grid points near $%
\Gamma $ by solving a small nonlinear system which is determined from the
jumps at $\Gamma $. Numerical examples show that we can compute solutions to
this problem with second-order accuracy.

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