Improving the conditioning of the immersed boundary projection method

The immersed boundary projection method (IBPM) is a continuous forcing variant of the immersed boundary method, used to model flow over arbitrary surfaces that do not conform to the underlying computational fluid grid. This method offers several advantages, such as the ability to use fast solvers and avoid the need for re-meshing. However, the IBPM is generally limited by its first-order spatial accuracy, the presence of numerical noise in the surface forcing, and the computational expense associated with non-stationary surfaces. This work aims to address the issue of ill-conditioning of IBPM's formulation, which causes the noise in the surface forcing. We introduce higher-order Taylor expansion terms within the support of the smooth delta functions used in the governing and constraint equations. This modification allows the unknown surface forcing to be incorporated into the constraint equation, transforming the governing equation for the forcing from an ill-posed Fredholm equation of the first kind into a well-posed Fredholm equation of the second kind. This approach not only improves the conditioning of the system but also increases the accuracy of the method in one-dimensional cases from first to second order, with ongoing investigations into its efficacy in higher dimensions. We demonstrate the effectiveness of our approach through its application to Dirichlet Poisson problems and discuss its extension to the incompressible Navier-Stokes equations.