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

Numerical methods are studied for the one-dimensional heat equation with a singular forcing term, uₜ = uₗₗ + c (t) (x - (t) ). The delta function (x) is replaced by a discrete approximation dₕ (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 dₕ. 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.