# 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}
}
• Published 1 April 1992
• Mathematics
• SIAM Journal on Numerical Analysis
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…
262 Citations

## Figures and Tables from this paper

• Mathematics, Computer Science
• 2007
This work uses a linearization of the Navier–Stokes equations and a linear elasticity model to prove the unconditional stability of the fully implicit discretization, achieved with the use of a backward Euler method for both the fluid and the structure evolution.
• Physics
• 2023
In the present study, a discrete forcing Immersed Boundary Method (IBM) is proposed for the numerical simulation of high-speed ﬂow problems including heat exchange. The ﬂow ﬁeld is governed by the
The study of three new and two existing 1D methods and their derived variants reveals their accuracy on different grids and shows that this accuracy can be substantially affected by the position of an immersed boundary with respect to the neighboring grid points.
• Computer Science
ArXiv
• 2022
A new, well-conditioned IB formulation for boundary value problems, which is called the Immersed Boundary Double Layer (IBDL) method, and is presented as it applies to Poisson and Helmholtz problems to demonstrate its eﬃciency over the original constraint method.
• Mathematics
• 2016
In this paper, we propose a new methodology for numerically solving elliptic and parabolic equations with discontinuous coefficients and singular source terms. This new scheme is obtained by clubbing
• Mathematics
• 2021
Peskin’s Immersed Boundary (IB) model and method are among one of the most important modeling tools and numerical methods. The IB method has been known to be ﬁrst order accurate in the velocity.
The well known discretization of the Dirichlet boundary condition for the Laplace equation for grid–aligned boundaries is shown to be a special case of the Explicit Jump Immersed Interface Method.
• Zhilin Li
• Mathematics, Computer Science
Math. Comput.
• 2015
Peskin’s Immersed Boundary (IB) method is one of the most popular numerical methods for many years and has been applied to problems in mathematical biology, uid mechanics, material sciences, and many

## References

SHOWING 1-8 OF 8 REFERENCES

• A. Mayo
• 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.

### 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