On Reciprocity and Splitting Errors in Direct-Forcing Immersed Boundary Methods

Direct-forcing immersed boundary methods (IBMs) are widely used to enforce no-slip boundary conditions in incompressible flow solvers owing to their flexibility and ease of implementation. In this work we introduce a semi-discrete framework that encompasses both Lagrangian and Eulerian direct-forcing IBMs. Within this formulation, classical algorithms can be interpreted as approximate block-LU factorizations of the fully coupled monolithic system. This perspective reveals two distinct sources of error in the enforcement of the no-slip condition: a reciprocity error, associated with the non-exact adjoint relationship between interpolation and spreading operators, and a splitting error, arising from the pressure predictor used in projection-based time integration. The framework clarifies how the relative importance of these mechanisms depends on the interpolation kernel, the time step, and the physical configuration. Representative numerical experiments illustrate the behavior of the two error mechanisms and confirm the predictions of the analysis. In particular, a simple oscillating-balloon configuration is proposed as a diagnostic benchmark that directly exposes inconsistencies between boundary motion and the Eulerian mass balance. The analysis provides a unified interpretation of previously observed inaccuracies in direct-forcing IBMs and suggests possible directions for improvement, including the development of stable higher-order pressure predictors and improved constructions of interpolation-spreading operators.