# 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…

## 258 Citations

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## References

SHOWING 1-8 OF 8 REFERENCES

### The Fast Solution of Poisson’s and the Biharmonic Equations on Irregular Regions

- Mathematics, Computer Science
- 1984

The main idea is to use the integral equation formulation to define a discontinuous extension of the solution to the rest of the rectangular region to solve Laplace's and the biharmonic equations on irregular regions with smooth boundaries.

### A MATHEMATICAL MODEL AND NUMERICAL METHOD FOR STUDYING PLATELET ADHESION AND AGGREGATION DURING BLOOD CLOTTING

- Engineering
- 1984

### GLAZ, A second-order projection method for the incompressible Navier-Stokes equations

- J. Comput. Phys.,
- 1989

### BEYER, A computational model of the cochlea using the immersed boundary method

- Ph.D. thesis,
- 1989