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

@article{Beyer1992AnalysisOA,
  title={Analysis of a one-dimensional model for the immersed boundary method},
  author={Richard P. Beyer and Randall J. LeVeque},
  journal={SIAM Journal on Numerical Analysis},
  year={1992},
  volume={29},
  pages={332-364}
}
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 Crank–Nicolson 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… 

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