An Analysis of the Numerical Stability of the Immersed Boundary Method

TL;DR

This paper conducts a Fourier-based stability analysis of the immersed boundary method, considering the effects of the spreading kernel, and validates the theoretical predictions with numerical results.

Contribution

It provides a novel stability analysis framework for the immersed boundary method that accounts for the spreading kernel's effects, applicable to various boundary conditions.

Findings

The stability boundary is accurately predicted by the analysis.

Numerical results confirm the theoretical stability predictions.

The framework can be extended to other boundary and membrane types.

Abstract

We present a numerical stability analysis of the immersed boundary(IB) method for a special case which is constructed so that Fourier analysis is applicable. We examine the stability of the immersed boundary method with the discrete Fourier transforms defined differently on the fluid grid and the boundary grid. This approach gives accurate theoretical results about the stability boundary since it takes the effects of the spreading kernel of the immersed boundary method on the numerical stability into account. In this paper, the spreading kernel is the standard 4-point IB delta function. A three-dimensional incompressible viscous flow and a no-slip planar boundary are considered. The case of a planar elastic membrane is also analyzed using the same analysis framework and it serves as an example of many possible generalizations of our theory. We present some numerical results and show that the observed stability behaviors are consistent with what are predicted by our theory.