Analysis of a One-Dimensional Model for the Immersed Boundary Method

TLDR

The study examines numerical solutions of a one‑dimensional heat equation with a singular forcing term, modeling the immersed boundary method for incompressible flow in irregular domains. The authors replace the delta function with a discrete approximation d_h(x), solve the resulting equation using a Crank–Nicolson scheme on a uniform grid, and analyze its accuracy for various d_h choices, including cases with specified or implicitly determined c(t). The analysis shows that accuracy depends on the choice of discrete delta function, highlighting the importance of selecting appropriate approximations.

Abstract

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 at the point a are studied. These problems serve as a model for the immersed boundary method of Peskin for incompressible flow problems in irregular regions. Some insight is gained into the accuracy that can be achieved and the importance of choosing appropriate discrete delta functions.